Network and Devices

MODULE IV : NETWORKS AND DEVICESNetwork theorems: superposition, Thevenin and Norton’s maximum power transfer, time domain analysis of simple RC, RL and RLC circuits, solution of network equations using Laplace transform: frequency domain analysis of RL, RC and RLC circuits, 2-port network parameters: driving point and transfer functions. Electronic Devices: Energy bands in silicon, carrier transport in silicon, diffusion current, drift current, mobility and resistivity, generation and recombination of carriers, working principles of p-n junction diode, Zener diode, tunnel diode, BJT, JFET, MOSFET, LED and photo diode.

Network theorems

Anyone who’s studied geometry should be familiar with the concept of a theorem: a relatively simple rule used to solve a problem, derived from a more intensive analysis using fundamental rules of mathematics. At least hypothetically, any problem in math can be solved just by using the simple rules of arithmetic (in fact, this is how modern digital computers carry out the most complex mathematical calculations: by repeating many cycles of additions and subtractions!), but human beings aren’t as consistent or as fast as a digital computer. We need “shortcut” methods in order to avoid procedural errors.

In electric network analysis, the fundamental rules are Ohm’s Law and Kirchhoff’s Laws. While these humble laws may be applied to analyze just about any circuit configuration (even if we have to resort to complex algebra to handle multiple unknowns), there are some “shortcut” methods of analysis to make the math easier for the average human.

As with any theorem of geometry or algebra, these network theorems are derived from fundamental rules. In this chapter, I’m not going to delve into the formal proofs of any of these theorems. If you doubt their validity, you can always empirically test them by setting up example circuits and calculating values using the “old” (simultaneous equation) methods versus the “new” theorems, to see if the answers coincide. They always should!

SUPER POSITION THEOREM

Superposition theorem is one of those strokes of genius that takes a complex subject and simplifies it in a way that makes perfect sense. A theorem like Millman’s certainly works well, but it is not quite obvious why it works so well. Superposition, on the other hand, is obvious.

The strategy used in the Superposition Theorem is to eliminate all but one source of power within a network at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately. Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all “superimposed” on top of each other (added algebraically) to find the actual voltage drops/currents with all sources active. Let’s look at our example circuit again and apply Superposition Theorem to it:

Since we have two sources of power in this circuit, we will have to calculate two sets of values for voltage drops and/or currents, one for the circuit with only the 28 volt battery in effect. . .

. . . and one for the circuit with only the 7 volt battery in effect:

When re-drawing the circuit for series/parallel analysis with one source, all other voltage sources are replaced by wires (shorts), and all current sources with open circuits (breaks). Since we only have voltage sources (batteries) in our example circuit, we will replace every inactive source during analysis with a wire.

Analyzing the circuit with only the 28 volt battery, we obtain the following values for voltage and current:

Analyzing the circuit with only the 7 volt battery, we obtain another set of values for voltage and current:

When superimposing these values of voltage and current, we have to be very careful to consider polarity (voltage drop) and direction (electron flow), as the values have to be added algebraically.

Applying these superimposed voltage figures to the circuit, the end result looks something like this:

Currents add up algebraically as well, and can either be superimposed as done with the resistor voltage drops, or simply calculated from the final voltage drops and respective resistances (I=E/R). Either way, the answers will be the same. Here I will show the superposition method applied to current:

Once again applying these superimposed figures to our circuit:

Quite simple and elegant, don’t you think? It must be noted, though, that the Superposition Theorem works only for circuits that are reducible to series/parallel combinations for each of the power sources at a time (thus, this theorem is useless for analyzing an unbalanced bridge circuit), and it only works where the underlying equations are linear (no mathematical powers or roots). The requisite of linearity means that Superposition Theorem is only applicable for determining voltage and current, not power!!! Power dissipations, being nonlinear functions, do not algebraically add to an accurate total when only one source is considered at a time. The need for linearity also means this Theorem cannot be applied in circuits where the resistance of a component changes with voltage or current. Hence, networks containing components like lamps (incandescent or gas-discharge) or varistors could not be analyzed.

Another prerequisite for Superposition Theorem is that all components must be “bilateral,” meaning that they behave the same with electrons flowing either direction through them. Resistors have no polarity-specific behavior, and so the circuits we’ve been studying so far all meet this criterion.

The Superposition Theorem finds use in the study of alternating current (AC) circuits, and semiconductor (amplifier) circuits, where sometimes AC is often mixed (superimposed) with DC. Because AC voltage and current equations (Ohm’s Law) are linear just like DC, we can use Superposition to analyze the circuit with just the DC power source, then just the AC power source, combining the results to tell what will happen with both AC and DC sources in effect. For now, though, Superposition will suffice as a break from having to do simultaneous equations to analyze a circuit.

REVIEW:

The Superposition Theorem states that a circuit can be analyzed with only one source of power at a time, the corresponding component voltages and currents algebraically added to find out what they’ll do with all power sources in effect.

To negate all but one power source for analysis, replace any source of voltage (batteries) with a wire; replace any current source with an open (break).

THEVENIN’S THEOREM

Thevenin’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots). If we’re dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are some components (especially certain gas-discharge and semiconductor components) which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits.

Thevenin’s Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it. Let’s take another look at our example circuit:

Let’s suppose that we decide to designate R2 as the “load” resistor in this circuit. We already have four methods of analysis at our disposal (Branch Current, Mesh Current, Millman’s Theorem, and Superposition Theorem) to use in determining voltage across R2 and current through R2, but each of these methods are time-consuming. Imagine repeating any of these methods over and over again to find what would happen if the load resistance changed (changing load resistance is very common in power systems, as multiple loads get switched on and off as needed. the total resistance of their

parallel connections changing depending on how many are connected at a time). This could potentially involve a lot of work!

Thevenin’s Theorem makes this easy by temporarily removing the load resistance from the original circuit and reducing what’s left to an equivalent circuit composed of a single voltage source and series resistance. The load resistance can then be re-connected to this “Thevenin equivalent circuit” and calculations carried out as if the whole network were nothing but a simple series circuit:

. . . after Thevenin conversion . . .

The “Thevenin Equivalent Circuit” is the electrical equivalent of B 1, R1, R3, and B2 as seen from the two points where our load resistor (R2) connects.

The Thevenin equivalent circuit, if correctly derived, will behave exactly the same as the original circuit formed by B1, R1, R3, and B2. In other words, the load resistor (R2) voltage and current should be exactly the same for the same value of load resistance in the two circuits. The load resistor R2 cannot “tell the difference” between the original network of B1, R1, R3, and B2, and the Thevenin equivalent circuit of EThevenin, and RThevenin, provided that the values for EThevenin and RThevenin have been calculated correctly.

The advantage in performing the “Thevenin conversion” to the simpler circuit, of course, is that it makes load voltage and load current so much easier to solve than in the original network. Calculating

the equivalent Thevenin source voltage and series resistance is actually quite easy. First, the chosen load resistor is removed from the original circuit, replaced with a break (open circuit):

Next, the voltage between the two points where the load resistor used to be attached is determined. Use whatever analysis methods are at your disposal to do this. In this case, the original circuit with the load resistor removed is nothing more than a simple series circuit with opposing batteries, and so we can determine the voltage across the open load terminals by applying the rules of series circuits, Ohm’s Law, and Kirchhoff’s Voltage Law:

The voltage between the two load connection points can be figured from the one of the battery’s voltage and one of the resistor’s voltage drops, and comes out to 11.2 volts. This is our “Thevenin voltage” (EThevenin) in the equivalent circuit:

To find the Thevenin series resistance for our equivalent circuit, we need to take the original circuit (with the load resistor still removed), remove the power sources (in the same style as we did with the Superposition Theorem: voltage sources replaced with wires and current sources replaced with breaks), and figure the resistance from one load terminal to the other:

With the removal of the two batteries, the total resistance measured at this location is equal to R1 and R3 in parallel: 0.8 Ω. This is our “Thevenin resistance” (RThevenin) for the equivalent circuit:

With the load resistor (2 Ω) attached between the connection points, we can determine voltage across it and current through it as though the whole network were nothing more than a simple series circuit:

Notice that the voltage and current figures for R2 (8 volts, 4 amps) are identical to those found using other methods of analysis. Also notice that the voltage and current figures for the Thevenin series resistance and the Thevenin source (total) do not apply to any component in the original, complex circuit. Thevenin’s Theorem is only useful for determining what happens to a single resistor in a network: the load.

The advantage, of course, is that you can quickly determine what would happen to that single resistor if it were of a value other than 2 Ω without having to go through a lot of analysis again. Just plug in that other value for the load resistor into the Thevenin equivalent circuit and a little bit of series circuit calculation will give you the result.

REVIEW:

Thevenin’s Theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source, series resistance, and series load.

Steps to follow for Thevenin’s Theorem:

(1) Find the Thevenin source voltage by removing the load resistor from the original circuit and calculating voltage across the open connection points where the load resistor used to be.

(2) Find the Thevenin resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.

(3) Draw the Thevenin equivalent circuit, with the Thevenin voltage source in series with the Thevenin resistance. The load resistor re-attaches between the two open points of the equivalent circuit.

(4) Analyze voltage and current for the load resistor following the rules for series circuits.

NORTON’S THEOREM

Norton’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Just as with Thevenin’s Theorem, the qualification of “linear” is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots).

Contrasting our original example circuit against the Norton equivalent: it looks something like this:

. . . after Norton conversion . . .

Remember that a current source is a component whose job is to provide a constant amount of current, outputting as much or as little voltage necessary to maintain that constant current.

As with Thevenin’s Theorem, everything in the original circuit except the load resistance has been reduced to an equivalent circuit that is simpler to analyze. Also similar to Thevenin’s Theorem are the steps used in Norton’s Theorem to calculate the Norton source current (INorton) and Norton resistance (RNorton).

As before, the first step is to identify the load resistance and remove it from the original circuit:

Then, to find the Norton current (for the current source in the Norton equivalent circuit), place a direct wire (short) connection between the load points and determine the resultant current. Note that this step is exactly opposite the respective step in Thevenin’s Theorem, where we replaced the load resistor with a break (open circuit):

With zero voltage dropped between the load resistor connection points, the current through R1 is strictly a function of B1‘s voltage and R1‘s resistance: 7 amps (I=E/R). Likewise, the current through R3 is now strictly a function of B2‘s voltage and R3‘s resistance: 7 amps (I=E/R). The total current through the short between the load connection points is the sum of these two currents: 7 amps + 7 amps = 14 amps. This figure of 14 amps becomes the Norton source current (I Norton) in our equivalent circuit:

Remember, the arrow notation for a current source points in the direction opposite that of electron flow. Again, apologies for the confusion. For better or for worse, this is standard electronic symbol notation. Blame Mr. Franklin again!

To calculate the Norton resistance (RNorton), we do the exact same thing as we did for calculating Thevenin resistance (RThevenin): take the original circuit (with the load resistor still removed), remove the power sources (in the same style as we did with the Superposition Theorem: voltage sources replaced with wires and current sources replaced with breaks), and figure total resistance from one load connection point to the other:

Now our Norton equivalent circuit looks like this:

If we re-connect our original load resistance of 2 Ω, we can analyze the Norton circuit as a simple parallel arrangement:

As with the Thevenin equivalent circuit, the only useful information from this analysis is the voltage and current values for R2; the rest of the information is irrelevant to the original circuit. However, the same advantages seen with Thevenin’s Theorem apply to Norton’s as well: if we wish to analyze load resistor voltage and current over several different values of load resistance, we can use the Norton equivalent circuit again and again, applying nothing more complex than simple parallel circuit analysis to determine what’s happening with each trial load.

REVIEW:

Norton’s Theorem is a way to reduce a network to an equivalent circuit composed of a single current source, parallel resistance, and parallel load.

Steps to follow for Norton’s Theorem:

(1) Find the Norton source current by removing the load resistor from the original circuit and calculating current through a short (wire) jumping across the open connection points where the load resistor used to be.

(2) Find the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.

(3) Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance. The load resistor re-attaches between the two open points of the equivalent circuit.

(4) Analyze voltage and current for the load resistor following the rules for parallel circuits.

MAXIMUM POWER TRANSFER THEOREM

The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than maximum.

This is essentially what is aimed for in radio transmitter design , where the antenna or transmission line “impedance” is matched to final power amplifier “impedance” for maximum radio frequency power output. Impedance, the overall opposition to AC and DC current, is very similar to resistance, and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output, but possibly overheating of the amplifier due to the power dissipated in its internal (Thevenin or Norton) impedance.

Taking our Thevenin equivalent example circuit, the Maximum Power Transfer Theorem tells us that the load resistance resulting in greatest power dissipation is equal in value to the Thevenin resistance (in this case, 0.8 Ω):

With this value of load resistance, the dissipated power will be 39.2 watts:

If we were to try a lower value for the load resistance (0.5 Ω instead of 0.8 Ω, for example), our power dissipated by the load resistance would decrease:

Power dissipation increased for both the Thevenin resistance and the total circuit, but it decreased for the load resistor. Likewise, if we increase the load resistance (1.1 Ω instead of 0.8 Ω, for example), power dissipation will also be less than it was at 0.8 Ω exactly:

If you were designing a circuit for maximum power dissipation at the load resistance, this theorem would be very useful. Having reduced a network down to a Thevenin voltage and resistance (or Norton current and resistance), you simply set the load resistance equal to that Thevenin or Norton equivalent (or vice versa) to ensure maximum power dissipation at the load. Practical applications of this might include radio transmitter final amplifier stage design (seeking to maximize power delivered to the antenna or transmission line), a grid tied inverter loading a solar array, or electric vehicle design (seeking to maximize power delivered to drive motor).

The Maximum Power Transfer Theorem is not: Maximum power transfer does not coincide with maximum efficiency. Application of The Maximum Power Transfer theorem to AC power distribution will not result in maximum or even high efficiency. The goal of high efficiency is more important for AC power distribution, which dictates a relatively low generator impedance compared to load impedance.

Similar to AC power distribution, high fidelity audio amplifiers are designed for a relatively low output impedance and a relatively high speaker load impedance. As a ratio, “output impdance” : “load impedance” is known as damping factor, typically in the range of 100 to 1000.

Maximum power transfer does not coincide with the goal of lowest noise. For example, the low-level radio frequency amplifier between the antenna and a radio receiver is often designed for lowest possible noise. This often requires a mismatch of the amplifier input impedance to the antenna as compared with that dictated by the maximum power transfer theorem.

REVIEW:

The Maximum Power Transfer Theorem states that the maximum amount of power will be dissipated by a load resistance if it is equal to the Thevenin or Norton resistance of the network supplying power.

The Maximum Power Transfer Theorem does not satisfy the goal of maximum efficiency.

RC AND RL CIRCUITS

Electrical transients

This chapter explores the response of capacitors and inductors to sudden changes in DC voltage (called a transient voltage), when wired in series with a resistor. Unlike resistors, which respond instantaneously to applied voltage, capacitors and inductors react over time as they absorb and release energy.

Capacitor transient response

Because capacitors store energy in the form of an electric field, they tend to act like small secondary-cell batteries, being able to store and release electrical energy. A fully discharged capacitor maintains zero volts across its terminals, and a charged capacitor maintains a steady quantity of voltage across its terminals, just like a battery. When capacitors are placed in a circuit with other sources of voltage, they will absorb energy from those sources, just as a secondary-cell battery will become charged as a result of being connected to a generator. A fully discharged capacitor, having a terminal voltage of zero, will initially act as a short-circuit when attached to a source of voltage, drawing maximum current as it begins to build a charge. Over time, the capacitor's terminal voltage rises to meet the applied voltage from the source, and the current through the capacitor decreases correspondingly. Once the capacitor has reached the full voltage of the source, it will stop drawing current from it, and behave essentially as an open-circuit.

When the switch is first closed, the voltage across the capacitor (which we were told was fully discharged) is zero volts; thus, it first behaves as though it were a short-circuit. Over time, the capacitor voltage will rise to equal battery voltage, ending in a condition where the capacitor

behaves as an open-circuit. Current through the circuit is determined by the difference in voltage between the battery and the capacitor, divided by the resistance of 10 kΩ. As the capacitor voltage approaches the battery voltage, the current approaches zero. Once the capacitor voltage has reached 15 volts, the current will be exactly zero. Let's see how this works using real values:

---------------------------------------------

| Time | Battery | Capacitor | Current |

|(seconds) | voltage | voltage | |

|-------------------------------------------|

| 0 | 15 V | 0 V | 1500 uA |

|-------------------------------------------|

| 0.5 | 15 V | 5.902 V | 909.8 uA |

|-------------------------------------------|

| 1 | 15 V | 9.482 V | 551.8 uA |

|-------------------------------------------|

| 2 | 15 V | 12.970 V | 203.0 uA |

|-------------------------------------------|

| 3 | 15 V | 14.253 V | 74.68 uA |

|-------------------------------------------|

| 4 | 15 V | 14.725 V | 27.47 uA |

|-------------------------------------------|

| 5 | 15 V | 14.899 V | 10.11 uA |

|-------------------------------------------|

| 6 | 15 V | 14.963 V | 3.718 uA |

|-------------------------------------------|

| 10 | 15 V | 14.999 V | 0.068 uA |

---------------------------------------------

The capacitor voltage's approach to 15 volts and the current's approach to zero over time is what a mathematician would call asymptotic: that is, they both approach their final values, getting closer and closer over time, but never exactly reaches their destinations. For all practical purposes, though, we can say that the capacitor voltage will eventually reach 15 volts and that the current will eventually equal zero.

Using the SPICE circuit analysis program, we can chart this asymptotic buildup of capacitor voltage and decay of capacitor current in a more graphical form (capacitor current is plotted in terms of voltage drop across the resistor, using the resistor as a shunt to measure current):

capacitor charging

v1 1 0 dc 15

r1 1 2 10k

c1 2 0 100u ic=0

.tran .5 10 uic

.plot tran v(2,0) v(1,2)

.end

legend:

*: v(2) Capacitor voltage

+: v(1,2) Capacitor current

time v(2)

(*+)----------- 0.000E+00 5.000E+00 1.000E+01 1.500E+01

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0.000E+00 5.976E-05 * . . +

5.000E-01 5.881E+00 . . * + . .

1.000E+00 9.474E+00 . .+ *. .

1.500E+00 1.166E+01 . + . . * .

2.000E+00 1.297E+01 . + . . * .

2.500E+00 1.377E+01 . + . . * .

3.000E+00 1.426E+01 . + . . * .

3.500E+00 1.455E+01 .+ . . *.

4.000E+00 1.473E+01 .+ . . *.

4.500E+00 1.484E+01 + . . *

5.000E+00 1.490E+01 + . . *

5.500E+00 1.494E+01 + . . *

6.000E+00 1.496E+01 + . . *

6.500E+00 1.498E+01 + . . *

7.000E+00 1.499E+01 + . . *

7.500E+00 1.499E+01 + . . *

8.000E+00 1.500E+01 + . . *

8.500E+00 1.500E+01 + . . *

9.000E+00 1.500E+01 + . . *

9.500E+00 1.500E+01 + . . *

1.000E+01 1.500E+01 + . . *

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

As you can see, I have used the .plot command in the netlist instead of the more familiar .print command. This generates a pseudo-graphic plot of figures on the computer screen using text characters. SPICE plots graphs in such a way that time is on the vertical axis (going down) and amplitude (voltage/current) is plotted on the horizontal (right=more; left=less). Notice how the voltage increases (to the right of the plot) very quickly at first, then tapering off as time goes on.

Current also changes very quickly at first then levels off as time goes on, but it is approaching minimum (left of scale) while voltage approaches maximum.

REVIEW:

Capacitors act somewhat like secondary-cell batteries when faced with a sudden change in applied voltage: they initially react by producing a high current which tapers off over time.

A fully discharged capacitor initially acts as a short circuit (current with no voltage drop) when faced with the sudden application of voltage. After charging fully to that level of voltage, it acts as an open circuit (voltage drop with no current).

In a resistor-capacitor charging circuit, capacitor voltage goes from nothing to full source voltage while current goes from maximum to zero, both variables changing most rapidly at first, approaching their final values slower and slower as time goes on.

Inductor transient response

Inductors have the exact opposite characteristics of capacitors. Whereas capacitors store energy in an electric field (produced by the voltage between two plates), inductors store energy in a magnetic field (produced by the current through wire). Thus, while the stored energy in a capacitor tries to maintain a constant voltage across its terminals, the stored energy in an inductor tries to maintain a constant current through its windings. Because of this, inductors oppose changes in current, and act precisely the opposite of capacitors, which oppose changes in voltage. A fully discharged inductor (no magnetic field), having zero current through it, will initially act as an open-circuit when attached to a source of voltage (as it tries to maintain zero current), dropping maximum voltage across its leads. Over time, the inductor's current rises to the maximum value allowed by the circuit, and the terminal voltage decreases correspondingly. Once the inductor's terminal voltage has decreased to a minimum (zero for a "perfect" inductor), the current will stay at a maximum level, and it will behave essentially as a short-circuit.

When the switch is first closed, the voltage across the inductor will immediately jump to battery voltage (acting as though it were an open-circuit) and decay down to zero over time (eventually acting as though it were a short-circuit). Voltage across the inductor is determined by calculating how much voltage is being dropped across R, given the current through the inductor, and subtracting that voltage value from the battery to see what's left. When the switch is first closed, the current is zero, then it increases over time until it is equal to the battery voltage divided by the series resistance of 1 Ω. This behavior is precisely opposite that of the series resistor-capacitor circuit, where current started at a maximum and capacitor voltage at zero. Let's see how this works using real values:

---------------------------------------------

| Time | Battery | Inductor | Current |

|(seconds) | voltage | voltage | |

|-------------------------------------------|

| 0 | 15 V | 15 V | 0 |

|-------------------------------------------|

| 0.5 | 15 V | 9.098 V | 5.902 A |

|-------------------------------------------|

| 1 | 15 V | 5.518 V | 9.482 A |

|-------------------------------------------|

| 2 | 15 V | 2.030 V | 12.97 A |

|-------------------------------------------|

| 3 | 15 V | 0.747 V | 14.25 A |

|-------------------------------------------|

| 4 | 15 V | 0.275 V | 14.73 A |

|-------------------------------------------|

| 5 | 15 V | 0.101 V | 14.90 A |

|-------------------------------------------|

| 6 | 15 V | 37.181 mV | 14.96 A |

|-------------------------------------------|

| 10 | 15 V | 0.681 mV | 14.99 A |

---------------------------------------------

Just as with the RC circuit, the inductor voltage's approach to 0 volts and the current's approach to 15 amps over time is asymptotic. For all practical purposes, though, we can say that the inductor voltage will eventually reach 0 volts and that the current will eventually equal the maximum of 15 amps.

Again, we can use the SPICE circuit analysis program to chart this asymptotic decay of inductor voltage and buildup of inductor current in a more graphical form (inductor current is plotted in terms of voltage drop across the resistor, using the resistor as a shunt to measure current):

inductor charging

v1 1 0 dc 15

r1 1 2 1

l1 2 0 1 ic=0

.tran .5 10 uic

.plot tran v(2,0) v(1,2)

.end

legend:

*: v(2) Inductor voltage

+: v(1,2) Inductor current

time v(2)

(*+)------------ 0.000E+00 5.000E+00 1.000E+01 1.500E+01

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0.000E+00 1.500E+01 + . . *

5.000E-01 9.119E+00 . . + * . .

1.000E+00 5.526E+00 . .* +. .

1.500E+00 3.343E+00 . * . . + .

2.000E+00 2.026E+00 . * . . + .

2.500E+00 1.226E+00 . * . . + .

3.000E+00 7.429E-01 . * . . + .

3.500E+00 4.495E-01 .* . . +.

4.000E+00 2.724E-01 .* . . +.

4.500E+00 1.648E-01 * . . +

5.000E+00 9.987E-02 * . . +

5.500E+00 6.042E-02 * . . +

6.000E+00 3.662E-02 * . . +

6.500E+00 2.215E-02 * . . +

7.000E+00 1.343E-02 * . . +

7.500E+00 8.123E-03 * . . +

8.000E+00 4.922E-03 * . . +

8.500E+00 2.978E-03 * . . +

9.000E+00 1.805E-03 * . . +

9.500E+00 1.092E-03 * . . +

1.000E+01 6.591E-04 * . . +

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Notice how the voltage decreases (to the left of the plot) very quickly at first, then tapering off as time goes on. Current also changes very quickly at first then levels off as time goes on, but it is approaching maximum (right of scale) while voltage approaches minimum.

REVIEW:

A fully "discharged" inductor (no current through it) initially acts as an open circuit (voltage drop with no current) when faced with the sudden application of voltage. After "charging" fully to the final level of current, it acts as a short circuit (current with no voltage drop).

In a resistor-inductor "charging" circuit, inductor current goes from nothing to full value while voltage goes from maximum to zero, both variables changing most rapidly at first, approaching their final values slower and slower as time goes on.

Voltage and current calculations

There's a sure way to calculate any of the values in a reactive DC circuit over time. The first step is to identify the starting and final values for whatever quantity the capacitor or inductor opposes change in; that is, whatever quantity the reactive component is trying to hold constant. For capacitors, this quantity is voltage; for inductors, this quantity is current. When the switch in a circuit is closed (or opened), the reactive component will attempt to maintain that quantity at the same level as it was before the switch transition, so that value is to be used for the "starting" value. The final value for this quantity is whatever that quantity will be after an infinite amount of time. This can be determined by analyzing a capacitive circuit as though the capacitor was an open-circuit, and an inductive circuit as though the inductor was a short-circuit, because that is what these components behave as when they've reached "full charge," after an infinite amount of time.

The next step is to calculate the time constant of the circuit: the amount of time it takes for voltage or current values to change approximately 63 percent from their starting values to their final values in a transient situation. In a series RC circuit, the time constant is equal to the total resistance in ohms multiplied by the total capacitance in farads. For a series L/R circuit, it is the total inductance in henrys divided by the total resistance in ohms. In either case, the time constant is expressed in units of seconds and symbolized by the Greek letter "tau" (τ):

The rise and fall of circuit values such as voltage and current in response to a transient is, as was mentioned before, asymptotic. Being so, the values begin to rapidly change soon after the transient and settle down over time. If plotted on a graph, the approach to the final values of voltage and current form exponential curves.

As was stated before, one time constant is the amount of time it takes for any of these values to change about 63 percent from their starting values to their (ultimate) final values. For every time constant, these values move (approximately) 63 percent closer to their eventual goal. The mathematical formula for determining the precise percentage is quite simple:

The letter e stands for Euler's constant, which is approximately 2.7182818. It is derived from calculus techniques, after mathematically analyzing the asymptotic approach of the circuit values. After one time constant's worth of time, the percentage of change from starting value to final value is:

After two time constant's worth of time, the percentage of change from starting value to final value is:

After ten time constant's worth of time, the percentage is:

The more time that passes since the transient application of voltage from the battery, the larger the value of the denominator in the fraction, which makes for a smaller value for the whole fraction, which makes for a grand total (1 minus the fraction) approaching 1, or 100 percent.

We can make a more universal formula out of this one for the determination of voltage and current values in transient circuits, by multiplying this quantity by the difference between the final and starting circuit values:

Let's analyze the voltage rise on the series resistor-capacitor circuit shown at the beginning of the chapter.

Note that we're choosing to analyze voltage because that is the quantity capacitors tend to hold constant. Although the formula works quite well for current, the starting and final values for current are actually derived from the capacitor's voltage, so calculating voltage is a more direct method. The resistance is 10 kΩ, and the capacitance is 100 µF (microfarads). Since the time constant (τ) for an RC circuit is the product of resistance and capacitance, we obtain a value of 1 second:

If the capacitor starts in a totally discharged state (0 volts), then we can use that value of voltage for a "starting" value. The final value, of course, will be the battery voltage (15 volts). Our universal formula for capacitor voltage in this circuit looks like this:

So, after 7.25 seconds of applying voltage through the closed switch, our capacitor voltage will have increased by:

Since we started at a capacitor voltage of 0 volts, this increase of 14.989 volts means that we have 14.989 volts after 7.25 seconds.

The same formula will work for determining current in that circuit, too. Since we know that a discharged capacitor initially acts like a short-circuit, the starting current will be the maximum

amount possible: 15 volts (from the battery) divided by 10 kΩ (the only opposition to current in the circuit at the beginning):

We also know that the final current will be zero, since the capacitor will eventually behave as an open-circuit, meaning that eventually no electrons will flow in the circuit. Now that we know both the starting and final current values, we can use our universal formula to determine the current after 7.25 seconds of switch closure in the same RC circuit:

Note that the figure obtained for change is negative, not positive! This tells us that current has decreased rather than increased with the passage of time. Since we started at a current of 1.5 mA, this decrease (-1.4989 mA) means that we have 0.001065 mA (1.065 µA) after 7.25 seconds.

We could have also determined the circuit current at time=7.25 seconds by subtracting the capacitor's voltage (14.989 volts) from the battery's voltage (15 volts) to obtain the voltage drop across the 10 kΩ resistor, then figuring current through the resistor (and the whole series circuit) with Ohm's Law (I=E/R). Either way, we should obtain the same answer:

The universal time constant formula also works well for analyzing inductive circuits. Let's apply it to our example L/R circuit in the beginning of the chapter:

With an inductance of 1 henry and a series resistance of 1 Ω, our time constant is equal to 1 second:

Because this is an inductive circuit, and we know that inductors oppose change in current, we'll set up our time constant formula for starting and final values of current. If we start with the switch in the open position, the current will be equal to zero, so zero is our starting current value. After the switch has been left closed for a long time, the current will settle out to its final value, equal to the source voltage divided by the total circuit resistance (I=E/R), or 15 amps in the case of this circuit.

If we desired to determine the value of current at 3.5 seconds, we would apply the universal time constant formula as such:

Given the fact that our starting current was zero, this leaves us at a circuit current of 14.547 amps at 3.5 seconds' time.

Determining voltage in an inductive circuit is best accomplished by first figuring circuit current and then calculating voltage drops across resistances to find what's left to drop across the inductor. With only one resistor in our example circuit (having a value of 1 Ω), this is rather easy:

Subtracted from our battery voltage of 15 volts, this leaves 0.453 volts across the inductor at time=3.5 seconds.

REVIEW:

Universal Time Constant Formula:

To analyze an RC or L/R circuit, follow these steps:

(1): Determine the time constant for the circuit (RC or L/R).

(2): Identify the quantity to be calculated (whatever quantity whose change is directly opposed by the reactive component. For capacitors this is voltage; for inductors this is current).

(3): Determine the starting and final values for that quantity.

(4): Plug all these values (Final, Start, time, time constant) into the universal time constant formula and solve for change in quantity.

(5): If the starting value was zero, then the actual value at the specified time is equal to the calculated change given by the universal formula. If not, add the change to the starting value to find out where you're at.

Why L/R and not LR?

It is often perplexing to new students of electronics why the time-constant calculation for an inductive circuit is different from that of a capacitive circuit. For a resistor-capacitor circuit, the time constant (in seconds) is calculated from the product (multiplication) of resistance in ohms and

capacitance in farads: τ=RC. However, for a resistor-inductor circuit, the time constant is calculated from the quotient (division) of inductance in henrys over the resistance in ohms: τ=L/R.

This difference in calculation has a profound impact on the qualitative analysis of transient circuit response. Resistor-capacitor circuits respond quicker with low resistance and slower with high resistance; resistor-inductor circuits are just the opposite, responding quicker with high resistance and slower with low resistance. While capacitive circuits seem to present no intuitive trouble for the new student, inductive circuits tend to make less sense.

Key to the understanding of transient circuits is a firm grasp on the concept of energy transfer and the electrical nature of it. Both capacitors and inductors have the ability to store quantities of energy, the capacitor storing energy in the medium of an electric field and the inductor storing energy in the medium of a magnetic field. A capacitor's electrostatic energy storage manifests itself in the tendency to maintain a constant voltage across the terminals. An inductor's electromagnetic energy storage manifests itself in the tendency to maintain a constant current through it.

Let's consider what happens to each of these reactive components in a condition of discharge: that is, when energy is being released from the capacitor or inductor to be dissipated in the form of heat by a resistor:

In either case, heat dissipated by the resistor constitutes energy leaving the circuit, and as a consequence the reactive component loses its store of energy over time, resulting in a measurable decrease of either voltage (capacitor) or current (inductor) expressed on the graph. The more power dissipated by the resistor, the faster this discharging action will occur, because power is by definition the rate of energy transfer over time.

Therefore, a transient circuit's time constant will be dependent upon the resistance of the circuit. Of course, it is also dependent upon the size (storage capacity) of the reactive component, but since the relationship of resistance to time constant is the issue of this section, we'll focus on the effects of resistance alone. A circuit's time constant will be less (faster discharging rate) if the resistance value is such that it maximizes power dissipation (rate of energy transfer into heat). For a capacitive circuit where stored energy manifests itself in the form of a voltage, this means the resistor must have a

low resistance value so as to maximize current for any given amount of voltage (given voltage times high current equals high power). For an inductive circuit where stored energy manifests itself in the form of a current, this means the resistor must have a high resistance value so as to maximize voltage drop for any given amount of current (given current times high voltage equals high power).

This may be analogously understood by considering capacitive and inductive energy storage in mechanical terms. Capacitors, storing energy electrostatically, are reservoirs of potential energy. Inductors, storing energy electromagnetically (electrodynamically), are reservoirs of kinetic energy. In mechanical terms, potential energy can be illustrated by a suspended mass, while kinetic energy can be illustrated by a moving mass. Consider the following illustration as an analogy of a capacitor:

The cart, sitting at the top of a slope, possesses potential energy due to the influence of gravity and its elevated position on the hill. If we consider the cart's braking system to be analogous to the resistance of the system and the cart itself to be the capacitor, what resistance value would facilitate rapid release of that potential energy? Minimum resistance (no brakes) would diminish the cart's altitude quickest, of course! Without any braking action, the cart will freely roll downhill, thus expending that potential energy as it loses height. With maximum braking action (brakes firmly set), the cart will refuse to roll (or it will roll very slowly) and it will hold its potential energy for a long period of time. Likewise, a capacitive circuit will discharge rapidly if its resistance is low and discharge slowly if its resistance is high.

Now let's consider a mechanical analogy for an inductor, showing its stored energy in kinetic form:

This time the cart is on level ground, already moving. Its energy is kinetic (motion), not potential (height). Once again if we consider the cart's braking system to be analogous to circuit resistance and the cart itself to be the inductor, what resistance value would facilitate rapid release of that kinetic energy? Maximum resistance (maximum braking action) would slow it down quickest, of course! With maximum braking action, the cart will quickly grind to a halt, thus expending its kinetic energy as it slows down. Without any braking action, the cart will be free to roll on indefinitely

(barring any other sources of friction like aerodynamic drag and rolling resistance), and it will hold its kinetic energy for a long period of time. Likewise, an inductive circuit will discharge rapidly if its resistance is high and discharge slowly if its resistance is low.

Hopefully this explanation sheds more light on the subject of time constants and resistance, and why the relationship between the two is opposite for capacitive and inductive circuits.

Complex voltage and current calculations

There are circumstances when you may need to analyze a DC reactive circuit when the starting values of voltage and current are not respective of a fully "discharged" state. In other words, the capacitor might start at a partially-charged condition instead of starting at zero volts, and an inductor might start with some amount of current already through it, instead of zero as we have been assuming so far. Take this circuit as an example, starting with the switch open and finishing with the switch in the closed position:

Since this is an inductive circuit, we'll start our analysis by determining the start and end values for current. This step is vitally important when analyzing inductive circuits, as the starting and ending voltagecan only be known after the current has been determined! With the switch open (starting condition), there is a total (series) resistance of 3 Ω, which limits the final current in the circuit to 5 amps:

So, before the switch is even closed, we have a current through the inductor of 5 amps, rather than starting from 0 amps as in the previous inductor example. With the switch closed (the final condition), the 1 Ω resistor is shorted across (bypassed), which changes the circuit's total resistance to 2 Ω. With the switch closed, the final value for current through the inductor would then be:

So, the inductor in this circuit has a starting current of 5 amps and an ending current of 7.5 amps. Since the "timing" will take place during the time that the switch is closed and R 2 is shorted past, we need to calculate our time constant from L1 and R1: 1 Henry divided by 2 Ω, or τ = 1/2 second. With these values, we can calculate what will happen to the current over time. The voltage across the inductor will be calculated by multiplying the current by 2 (to arrive at the voltage across the 2 Ω resistor), then subtracting that from 15 volts to see what's left. If you realize that the voltage across the inductor starts at 5 volts (when the switch is first closed) and decays to 0 volts over time, you can also use these figures for starting/ending values in the general formula and derive the same results:

---------------------------------------------

| Time | Battery | Inductor | Current |

|(seconds) | voltage | voltage | |

|-------------------------------------------|

| 0 | 15 V | 5 V | 5 A |

|-------------------------------------------|

| 0.1 | 15 V | 4.094 V | 5.453 A |

|-------------------------------------------|

| 0.25 | 15 V | 3.033 V | 5.984 A |

|-------------------------------------------|

| 0.5 | 15 V | 1.839 V | 6.580 A |

|-------------------------------------------|

| 1 | 15 V | 0.677 V | 7.162 A |

|-------------------------------------------|

| 2 | 15 V | 0.092 V | 7.454 A |

|-------------------------------------------|

| 3 | 15 V | 0.012 V | 7.494 A |

---------------------------------------------

Complex circuits

What do we do if we come across a circuit more complex than the simple series configurations we've seen so far? Take this circuit as an example:

The simple time constant formula (τ=RC) is based on a simple series resistance connected to the capacitor. For that matter, the time constant formula for an inductive circuit (τ=L/R) is also based on the assumption of a simple series resistance. So, what can we do in a situation like this, where resistors are connected in a series-parallel fashion with the capacitor (or inductor)?

The answer comes from our studies in network analysis. Thevenin's Theorem tells us that we can reduce any linear circuit to an equivalent of one voltage source, one series resistance, and a load component through a couple of simple steps. To apply Thevenin's Theorem to our scenario here, we'll regard the reactive component (in the above example circuit, the capacitor) as the load and remove it temporarily from the circuit to find the Thevenin voltage and Thevenin resistance. Then, once we've determined the Thevenin equivalent circuit values, we'll re-connect the capacitor and solve for values of voltage or current over time as we've been doing so far.

After identifying the capacitor as the "load," we remove it from the circuit and solve for voltage across the load terminals (assuming, of course, that the switch is closed):

This step of the analysis tells us that the voltage across the load terminals (same as that across resistor R2) will be 1.8182 volts with no load connected. With a little reflection, it should be clear that this will be our final voltage across the capacitor, seeing as how a fully-charged capacitor acts like an open circuit, drawing zero current. We will use this voltage value for our Thevenin equivalent circuit source voltage.

Now, to solve for our Thevenin resistance, we need to eliminate all power sources in the original circuit and calculate resistance as seen from the load terminals:

Re-drawing our circuit as a Thevenin equivalent, we get this:

Our time constant for this circuit will be equal to the Thevenin resistance times the capacitance (τ=RC). With the above values, we calculate:

Now, we can solve for voltage across the capacitor directly with our universal time constant formula. Let's calculate for a value of 60 milliseconds. Because this is a capacitive formula, we'll set our calculations up for voltage:

Again, because our starting value for capacitor voltage was assumed to be zero, the actual voltage across the capacitor at 60 milliseconds is equal to the amount of voltage change from zero, or 1.3325 volts.

We could go a step further and demonstrate the equivalence of the Thevenin RC circuit and the original circuit through computer analysis. I will use the SPICE analysis program to demonstrate this:

Comparison RC analysis

* first, the netlist for the original circuit:

v1 1 0 dc 20

r1 1 2 2k

r2 2 3 500

r3 3 0 3k

c1 2 3 100u ic=0

* then, the netlist for the thevenin equivalent:

v2 4 0 dc 1.818182

r4 4 5 454.545

c2 5 0 100u ic=0

* now, we analyze for a transient, sampling every .005 seconds

* over a time period of .37 seconds total, printing a list of

* values for voltage across the capacitor in the original

* circuit (between modes 2 and 3) and across the capacitor in

* the thevenin equivalent circuit (between nodes 5 and 0)

.tran .005 0.37 uic

.print tran v(2,3) v(5,0)

.end

time v(2,3) v(5)

0.000E+00 4.803E-06 4.803E-06

5.000E-03 1.890E-01 1.890E-01

1.000E-02 3.580E-01 3.580E-01

1.500E-02 5.082E-01 5.082E-01

2.000E-02 6.442E-01 6.442E-01

2.500E-02 7.689E-01 7.689E-01

3.000E-02 8.772E-01 8.772E-01

3.500E-02 9.747E-01 9.747E-01

4.000E-02 1.064E+00 1.064E+00

4.500E-02 1.142E+00 1.142E+00

5.000E-02 1.212E+00 1.212E+00

5.500E-02 1.276E+00 1.276E+00

6.000E-02 1.333E+00 1.333E+00

6.500E-02 1.383E+00 1.383E+00

7.000E-02 1.429E+00 1.429E+00

7.500E-02 1.470E+00 1.470E+00

8.000E-02 1.505E+00 1.505E+00

8.500E-02 1.538E+00 1.538E+00

9.000E-02 1.568E+00 1.568E+00

9.500E-02 1.594E+00 1.594E+00

1.000E-01 1.617E+00 1.617E+00

1.050E-01 1.638E+00 1.638E+00

1.100E-01 1.657E+00 1.657E+00

1.150E-01 1.674E+00 1.674E+00

1.200E-01 1.689E+00 1.689E+00

1.250E-01 1.702E+00 1.702E+00

1.300E-01 1.714E+00 1.714E+00

1.350E-01 1.725E+00 1.725E+00

1.400E-01 1.735E+00 1.735E+00

1.450E-01 1.744E+00 1.744E+00

1.500E-01 1.752E+00 1.752E+00

1.550E-01 1.758E+00 1.758E+00

1.600E-01 1.765E+00 1.765E+00

1.650E-01 1.770E+00 1.770E+00

1.700E-01 1.775E+00 1.775E+00

1.750E-01 1.780E+00 1.780E+00

1.800E-01 1.784E+00 1.784E+00

1.850E-01 1.787E+00 1.787E+00

1.900E-01 1.791E+00 1.791E+00

1.950E-01 1.793E+00 1.793E+00

2.000E-01 1.796E+00 1.796E+00

2.050E-01 1.798E+00 1.798E+00

2.100E-01 1.800E+00 1.800E+00

2.150E-01 1.802E+00 1.802E+00

2.200E-01 1.804E+00 1.804E+00

2.250E-01 1.805E+00 1.805E+00

2.300E-01 1.807E+00 1.807E+00

2.350E-01 1.808E+00 1.808E+00

2.400E-01 1.809E+00 1.809E+00

2.450E-01 1.810E+00 1.810E+00

2.500E-01 1.811E+00 1.811E+00

2.550E-01 1.812E+00 1.812E+00

2.600E-01 1.812E+00 1.812E+00

2.650E-01 1.813E+00 1.813E+00

2.700E-01 1.813E+00 1.813E+00

2.750E-01 1.814E+00 1.814E+00

2.800E-01 1.814E+00 1.814E+00

2.850E-01 1.815E+00 1.815E+00

2.900E-01 1.815E+00 1.815E+00

2.950E-01 1.815E+00 1.815E+00

3.000E-01 1.816E+00 1.816E+00

3.050E-01 1.816E+00 1.816E+00

3.100E-01 1.816E+00 1.816E+00

3.150E-01 1.816E+00 1.816E+00

3.200E-01 1.817E+00 1.817E+00

3.250E-01 1.817E+00 1.817E+00

3.300E-01 1.817E+00 1.817E+00

3.350E-01 1.817E+00 1.817E+00

3.400E-01 1.817E+00 1.817E+00

3.450E-01 1.817E+00 1.817E+00

3.500E-01 1.817E+00 1.817E+00

3.550E-01 1.817E+00 1.817E+00

3.600E-01 1.818E+00 1.818E+00

3.650E-01 1.818E+00 1.818E+00

3.700E-01 1.818E+00 1.818E+00

At every step along the way of the analysis, the capacitors in the two circuits (original circuit versus Thevenin equivalent circuit) are at equal voltage, thus demonstrating the equivalence of the two circuits.

REVIEW:

To analyze an RC or L/R circuit more complex than simple series, convert the circuit into a Thevenin equivalent by treating the reactive component (capacitor or inductor) as the "load" and reducing everything else to an equivalent circuit of one voltage source and one series resistor. Then, analyze what happens over time with the universal time constant formula.

Solving for unknown time

Sometimes it is necessary to determine the length of time that a reactive circuit will take to reach a predetermined value. This is especially true in cases where we're designing an RC or L/R circuit to perform a precise timing function. To calculate this, we need to modify our "Universal time constant formula." The original formula looks like this:

However, we want to solve for time, not the amount of change. To do this, we algebraically manipulate the formula so that time is all by itself on one side of the equal sign, with all the rest on the other side:

The ln designation just to the right of the time constant term is the natural logarithm function: the exact reverse of taking the power of e. In fact, the two functions (powers of e and natural logarithms) can be related as such:

If ex = a, then ln a = x.

If ex = a, then the natural logarithm of a will give you x: the power that e must be was raised to in order to produce a.

Let's see how this all works on a real example circuit. Taking the same resistor-capacitor circuit from the beginning of the chapter, we can work "backwards" from previously determined values of voltage to find how long it took to get there.

The time constant is still the same amount: 1 second (10 kΩ times 100 µF), and the starting/final values remain unchanged as well (EC = 0 volts starting and 15 volts final). According to our chart at the beginning of the chapter, the capacitor would be charged to 12.970 volts at the end of 2 seconds. Let's plug 12.970 volts in as the "Change" for our new formula and see if we arrive at an answer of 2 seconds:

Indeed, we end up with a value of 2 seconds for the time it takes to go from 0 to 12.970 volts across the capacitor. This variation of the universal time constant formula will work for all capacitive and inductive circuits, both "charging" and "discharging," provided the proper values of time constant, Start, Final, and Change are properly determined beforehand. Remember, the most important step in solving these problems is the initial set-up. After that, its just a lot of button-pushing on your calculator!

REVIEW:

To determine the time it takes for an RC or L/R circuit to reach a certain value of voltage or current, you'll have to modify the universal time constant formula to solve for time instead of change.

The mathematical function for reversing an exponent of "e" is the natural logarithm (ln), provided on any scientific calculator.

RLC CIRCUITS

Series RLC Circuit

The series RLC circuit above has a single loop with the instantaneous current flowing through the loop being the same for each circuit element. Since the inductive and capacitive reactance’s XL andXC are a function of the supply frequency, the sinusoidal response of a series RLC circuit will therefore vary with frequency, ƒ. Then the individual voltage drops across each circuit element of R,L and C element will be “out-of-phase” with each other as defined by:

i(t) = Imax sin(ωt)

The instantaneous voltage across a pure resistor, VR is “in-phase” with the current.

The instantaneous voltage across a pure inductor, VL “leads” the current by 90o

The instantaneous voltage across a pure capacitor, VC “lags” the current by 90o

Therefore, VL and VC are 180o “out-of-phase” and in opposition to each other.

For the series RLC circuit above, this can be shown as:

The amplitude of the source voltage across all three components in a series RLC circuit is made up of the three individual component voltages, VR, VL and VC with the current common to all three components. The vector diagrams will therefore have the current vector as their reference with the three voltage vectors being plotted with respect to this reference as shown below.

Individual Voltage Vectors

This means then that we can not simply add together VR, VL and VC to find the supply voltage, VSacross all three components as all three voltage vectors point in different directions with regards to the current vector. Therefore we will have to find the supply voltage, VS as the Phasor Sum of the three component voltages combined together vectorially.

Kirchoff’s voltage law ( KVL ) for both loop and nodal circuits states that around any closed loop the sum of voltage drops around the loop equals the sum of the EMF’s. Then applying this law to the these three voltages will give us the amplitude of the source voltage, VS as.

Instantaneous Voltages for a Series RLC Circuit

The phasor diagram for a series RLC circuit is produced by combining together the three individual phasors above and adding these voltages vectorially. Since the current flowing through the circuit is common to all three circuit elements we can use this as the reference vector with the three voltage vectors drawn relative to this at their corresponding angles.

The resulting vector VS is obtained by adding together two of the vectors, VL and VC and then adding this sum to the remaining vector VR. The resulting angle obtained between VS and i will be the circuits phase angle as shown below.

Phasor Diagram for a Series RLC Circuit

We can see from the phasor diagram on the right hand side above that the voltage vectors produce a rectangular triangle, comprising of hypotenuse VS, horizontal axis VR and vertical axis VL – VC Hopefully you will notice then, that this forms our old favourite the Voltage Triangle and we can therefore use Pythagoras’s theorem on this voltage triangle to mathematically obtain the value of VS as shown.

Voltage Triangle for a Series RLC Circuit

Please note that when using the above equation, the final reactive voltage must always be positive in value, that is the smallest voltage must always be taken away from the largest voltage we can not have a negative voltage added to VR so it is correct to have VL – VC or VC – VL. The smallest value from the largest otherwise the calculation of VS will be incorrect.

We know from above that the current has the same amplitude and phase in all the components of a series RLC circuit. Then the voltage across each component can also be described mathematically according to the current flowing through, and the voltage across each element as.

By substituting these values into Pythagoras’s equation above for the voltage triangle will give us:

So we can see that the amplitude of the source voltage is proportional to the amplitude of the current flowing through the circuit. This proportionality constant is called the Impedance of the circuit which ultimately depends upon the resistance and the inductive and capacitive reactance’s.

Then in the series RLC circuit above, it can be seen that the opposition to current flow is made up of three components, XL, XC and R with the reactance, XT of any series RLC circuit being defined as:XT = XL – XC or XT = XC – XL with the total impedance of the circuit being thought of as the voltage source required to drive a current through it.

The Impedance of a Series RLC Circuit

As the three vector voltages are out-of-phase with each other, XL, XC and R must also be “out-of-phase” with each other with the relationship between R, XL and XC being the vector sum of these three components thereby giving us the circuits overall impedance, Z. These circuit impedance’s can be drawn and represented by an Impedance Triangle as shown below.

The Impedance Triangle for a Series RLC Circuit

The impedance Z of a series RLC circuit depends upon the angular frequency, ω as do XL and XC If the capacitive reactance is greater than the inductive reactance, XC > XL then the overall circuit reactance is capacitive giving a leading phase angle. Likewise, if the inductive reactance is greater than the capacitive reactance, XL > XC then the overall circuit reactance is inductive giving the series circuit a lagging phase angle. If the two reactance’s are the same and XL = XC then the angular frequency at which this occurs is called the resonant frequency and produces the effect ofresonance which we will look at in more detail in another tutorial.

Then the magnitude of the current depends upon the frequency applied to the series RLC circuit. When impedance, Z is at its maximum, the current is a minimum and likewise, when Z is at its minimum, the current is at maximum. So the above equation for impedance can be re-written as:

The phase angle, θ between the source voltage, VS and the current, i is the same as for the angle between Z and R in the impedance triangle. This phase angle may be positive or negative in value depending on whether the source voltage leads or lags the circuit current and can be calculated mathematically from the ohmic values of the impedance triangle as:

Series RLC Circuit Example No1

A series RLC circuit containing a resistance of 12Ω, an inductance of 0.15H and a capacitor of 100uFare connected in series across a 100V, 50Hz supply. Calculate the total circuit impedance, the circuits current, power factor and draw the voltage phasor diagram.

Inductive Reactance, XL.

Capacitive Reactance, XC.

Circuit Impedance, Z.

Circuits Current, I.

Voltages across the Series RLC Circuit, VR, VL, VC.

Circuits Power factor and Phase Angle, θ.

Phasor Diagram.

Electrical Circuit Theory and Technology

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Since the phase angle θ is calculated as a positive value of 51.8o the overall reactance of the circuit must be inductive. As we have taken the current vector as our reference vector in a series RLC circuit, then the current “lags” the source voltage by 51.8o so we can say that the phase angle is lagging as confirmed by our mnemonic expression “ELI”.

Series RLC Circuit Summary

In a series RLC circuit containing a resistor, an inductor and a capacitor the source voltage VS is the phasor sum made up of three components, VR, VL and VC with the current common to all three. Since the current is common to all three components it is used as the horizontal reference when constructing a voltage triangle.

The impedance of the circuit is the total opposition to the flow of current. For a series RLC circuit, and impedance triangle can be drawn by dividing each side of the voltage triangle by its current, I. The voltage drop across the resistive element is equal to I x R, the voltage across the two reactive

elements is I x X = I x XL – I x XC while the source voltage is equal to I x Z. The angle between VS and I will be the phase angle, θ.

When working with a series RLC circuit containing multiple resistances, capacitance’s or inductance’s either pure or impure, they can be all added together to form a single component. For example all resistances are added together, RT = ( R1 + R2 + R3 )…etc or all the inductance’sLT = ( L1 + L2 + L3 )…etc this way a circuit containing many elements can be easily reduced to a single impedance.

PARALLEL RLC CIRCUIT

This time instead of the current being common to the circuit components, the applied voltage is now common to all so we need to find the individual branch currents through each element. The total impedance, Z of a parallel RLC circuit is calculated using the current of the circuit similar to that for a DC parallel circuit, the difference this time is that admittance is used instead of impedance. Consider the parallel RLC circuit below.

Parallel RLC Circuit

In the above parallel RLC circuit, we can see that the supply voltage, VS is common to all three components whilst the supply current IS consists of three parts. The current flowing through the resistor, IR, the current flowing through the inductor, IL and the current through the capacitor, IC.

But the current flowing through each branch and therefore each component will be different to each other and to the supply current, IS. The total current drawn from the supply will not be the mathematical sum of the three individual branch currents but their vector sum.

Like the series RLC circuit, we can solve this circuit using the phasor or vector method but this time the vector diagram will have the voltage as its reference with the three current vectors plotted with respect to the voltage. The phasor diagram for a parallel RLC circuit is produced by combining together the three individual phasors for each component and adding the currents vectorially.

Since the voltage across the circuit is common to all three circuit elements we can use this as the reference vector with the three current vectors drawn relative to this at their corresponding angles. The resulting vector IS is obtained by adding together two of the vectors, IL and IC and then adding this sum to the remaining vector IR. The resulting angle obtained between V and IS will be the circuits phase angle as shown below.

Phasor Diagram for a Parallel RLC Circuit

We can see from the phasor diagram on the right hand side above that the current vectors produce a rectangular triangle, comprising of hypotenuse IS, horizontal axis IR and vertical axis IL – IC Hopefully you will notice then, that this forms a Current Triangle and we can therefore use Pythagoras’s theorem on this current triangle to mathematically obtain the magnitude of the branch currents along the x-axis and y-axis and then determine the total current IS of these components as shown.

Current Triangle for a Parallel RLC Circuit

Since the voltage across the circuit is common to all three circuit elements, the current through each branch can be found using Kirchoff’s Current Law, (KCL). Kirchoff’s current law or junction law states that “the total current entering a junction or node is exactly equal to the current leaving that node”, so the currents entering and leaving node “A” above are given as:

Taking the derivative, dividing through the above equation by C and rearranging gives us the following Second-order equation for the circuit current. It becomes a second-order equation because there are two reactive elements in the circuit, the inductor and the capacitor.

The opposition to current flow in this type of AC circuit is made up of three components: XL XC andR and the combination of these three gives the circuit impedance, Z. We know from above that the voltage has the same amplitude and phase in all the components of a parallel RLC circuit. Then the impedance across each component can also be described mathematically according to the current flowing through, and the voltage across each element as.

Impedance of a Parallel RLC Circuit

You will notice that the final equation for a parallel RLC circuit produces complex impedance’s for each parallel branch as each element becomes the reciprocal of impedance, ( 1/Z ) with the reciprocal of impedance being called Admittance.

In parallel AC circuits it is more convenient to use admittance, symbol ( Y ) to solve complex branch impedance’s especially when two or more parallel branch impedance’s are involved (helps with the math’s). The total admittance of the circuit can simply be found by the addition of the parallel admittances. Then the total impedance, ZT of the circuit will therefore be 1/YT Siemens as shown.

Admittance of a Parallel RLC Circuit

The new unit for admittance is the Siemens, abbreviated as S, ( old unit mho’s ℧, ohm’s in reverse ). Admittances are added together in parallel branches, whereas impedance’s are added together in series branches. But if we can have a reciprocal of impedance, we can also have a reciprocal of resistance and reactance as impedance consists of two components, R and X. Then the reciprocal of resistance is called Conductance and the reciprocal of reactance is called Susceptance.

Conductance, Admittance and Susceptance

The units used for conductance, admittance and susceptance are all the same namely Siemens ( S ), which can also be thought of as the reciprocal of Ohms or ohm -1, but the symbol used for each element is different and in a pure component this is given as:

Admittance ( Y ) :

Admittance is the reciprocal of impedance, Z and is given the symbol Y. In AC circuits admittance is defined as the ease at which a circuit composed of resistances and reactances allows current to flow when a voltage is applied taking into account the phase difference between the voltage and the current.

The admittance of a parallel circuit is the ratio of phasor current to phasor voltage with the angle of the admittance being the negative to that of impedance.

Conductance ( G ) :

Conductance is the reciprocal of resistance, R and is given the symbol G. Conductance is defined as the ease at which a resistor (or a set of resistors) allows current to flow when a voltage, either AC or DC is applied.

Susceptance ( B ) :

Susceptance is the reciprocal of of a pure reactance, X and is given the symbol B. In AC circuits susceptance is defined as the ease at which a reactance (or a set of reactances) allows an alternating current to flow when a voltage of a given frequency is applied.

Susceptance has the opposite sign to reactance so capacitive susceptance BC is positive, +ve in value and inductive susceptance BL is negative, -ve in value.

We can therefore define inductive and capacitive susceptance as being:

In AC series circuits the opposition to current flow is impedance, Z which has two components, resistance R and reactance, X and from these two components we can construct an impedance triangle. Similarly, in a parallel RLC circuit, admittance, Y also has two components, conductance, Gand susceptance, B. This makes it possible to construct an admittance triangle that has a horizontal conductance axis, G and a vertical susceptance axis, jB as shown.

Admittance Triangle for a Parallel RLC Circuit

Now that we have an admittance triangle, we can use Pythagoras to calculate the magnitudes of all three sides as well as the phase angle as shown.

from Pythagoras,

Then we can define both the admittance of the circuit and the impedance with respect to admittance as:

Giving us a power factor angle of:

As the admittance, Y of a parallel RLC circuit is a complex quantity, the admittance corresponding to the general form of impedance Z = R + jX for series circuits will be written as Y = G - jB for parallel circuits where the real part G is the conductance and the imaginary part jB is the susceptance. In polar form this will be given as:

Parallel RLC Circuit Example No1

A 1kΩ resistor, a 142mH coil and a 160uF capacitor are all connected in parallel across a 240V, 60Hz supply. Calculate the impedance of the parallel RLC circuit and the current drawn from the supply.

Impedance of a Parallel RLC Circuit

In an AC circuit, the resistor is unaffected by frequency therefore R = 1kΩ’s

Inductive Reactance, ( XL ):

Capacitive Reactance, ( XC ):

Impedance, ( Z ):

Supply Current, ( Is ):

Parallel RLC Circuit Example No2

A 50Ω resistor, a 20mH coil and a 5uF capacitor are all connected in parallel across a 50V, 100Hz supply. Calculate the total current drawn from the supply, the current for each branch, the total impedance of the circuit and the phase angle. Also construct the current and admittance triangles representing the circuit.

Parallel RLC Circuit

1). Inductive Reactance, ( XL ):

2). Capacitive Reactance, ( XC ):

3). Impedance, ( Z ):

4). Current through resistance, R ( IR ):

5). Current through inductor, L ( IL ):

6). Current through capacitor, C ( IC ):

7). Total supply current, ( IS ):

8). Conductance, ( G ):

9). Inductive Susceptance, ( BL ):

10). Capacitive Susceptance, ( BC ):

11). Admittance, ( Y ):

12). Phase Angle, ( φ ) between the resultant current and the supply voltage:

Current and Admittance Triangles

Parallel RLC Circuit Summary

In a parallel RLC circuit containing a resistor, an inductor and a capacitor the circuit current IS is the phasor sum made up of three components, IR, IL and IC with the supply voltage common to all three. Since the supply voltage is common to all three components it is used as the horizontal reference when constructing a current triangle.

Parallel RLC networks can be analysed using vector diagrams just the same as with series RLC circuits. However, the analysis of parallel RLC circuits is a little more mathematically difficult than for series RLC circuits when it contains two or more current branches. So an AC parallel circuit can be easily analysed using the reciprocal of impedance called Admittance.

Admittance is the reciprocal of impedance given the symbol, Y. Like impedance, it is a complex quantity consisting of a real part and an imaginary part. The real part is the reciprocal of resistance and is called Conductance, symbol Y while the imaginary part is the reciprocal of reactance and is called Susceptance, symbol B and expressed in complex form as: Y = G + jB with the duality between the two complex impedance’s being defined as:

Series Circuit Parallel Circuit

Voltage, (V) Current, (I)

Resistance, (R) Conductance, (G)

Reactance, (X) Susceptance, (B)

Impedance, (Z) Admittance, (Y)

As susceptance is the reciprocal of reactance, in an inductive circuit, inductive susceptance, BL will be negative in value and in a capacitive circuit, capacitive susceptance, BC will be positive in value. The exact opposite to XL and XC respectively.

We have seen so far that series and parallel RLC circuits contain both capacitive reactance and inductive reactance within the same circuit. If we vary the frequency across these circuits there must become a point where the capacitive reactance value equals that of the inductive reactance and

therefore, XC = XL. The frequency point at which this occurs is called resonance and in the next tutorial we will look at series resonance and how its presence alters the characteristics of the circuit.

SOLUTION OF NETWORK EQUATIONS USING LAPLACE TRANSFORM

Frequency response of linear amplifiers

Problem how to design of amplifiers which give uniform gain and phase delay over the frequency pass - band. For example: pass - band in region:

o 20 Hz - 100 kHz for audio amplifiers

o 0 - 4,5 MHz for video amplifiers.

RC (resistance - capacitance) coupling amplifier are mostly used. Otherwise direct coupled amplifiers (in integrated form).

Most circuits for pulse shaping are based on the chain RC and transmission lines. Before proceeding to the subject of pulse processing I will deal briefly with the RC passive filters and with other interesting subject of transmission lines

Passive Filters

Low - pass filter - pass low frequencies and reject high frequencies.

High - pass filter - pass high frequencies and reject low frequencies.

Band - pass filter - pass some particular band of frequencies and reject all frequencies outside the range.

Band - rejection filter - specifically designed to reject a particular band of frequencies and pass some other frequencies.

Typical or physically realized characteristic are different from ideal frequency characteristic ( to pass all frequencies to some frequency w0=2pf0 and pass no frequencies above that value. Reason why is not possible to design such ideal filter with linear circuits elements.

Break frequency - (1/(2)0.5 ) - half - power frequency. The name derived from the fact that voltage (U/(2)0.5 ) or current (I/(2)0.5 ) => power is proportional to the square of U or I, is one-half of its maximum value.

RC filter

The magnitude in dB [=> logarithmic scale]. Magnitude curve is flat for low frequency and rolls off at high frequency. The phase shifts from 0o to 90o at high frequencies.

Low-pass filter circuit (RC circuit) and frequency characteristic.

(Low-frequency asymptote 0dB/decade, high-frequency asymptote - 20dB/decade). Phase shift characteristic (arctangent curve) has pulse shift 45deg at frequencyw=1/t.

CR filter

Ideal high - pass filter passes all frequencies above some frequency w0 =1/t , j(w0)=45o and no frequencies bellow that value. At low frequencies the magnitude has slope +20dB/decade due to term wt in the numerator.

High-pass filter circuit (CR circuit) and frequency characteristic.

(Low-frequency asymptote 0dB/decade, high-frequency asymptote - 20dB/decade). Phase shift characteristic (arctangent curve) has pulse shift 45deg at frequencyw=1/t.

Simple networks are capable of realizing characteristics with w0 as a center frequency of pass or rejection band and the frequency wL0, wH0 at which the maximum or minimum occurs is called cutoff frequency ( lower / upper break frequency)

The width of the pass / rejection band - bandwidth BW = wH0 -wL0

Band-pass and band-rejection filters and characteristics

Frequency domain vs. time domain

Considering the response of pass CR filter to rectangular pulse in time domain (a rectangular pulse generated as a positive step function at t = 0 s and a negative step function to the end of duration of a pulse) we will obtain a sag of the pulse with duration t i. For small x<<1 is 1-exp(-x) ~x and the exponential function can be replaced with the first terms of its series

u(t)=U0(1-exp(-ti/td)~U0ti/td.

Allowable slope of the pulse hight after shaping rectangular pulse

Small sag is obtained by making td=CR large with respect to the pulse duration (=> t i/td is small).From low frequency band limit definition wd=2pfd=1/td=1/RC. The ratio terms of shape that sags d=DU0/U0=ti/td=ti2pfd.

Relation between frequency domain characteristic and time domain response characteristic

fd=d/2pti

=> for small sag (<< 10%) is necessary small fd.

The rise time dependents on the pulse transfer in the high frequencies characteristic region or rise time is due to frequency distortion. If the leading edge of the input pulse is a step function then the output rise time (measured at 10% level t10 and 90% level t90 is defined tra=t90-t10=2.3 ti-0.1ti=2.2ti. Considering that RC circuits is law-pass device with upper break frequency wi=2pfi=1/ti=1/RC and ti=RC=tra/(2.2), then relation between rise time tra and upper break frequency

fi=(2.2)/2ptra ~ (0.35)/tra

Determinating of the active pulse rise time tra

TWO PORT NETWORK

Two Port Networks - Z , Y , h , g , ABCD Parameters

A pair of terminals at which a signal (voltage or current) may enter or leave is called a port.

A network having only one such pair of terminals is called a one port network.

A two-port network (or four-terminal network, or quadripole) is an electrical circuit or device with two pairs of terminals.Examples include transistors, filters and matching networks. The analysis of two-port networks was pioneered in the 1920s by Franz Breisig, a German mathematician.

A two-port network basically consists in isolating either a complete circuit or part of it and finding its characteristic parameters. Once this is done, the isolated part of the circuit becomes a "black box" with a set of distinctive properties, enabling us to abstract away its specific physical buildup, thus simplifying analysis. Any circuit can be transformed into a two-port network provided that it does not contain an independent source.

A two-port network is represented by four external variables: voltage and

current at the input port, and voltage and current at the output port, so that the two-port network can be treated as a black box modeled by the the relationships between the four variables , , and . There exist six different ways to describe the relationships between these variables, depending on which two of the four variables are given, while the other two can always be derived.

Note: All voltages and currents below are complex variables and represented by phasors containing

both magnitude and phase angle. However, for convenience the phasor notation and are replaced by V andI respectively.

The parameters used in order to describe a two-port network are the following: Z, Y, A , h, g. They are usually expressed in matrix notation and they establish relations between the following parameters:Input voltage V1Output voltage V2Input current I1Output current I2

Z-model : In the Z-model or impedance model, the two currents I1 and I2 are assumed to be known, and the voltages V1and V2can be found by:

where

Here all four parameters Z11,Z12 ,Z21 , and Z22 represent impedance. In particular, Z21 and Z12 aretransfer impedances, defined as the ratio of a voltage V1(or V2) in one part of a network to a current I2(or I1) in another part . Z12 = V1 / I2 . Z is a 2 by 2 matrix containing all four parameters.

Y-model : In the Y-model or admittance model, the two voltages V1 and V2 are assumed to be known, and the currents I1 and I2 can be found by:

where

Here all four parameters Y11,Y12 ,Y21 , and Y22 represent admittance. In particular, Y21 and Y12 are transfer admittances. Y is the corresponding parameter matrix.

ABCD -model : In the A-model or transmission model, we assume V1 and I1 are known, and find V2 and I2by:

where

Here A and D are dimensionless coefficients, B is impedance and C is admittance. A negative sign is added to the output current I2 in the model, so that the direction of the current is out-ward, for easy analysis of a cascade of multiple network models.

H-model : In the H-model or hybrid model, we assume V2 and I1 are known, and find V1 and I2 by:

where

Here h12 and h21 are dimensionless coefficients, h11 is impedance and h22 is admittance.

g model :In g model or inverse hybrid model, we assume V1 and I2 are known, and find V2 and I1 by :

where

Here g12 and g21 are dimensionless coefficients, g22 is impedance and g11 is admittance.

ELECTRONIC DEVICES:

ENERGY BANDS IN SILICON

Silicon is a kind of semiconductor material whose number of free electrons is less than conductor but more than that of insulator. For having this unique characteristics, silicon has a vast application in the field of electronics. There are two kinds of energy band in silicon which are conduction band and valance band. A series of energy levels having valance electrons forms the valance band in the solid. At absolute 0°K temperature the energy levels of the valance band is filled with electrons. This band contains maximum amount of energy when the electrons are in valance band, no current flows due to such electrons. Conduction band is the higher energy level band which is of minimum amount of energy. This band is partially filled by the electrons which are known as the free electrons as they can move anywhere in the solid. These electrons are responsible for current flowing. There is a gap of energy between the conduction band and the valance band. This gap of energy is called forbidden energy gap. Actually this determines the nature of a solid. Whether a solid is metal, insulator or semiconductor in nature, the fact is determined by the amount of forbidden energy gap. Partially there is no gap for metals and very large gap for insulators. For semiconductors the gap is neither very large nor the bands get overlapped. Silicon has forbidden gap of 1.2 ev at 300°K temperature. We know that in silicon crystal, covalent bond exists. Silicon is neutrally charged. When an electron breaks away from its covalent bond, a hole is created behind it. As temperature increases more, more electrons jump into conduction band and more holes are created in the valance bond.

Energy Band Diagram of Silicon

Energy band diagram of silicon shows the levels of energies of electrons in the material. There are two kinds of energy band, valance band and conduction band. Valance electrons occupy the valance band with highest energy level. Free electrons are in conduction band with minimum amount of energy. Valance and conduction bands are separated by the amount of energy known as the forbidden energy gap. This amount is nearly 1.2 ev at 300° K. In intrinsic silicon, the Fermi level lies in the middle of the donor atoms, it becomes n-type when Fermi level moves higher i.e. closer to conduction band. When intrinsic silicon is doped with acceptor atoms, it becomes p - type and Fermi level moves towards valance

band. Energy Bands Diagram of Intrinsic Silicon

CARRIER TRANSPORT IN SILICON

Semiconductor

Semiconductors are electronic conductors whose conductivity lies between conductor and insulator.eg:- germanium, silicon. Both electrons and holes are carriers of electric current. These are known as charge carriers. The purest forms of semiconductors are known as intrinsic semiconductors. The process of adding impurities to the intrinsic semiconductors is known as doping. This will increase the number of charge carries and thus increases the conductivity of semiconductors. Usually pentavalent impurities such as Arsenic, Antimony or trivalent impurities such as Indium, Boron etc are added for doping. When majority carriers are electrons and minority carriers are holes, then it is called p-type semiconductors. When majority carriers are holes and minority carriers are electrons, then it is called n-type semiconductors.

Carrier Drift

Electrons and holes will move under the influence of applied electric field due to the force exerted by field on charge carriers such as electrons and holes.The process in which the movement of charge carriers occurred due to the applied electric field is known as drift.

The net force on carrier, F=qE

The movement of electrons and holes produces current known as drift current (Id).

Id = nqVdA

Where Id is the drift current, n is the number of charge carriers per unit volume, Vd is the drift velocity of charge carriers, q is the charge of electron and A is the area of semiconductor.

When an electric field is applied across the semiconductor, a current is produced due to the flow of electron and holes. Drift velocity is the average velocity of charge carriers in the drift current. The positively charged holes moved with the applied electric field but the negatively charged electrons moved against the electric field. When electric field is applied to an electron in free space, the electron will accelerate in a straight line from negative terminal to the positive terminal of the applied voltage. But in semiconductors, contains lots of electrons and accelerate in a random direction. The random movement of electrons in the straight line is known as drift current. The drift current depends upon mobility of electrons and holes.

Carrier Mobility

It refers to the movement of electrons and holes in semiconductors. The electron mobility means the movement of electrons move through semiconductor and hole mobility means the movement of holes through semiconductors.

When an electric field is applied across the material, the electrons will move with an average velocity called the drift velocity Vd. The mobility is defined as:-

Vd= μ/E

E - Magnitude of applied electric field.

Vd - Magnitude of drift velocity due to applied electric field. μ is the proportionality factor known as electron mobility. This can also be used for determining the hole mobility.

μ= Vd/E

The mobility determines how quick the charge carriers are or it is the measure of how the charge carriers are move under applied electric field.

Conductivity

It is proportional to the product of mobility and carrier concentration. i.e, large number of electrons with smaller mobility having the same as small number of electrons with high mobility. The mobility in semiconductors depends on temperature, concentration of charge carriers, impurity concentration etc.

Let n as number density of electrons, μe is the mobility of electrons. Then the electrical conductivity σ = ne μe

For the hole mobility, σ= peμh

P - Density of holes.

μh - hole mobility.

If electrons and holes are present, then the total conductivity is

σ= e (μe + μh )

Velocity Saturation

The drain velocity is linearly proportional to electric field at low fields. So the mobility is constant and is known as low field mobility. When the electric field increases, the carrier velocity will linearly increases and achieves maximum possible value. This is known as saturation velocity,Vsat. The linear relationship between average carrier velocity and applied electric field changes at high electric field which is shown in below graph. Another field effect is called Gunn Effect. This causes intervalley electron transfer and this cause the reduction of drift velocity. But in most cases, the drift velocity increases as electric field increases. Otherwise it remains unchanged.

The saturation velocity can be determined by the following equation. Using this equation an estimate of velocity saturation can be calculated.

Vsat = √((2Ephonon)/m*)

Carrier diffusion

It is the process of distributing of particles from high concentration region to low concentration region. It does not require any external forces. It is due to the thermal energy. This causes the random movement of charge carriers without applying any electric field. This movement of charge carriers causes diffusion current. Diffusion current is entirely different between the drift current. The drift current is occurred due to the applied electric field. No drift current occurs without any applied electric field. But the diffusion current does not depend on applied electric field. It occurs even if there is no applied electric field. The direction of diffusion current depends on carrier concentration as well as changes in charge carrier concentration. The total current in semiconductors is made by drift current and diffusion current.

Hall Effect

The Hall Effect describes the charge carriers behavior when applying an electric field or magnetic field. It is the voltage difference produced across the electrical conductor. When a current carrying semiconductor is placed in a magnetic field, the charge carriers of semiconductors experience a force in direction perpendicular to both magnetic field and current. At equilibrium, the charges are build up on the sides of conductor and it will balance the influence of magnetic field which produces a voltage developed at the semiconductor edge. It produces a measurable voltage between two edges of the conductors. The presence of this voltage is known as Hall Effect and it was discovered by Edwin Hall in 1879. The ratio of induced electric field to the product of current density and applied magnetic field is known as Hall coefficient. In semiconductors, electrons and holes are the charge carriers. It may present in different concentrations and different mobilities.

The Hall voltage is given by Vh= IB/qdn ; where I is the current and B is the magnetic field strength, n is the density of mobile charges, d is the thickness .

The Hall voltage is directly proportional to the current flowing and magnetic field strength and it is inversely proportional to the number density of mobile charges.

At moderate magnetic field, the Hall coefficient is

Where b = μe/ μh

n= electron concentration

p=hole concentration

μe= electron mobility

μh= hole mobility

e=elementary charge

The Hall Effect can be used to measure magnetic fields with Hall probe. Hall Effect is different for different charge carriers. It has different polarity for positive and negative charge carriers. It helps for the study of conduction in semiconductors. It measures the average drift velocity of charge carriers by using Hall probe.

. Carrier recombination and generation

2.8.1. Simple recombination-generation model 2.8.2. Band-to-band recombination 2.8.3. Trap assisted recombination 2.8.4. Surface recombination 2.8.5. Auger recombination 2.8.6. Generation due to light

Recombination of electrons and holes is a process by which both carriers annihilate each other: electrons occupy - through one or multiple steps - the empty state associated with a hole. Both carriers eventually disappear in the process. The energy difference between the initial and final state of the electron is released in the process. This leads to one possible classification of the recombination processes. In the case of radiative recombination, this energy is emitted in the form of a photon. In the case of non-radiative recombination, it is passed on to one or more phonons and in the case of Auger recombination it is given off in the form of kinetic energy to another electron. Another classification scheme considers the individual energy levels and particles involved. These different processes are further illustrated with Figure 2.8.1.

Figure 2.8.1 : Carrier recombination mechanisms in semiconductors

Band-to-band recombination occurs when an electron moves from its conduction band state into the empty valence band state associated with the hole. This band-to-band transition is typically also a radiative transition in direct bandgap semiconductors.

Trap-assisted recombination occurs when an electron falls into a "trap", an energy level within the

bandgap caused by the presence of a foreign atom or a structural defect. Once the trap is filled it cannot accept another electron. The electron occupying the trap, in a second step, moves into an empty valence band state, thereby completing the recombination process. One can envision this process as a two-step transition of an electron from the conduction band to the valence band or as the annihilation of the electron and hole, which meet each other in the trap. We will refer to this process as Shockley-Read-Hall (SRH) recombination.

Auger recombination is a process in which an electron and a hole recombine in a band-to-band transition, but now the resulting energy is given off to another electron or hole. The involvement of a third particle affects the recombination rate so that we need to treat Auger recombination differently from band-to-band recombination.

Each of these recombination mechanisms can be reversed leading to carrier generation rather than recombination. A single expression will be used to describe recombination as well as generation for each of the above mechanisms.

In addition, there are generation mechanisms, which do not have an associated recombination mechanism, such as generation of carriers by light absorption or by a high-energy electron/particle beam. These processes are referred to as ionization processes. Impact ionization, which is the generation mechanism associated with Auger recombination, also belongs to this category. The generation mechanisms are illustrated with Figure 2.8.2.

Figure 2.8.2 : Carrier generation due to light absorption and ionization due to high-energy particle beams

Carrier generation due to light absorption occurs if the photon energy is large enough to raise an electron from the valence band into an empty conduction band state, thereby generating one electron-hole pair. The photon energy needs to be larger than the bandgap energy to satisfy this condition. The photon is absorbed in this process and the excess energy, Eph - Eg, is added to the electron and the hole in the form of kinetic energy.

Carrier generation or ionization due to a high-energy beam consisting of charged particles is similar except that the available energy can be much larger than the bandgap energy so that multiple electron-hole pairs can be formed. The high-energy particle gradually loses its energy and eventually stops. This generation mechanism is used in semiconductor-based nuclear particle counters. As the number of ionized electron-hole pairs varies with the energy of the particle, one can also use such detector to measure the particle energy.

Finally, there is a generation process called impact ionization, the generation mechanism that is the counterpart of Auger recombination. Impact ionization is caused by an electron/hole with an energy, which is much larger/smaller than the conduction/valence band edge. The detailed mechanism is illustrated with Figure 2.8.3.

Figure 2.8.3: Impact ionization and avalanche multiplication of electrons and holes in the presence of a large electric field.

The excess energy is given off to generate an electron-hole pair through a band-to-band transition. This generation process causes avalanche multiplication in semiconductor diodes under high reverse bias: As one carrier accelerates in the electric field it gains energy. The kinetic energy is given off to an electron in the valence band, thereby creating an electron-hole pair. The resulting two electrons can create two more electrons which generate four more causing an avalanche multiplication effect. Electrons as well as holes contribute to avalanche multiplication.

2.8.1. Simple recombination-generation model

A simple model for the recombination-generation mechanisms states that the recombination-generation rate is proportional to the excess carrier density. It acknowledges the fact that no net recombination takes place if the carrier density equals the thermal equilibrium value. The resulting expression for the recombination of electrons in a p-type semiconductor is given by:

(2.8.1)

and similarly for holes in an n-type semiconductor:

(2.8.2)

where the parameter t can be interpreted as the average time after which an excess minority carrier recombines.

We will show for each of the different recombination mechanisms that the recombination rate can be simplified to this form when applied to minority carriers in a "quasi-neutral" semiconductor. The above expressions are therefore only valid under these conditions. The recombination rates of the majority carriers equals that of the minority carriers since in steady state recombination involves an equal number of holes and electrons. Therefore, the recombination rate of the majority carriers depends on the excess-minority-carrier-density as the minority carriers limit the recombination rate.

Recombination in a depletion region and in situations where the hole and electron density are close to each other cannot be described with the simple model and the more elaborate expressions for the individual recombination mechanisms must be used.

2.8.2. Band-to-band recombination

Band-to-band recombination depends on the density of available electrons and holes. Both carrier types need to be available in the recombination process. Therefore, the rate is expected to be proportional to the product of n and p. Also, in thermal equilibrium, the recombination rate must equal the generation rate since there is no net recombination or generation. As the product of n and p equals ni

2 in thermal equilibrium, the net recombination rate can be expressed as:

(2.8.3)

where b is the bimolecular recombination constant.

2.8.3. Trap assisted recombination

The net recombination rate for trap-assisted recombination is given by:

(2.8.4)

The derivation of this equation is beyond the scope of this text.

This expression can be further simplified for p >> n to:

(2.8.5)

and for n >> p to:

(2.8.6)

were

(2.8.7)

2.8.4. Surface recombination

Recombination at surfaces and interfaces can have a significant impact on the behavior of semiconductor devices. This is because surfaces and interfaces typically contain a large number of recombination centers because of the abrupt termination of the semiconductor crystal, which leaves a large number of electrically active states. In addition, the surfaces and interfaces are more likely to contain impurities since they are exposed during the device fabrication process. The net recombination rate due to trap-assisted recombination and generation is given by:

(2.8.8)

This expression is almost identical to that of Shockley-Hall-Read recombination. The only difference is that the recombination is due to a two-dimensional density of traps, Nts, as the traps only exist at the surface or interface.

This equation can be further simplified for minority carriers in a quasi-neutral region. For instance for electrons in a quasi-neutral p-type region p >> n and p >> ni so that for Ei = Est, it can be simplified to:

(2.8.9)

where the recombination velocity, vs, is given by:

(2.8.10)

2.8.5. Auger recombination

Auger recombination involves three particles: an electron and a hole, which recombine in a band-to-band transition and give off the resulting energy to another electron or hole. The expression for the net recombination rate is therefore similar to that of band-to-band recombination but includes the density of the electrons or holes, which receive the released energy from the electron-hole annihilation:

(2.8.11)

The two terms correspond to the two possible mechanisms.

2.8.6. Generation due to light

Carriers can be generated in semiconductors by illuminating the semiconductor with light. The energy of the incoming photons is used to bring an electron from a lower energy level to a higher energy level. In the case where an electron is removed from the valence band and added to the conduction band, an electron-hole pair is generated. A necessary condition is that the energy of the photon, Eph,is larger than the bandgap energy,Eg. As the energy of the photon is given off to the electron, the photon no longer exists.

If each absorbed photon creates one electron-hole pair, the electron and hole generation rates are given by:

(2.8.12)

where a is the absorption coefficient of the material at the energy of the incoming photon. The absorption of light in a semiconductor causes the optical power to decrease with distance. This effect is described mathematically by:

(2.8.13)

The calculation of the generation rate of carriers therefore requires first a calculation of the optical power within the structure from which the generation rate can then be obtained using (2.8.12).

Example 2.11

Calculate the electron and hole densities in an n-type silicon wafer (Nd = 1017 cm-3) illuminated uniformly with 10 mW/cm2 of red light (Eph = 1.8 eV). The absorption coefficient of red light in silicon is 10-3 cm-1. The minority carrier lifetime is 10 ms.

Solution The generation rate of electrons and holes equals:

where the photon energy was converted into Joules. The excess carrier densities are then obtained from:

The excess carrier densities are then obtained from: So that the electron and hole densities equal:

NOTE:-working principle of PN junction, Zener diode and BJT are given covered in the study material of Basic Electronics Engineering

TUNNEL DIODE

The application of transistors is very high in frequency range are hampered due to the transit time and other effects. Many devices use the negative conductance property of semiconductors for high frequency applications. Tunnel diode is one of the most commonly used negative conductance devices. It is also known as Esaki diode after L. Esaki for his work on this effect.

This diode is a two terminal device. The concentration of dopants in both p and n region is very high. It is about 1024 - 1025 m-3 the p-n junction is also abrupt. For this reasons, the depletion layer width is very small. In the current voltage characteristics of tunnel diode, we can find a negative slope region when forward bias is applied. Quantum mechanical tunneling is responsible for the phenomenon and thus this device is named as tunnel diode. The doping is very high so at absolute zero temperature the Fermi levels lies within the bias of the semiconductors. When no bias is applied any current flows through the junction.

Characteristics of Tunnel Diode

When reverse bias is applied the Fermi level of p - side becomes higher than the Fermi level of n-side. Hence, the tunneling of electrons from the balance band of p-side to the

conduction band of n-side takes place. With the interments of the reverse bias the tunnel current also increases. When forward junction is a applied the Fermi level of n - side becomes higher that the Fermi level of p - side thus the tunneling of electrons from the n - side to p - side takes place. The amount of the tunnel current is very large than the normal junction current. When the forward bias is increased, the tunnel current is increased up to certain limit. When the band edge of n - side is same with the Fermi level in p - side the tunnel current is maximum with the further increment in the forward bias the tunnel current decreases and we get the desired negative conduction region. When the forward bias is raised further, normal p-n junction current is obtained which is exponentially proportional to the applied voltage. The V - I characteristics of the tunnel diode is given,

The negative resistance is used to achieve oscillation and often Ck+ function is of very high frequency frequencies.

Tunnel Diode Symbol

Tunnel Diode Applications

Tunnel diode is a type of sc diode which is capable of very fast and in microwave frequency range. It was the quantum mechanical effect which is known as tunneling. It is ideal for fast oscillators and receivers for its negative slope characteristics. But it cannot be used in large integrated circuits – that’s why it’s an applications are limited.

When the voltage is first applied current stars flowing through it. The current increases with the increase of voltage. Once the voltage rises high enough suddenly the current again starts increasing and tunnel diode stars behaving like a normal diode. Because of this unusual behavior, it can be used in number of special applications started below.

Oscillator circuits :Tunnel diodes can be used as high frequency oscillators as the transition between the high electrical conductivity is very rapid. They can be used to create oscillation as high as 5Gz. Even they are capable of creativity oscillation up to 100 GHz in a appropriate digital circuits.

Used in microwave circuits: Normal diode transistors do not perform well in microwave operation. So, for microwave generators and amplifiers tunnel diode are. In microwave waves and satellite communication equipments they were used widely, but now a day’s their uses is decreasing rapidly as transistor for working in wave frequency area available in market.

Resistant to nuclear radiation :Tunnel diodes are resistant to the effects of magnetic fields, high temperature and radioactivity. That’s why these can be used in modern military equipment. These are used in nuclear magnetic resource machine also. But the most important field of its use satellite communication equipments.

Tunnel Diode Oscillator

Tunnel diode can make a very stable oscillator circuit when they are coupled to a tuned circuit or cavity, biased at the centre point of negative resistance region. Here is an example

of tunnel diode oscillatory circuit.

The tunnel diode is losing coupled to a tunable cavity. By using a short, antenna feed probe placed in the cavity off centre loose coupling is achieved. To increase the stability of oscillation and achieve o/p power over wider bandwidth loose coupling is used. The range of the output power produced is few hundred micro-watts. This is useful for many microwave application. The physical position of the tuner determining the frequency of operation. If the frequency of operation is changed by this method, that is called mechanical tuning. Tunnel diode oscillators can be tuned electronically also.

Tunnel diode oscillators which are meant to be operated at microwave frequencies, generally used some form of transmission lines as tunnel circuit. These oscillators are useful in application that requires a few millwatts of power, example- local oscillators for microwave super electrodyne receiver.

JFET OR JUNCTION FIELD EFFECT TRANSISTOR

The junction field effect transistor or JFET is one of the simplest transistors from the structural point of view. It is a voltage controlled semiconductor device. In this, the current is carried by only one type of carriers. So, it is a unipolar device. It has a very high input

electrical resistance. JFET consists of a doped Si or GaAs bar. There are ohmic contacts, the two ends of the bar and semiconductor junction on

its two sides. If the semiconductor bar is n - type, the two sides of the bar is heavily doped with p - type impurities and this is known as n - channel JFET. On the other hand if the semiconductor bar is p- type, the two sides of the bar is heavily doped with n - type impurities and this is known as p- channel JFET. When a voltage is applied between the two ends, a current which is carried by the majority carriers of the bar flows along the length of the bar.

There are several terminals in JFET. The terminal through which the majority carrier enter the bar and the terminal through which they leave are known as source (s) and drain (D) respectively. The heavily doped region on the two sides is known as the gate (G). In junction field effect transistor, the junction is a reverse biased. As a result, depletion regions form, which extend to the bar. By changing gate to source voltage, the depletion width can be controlled. So, the effective cross section area decreased with increasing reverse bias. So, the drain current is a function of the gate to the source voltage: Now days JFET is obsolete. Its applicants are limited to circuit design. Where it can be used an amplifier and as a switch both.

N-Channel JFET

A semiconductor bar of n-type material is taken & ohmic contacts are made on either ends of the bar. Terminals are brought out from these ohmic contacts and named as drain & source as shown in the figure below. On the other two sides of the n-type semiconductor bar, heavily doped p-type regions are formed to create a p-n junction. Both these p-type regions are connected together via ohmic contacts and the gate terminal is brought out as seen below. Figure below shows the n-channel & p-channel JFET with symbols. The arrow on the gate indicates the direction of the current. Current flows through the length of the n-type bar (channel) due to majority charge carries which in this case are electrons. When a voltage is applied between the two ends, a current which is carried by the majority carriers electrons flows along the length of a bar. The majority carriers enter the bar through the source terminal

and leave through the drain terminal. The heavily doped regions of the n-type bar are known as the gates. The gate source junctions is reverse is biased as a result depletion regions from which extend to the bar by changing gate to source voltage effective cross sectional area decreases with the function of the gate to source voltage.

P-Channel JEFT

p-channel JFET consists of a p-type silicon or GaAs. Two sides of the bar is heavily doped with n-type impurities. When a voltage is applied between the two ends, a current which is carried by the majority carrier holes flow along the length of a bar. The gate source junction is reverse biased as a result depletion regions form, which extend to the bar by changing gate to extend to source voltage the depletion width can be controlled. The effective cross sectional area decreased with increasing reverse bias, so the drain current is the function of the gate to source voltage.

Biasing of JFET

The gate to source p-n junction of a JFET is always reverse biased and supply voltage is given across the drain to source terminal.

Operation of Junction Field Effect Transistor or JFET

Operation with gate to source voltage = 0

If an n-channel JFET is biased as explained above and the gate to source voltage is kept zero, due to the positive drain to source voltage few electrons which are available for conduction in the n-type material will start flowing from the narrow passage (channel) from source to drain. This current is called as drain current. As the channel has some finite resistance it will cause some voltage drop across the channel. Hence the depletion region of the p-n junction starts increasing and penetrates more into the n-type material as it is lightly doped. Due to this the width of the channel available for conduction is reduced. The penetration of the depletion region into the n-type region depends on the reverse bias voltage. Maximum drain current ID(MAX) will flow through the device when the channel is widest i.e. when VGS is

zero. Operation with negative gate to source voltage As a negative voltage is applied to the gate to source p-n junction the depletion region increases and penetration of the depletion region into the n-type channel further increases. If the negative gate to source voltage is further increased the depletion region spreads more and more inside the n-type bar. Due to this less and less number of charge carries (electrons) can pass through the channel and the drain current reduces. Hence with increase in negative gate to source voltage drain current reduces. At a certain value of this voltage the depletion region from both the ends will increase and touch each other and the drain current will become zero. This gate to source voltage at which drain current is cutoff is called as VGS(OFF). As seen the VGS controls ID. Hence, JFET is a voltage controlled device. The relationship between ID and VGS is given by Shockley’s

equation Where, VP is the pinch off voltage which is the value of drain to source VDS at which drain current reaches its constant saturation value. Any further increase in VDS does not affect ID.

JEFT Characteristics or Junction Field Effect Transistor Characteristics

In this characteristics we can find three regions,

1) The linear or the ohmic region: Here the drain to source voltage is small and drain current in nearly proportional to the drain to source voltage. When a positive drain to source voltage is applied, this voltage increases from zero to a small value, the depletion region width remain very small and under this condition the semi conductor bar behaves just like a resistor. So, drain current increases almost linearly with drain to source voltage.

ii) The saturation of the active region: Here the drain current is almost constant and it is not dependent on the drain to source voltage actually. When the drain to source voltage continuous to increase the channel resistance increases and at some point, the depletion regions meet near the drain to pinch off the channel. Beyond that pinch off voltage , the drain, current attains saturation.

iii) The breakdown voltage: Here the drain current increases rapidly with a small increase of the drain to source voltage. Actually for large value of drain to source voltage, a breakdown of the gate junction takes place which results a sharp increase of the drain current.

Transfer characteristics The graphical characteristics plot of the saturation drain current against the gate to source voltage is known as the transfer characteristics of JFET. It can be obtained from static characteristics very easily. The transfer characteristics of an n- channel is shown below.

JFET as Switch

The junction field effect transistor (JFET) can be used as an electronically controlled switch to control electric power to a load.

JFET’s are normally on (NO) devices. They are normally saturated devices. When a reverse bias is applied between gate and source, the depletion regions of that junction expand and pinching off the channel through which current flowing takes place. If the channel is pinched the current does not flow the device will be in switched off condition.

By this process junction field effect transistor can be used as switches. But now days their application is obsolete. An example of JFETs acting as a switch and the corresponding circuit is given below.

Applications of JFET

The junction field effect transistor has many application in the field of electronics and communication.

Some of these applications are stated below.

1. Low noise and high input impedance amplifier:-

Noise is an undesirable disturbance which interferes with the signals information - greater the noise less the information. Energy electronics device cause some amount of noise. If FET s is used at the front end, we get less amount of amplified noise at the output. Now, it has very high input impedance. So, it can be used in high input impedance amplifier.

2. Buffer amplifier:- Buffer amplifier should have very high input impedance and low output impedance. Because of high i / p impedance and low output impedance, FET acts as great buffer amplifier. the common drain mode can be used in this purpose.

3. R.F. Amplifier:-

JFET is good in low current signal operation as it is a voltage controlled semiconductors device. It has very low noise level. So, it can be used as RF amplifier in receiver sections of communication field.

4. Current source:-

Here all the supply voltage appears across load. If the current tries to increase very much, the excessive load a current drives the JFET in to active region. Thus JFET acts as a current source .

5. Switch:-

JFET may be used as an on / off switch controlling electrical power to load. An example is given below

Chopper :- When a source wave is applied to the gate of JFET witch, the chopper operation can be done using JFET.

6. Multiplexer:-

Analog multiplexer circuit can be made using JFETs. An example is given below.

MOSFET WORKING

MOSFET stands for metal oxide semiconductor field effect transistor. It is capable of voltage gain and signal power gain. The MOSFET is the core of integrated circuit designed as thousands of these can be fabricated in a single chip because of its very small size. Every modern electronic system consists of VLST technology and without MOSFET, large scale integration is impossible.

It is a four terminals device. The drain and source terminals are connected to the heavily doped regions. The gate terminal is connected top on the oxide layer and the substrate or body terminal is connected to the intrinsic semiconductor. MOSFET has four terminals which is already stated above, they are gate, source drain and substrate or body. MOS capacity present in the device is the main part. The conduction and valance bands are position relative to the Fermi level at the surface is a function of MOS capacitor voltage. The metal of the gate terminal and the sc acts the parallel and the oxide layer acts as insulator of the state MOS capacitor. Between the drain and source terminal inversion layer is formed and due to the flow of carriers in it, the current flows in MOSFET the inversion layer is properties are controlled by gate voltage. Thus it is a voltage controlled device.

Two basic types of MOSFET are n channel and p channel MOSFETs. In n channel MOSFET is current is due to the flow of electrons in inversion layer and in p channel current is due to the flow of holes. Another type of characteristics of clarification can be made of those are enhancement type and depletion type MOSFETs. In enhancement mode, these are normally off and turned on by applying gate voltage. The opposite phenomenon happens in depletion type MOSFETs.

Working Principle of MOSFET

The working principle of MOSFET depends up on the MOS capacitor. The MOS capacitor is the main part. The semiconductor surface at below the oxide layer and between the drain and source terminal can be inverted from p-type to n-type by applying a positive or negative gate voltages respectively. When we apply positive gate voltage the holes present beneath the oxide layer experience repulsive force and the holes are pushed downward with the substrate. The depletion region is populated by the bound negative charges, which are associated with the acceptor atoms. The positive voltage also attracts electrons from the n+ source and drain regions in to the channel. The electron reach channel is formed. Now, if a voltage is applied between the source and the drain, current flows freely between the source and drain gate voltage controls the electrons concentration the channel. Instead of positive if apply negative voltage a hole channel will be formed beneath the oxide layer.

Now, the controlling of source to gate voltage is responsible for the conduction of current between source and the drain. If the gate voltage exceeds a given value, called the three voltage only then the conduction begins.

The current equation of MOSFET in triode region is -

Where, un = Mobility of the electrons Cox = Capacitance of the oxide layer W = Width of the gate area L = Length of the channel VGS = Gate to Source voltage VTH = Threshold voltage VDS

= Drain to Source voltage.

P-Channel MOSFET

MOSFET which has p - channel region between source any gate is known as p - channel MOSFET. It is a four terminal devices, the terminals are gate, drain, source and substrate or body. The drain and source are heavily doped p+ region and the substrate is in n-type. The current flows due to the flow of positively charged holes that’s why it is known as p-channel MOSFET. When we apply negative gate voltage, the electrons present beneath the oxide layer, experiences repulsive force and they are pushed downward in to the substrate, the depletion region is populated by the bound positive charges which are associated with the donor atoms. The negative gate voltage also attracts holes from p+ source and drain region in to the channel region. Thus hole which channel is formed now if a voltage between the source and the drain is applied current flows. The gate voltage controls the hole concentration of the channel. The diagram of p- channel enhancement and depletion MOSFET are given below.

N-Channel MOSFET

MOSFET having n-channel region between source and drain is known as n-channel MOSFET . It is a four terminal device, the terminals are gate, drain and source and substrate or body. The drain and source are heavily doped n+ region and the substrate is p-type. The current flows due to flow of the negatively charged electrons, that’s why it is known as n- channel MOSFET. When we apply the positive gate voltage the holes present beneath the oxide layer experiences repulsive force and the holes are pushed downwards in to the bound negative charges which are associated with the acceptor atoms. The positive gate voltage also attracts electrons from n+ source and drain region in to the channel thus an electron reach channel is formed, now if a voltage is applied between the source and drain. The gate voltage controls the electron concentration in the channel n-channel MOSFET is preferred over p-channel MOSFET as the mobility of electrons are higher than holes. The diagrams of enhancements mode and depletion mode are given below.

LED OR LIGHT EMITTING DIODE

A light emitting diode (LED) is known to be one of the best optoelectronic devices out of the lot. The device is capable of emitting a fairly narrow bandwidth of visible or invisible light when its internal diode junction attains a forward electric current or voltage. The visible lights that an LED emits are usually orange, red, yellow, or green. The invisible light includes the infrared light. The biggest advantage of this device is its high power to light conversion efficiency. That is, the efficiency is almost 50 times greater than a simple tungsten lamp. The response time of the LED is also known to be very fast in the range of 0.1 microseconds when compared with 100 milliseconds for a tungsten lamp. Due to these advantages, the device wide applications as visual indicators and as dancing light displays.

We know that a P-N junction can connect the absorbed light energy into its proportional electric current. The same process is reversed here. That is, the P-N junction emits light when energy is applied on it. This phenomenon is generally called electroluminance, which can be defined as the emission of light from a semi-conductor under the influence of an electric field. The charge carriers recombine in a forward P-N junction as the electrons cross from the N-region and recombine with the holes existing in the P-region. Free electrons are

in the conduction band of energy levels, while holes are in the valence energy band. Thus the energy level of the holes will be lesser than the energy levels of the electrons. Some part of the energy must be dissipated in order to recombine the electrons and the holes. This energy is emitted in the form of heat and light.

The electrons dissipate energy in the form of heat for silicon and germanium diodes. But in Galium- Arsenide-phosphorous (GaAsP) and Galium-phosphorous (GaP) semiconductors, the electrons dissipate energy by emitting photons. If the semiconductor is translucent, the junction becomes the source of light as it is emitted, thus becoming a light emitting diode (LED). But when the junction is reverse biased no light will be produced by the LED, and, on the contrary the device may also get damaged.

The constructional diagram of a LED is shown below.

LED Construction

All the semiconductors listed above can be used. An N-type epitaxial layer is grown upon a substrate, and the P-region is produced by diffusion . The P-region that includes the recombination of charge carriers is shown is the top. Thus the P-region becomes the device surface. Inorder to allow more surface area for the light to be emitted the metal anode connections are made at the outer edges of the P-layer. For the light t be reflected as much as possible towards the surface of the device, a gold film s applied to the surface bottom. This setting also enables to provide a cathode connection. The reabsorption problem is fixed by including domed lenses for the device. All the wires in the electronic circuits of the device is protected by encasing the device. The light emitted by the device depends on the type of semiconductor material used. Infrared light is produced by using Gallium Arsenide (GaAs) as semiconductor. Red or yellow light is produced by using Gallium-Arsenide-Phosphorus (GaAsP) as semiconductor. Red or green light is produced by using Gallium-Phosphorus (GaP) as semiconductor.

LED Circuit Symbol

The circuit symbol of LED consists of two arrow marks which indicate the radiation emitted by the diode.

LED Circuit Symbol

LED Characteristics

LED Characteristics

The forward bias Voltage-Current (V-I) curve and the output characteristics curve is shown in the figure above. The V-I curve is practically applicable in burglar alarms. Forward bias of approximately 1 volt is needed to give significant forward current. The second figure is used to represent a radiant power-forward current curve. The output power produced is very small and thus the efficiency in electrical-to-radiant energy conversion is very less.

The figure below shows a series resistor Rseries connected to the LED. Once the forward bias of the device exceeds, the current will increase at a greater rate in accordance to a small increase in voltage. This shows that the forward resistance of the device is very low. This

shows the importance of using an external series current limiting resistor. Series resistance is determined by the following equation.

Rseries = (Vsupply – V)/I

Vsupply – Supply Voltage

V – LED forward bias voltage

I – Current

LED Circuit

The commercially used LED’s have a typical voltage drop between 1.5 Volt to 2.5 Volt or current between 10 to 50 milliamperes. The exact voltage drop depends on the LED current, colour, tolerance, and so on.

LED as an Indicator

The circuit shown below is one of the main applications of LED. The circuit is designed by wiring it in inverse parallel with a normal diode, to prevent the device from being reverse biased. The value of the series resistance should be half, relative to that o f a DC circuit.

LED as an Indicator

LEDS displays are made to display numbers from segments. One such design is the seven-segment display as shown below. Any desired numerals from 0-9 can be displayed by passing current through the correct segments. To connect such segment a common anode or common cathode cathode configuration can be used. Both the connections are shown below. The LED’s are switched ON and OFF by using transistors.

Advantages of LED’s

Very low voltage and current are enough to drive the LED.

Voltage range – 1 to 2 volts.

Current – 5 to 20 milliamperes.

Total power output will be less than 150 milliwatts.

The response time is very less – only about 10 nanoseconds.

The device does not need any heating and warm up time.

Miniature in size and hence light weight.

Have a rugged construction and hence can withstand shock and vibrations.

An LED has a life span of more than 20 years.

Disadvantages

A slight excess in voltage or current can damage the device.

The device is known to have a much wider bandwidth compared to the laser.

The temperature depends on the radiant output power and wavelength.

PHOTO DIODE

photo-diode-construction-symbol

Photo-diode is a two-terminal semiconductor P-N junction device and is designed to operate with reverse bias. The basic biasing arrangement, construction and symbols for the device are given in figure. It is either mounted in translucent case or has its semiconductor junction mounted beneath an optical lens. The output voltage is taken from across a series-con-nected load resistor R. This resistance may be connected between the diode and ground or between the diode and the positive terminal of the supply, as illustrated in figure.

When the P-N junction is reverse-biased, a reverse saturation current flows due to thermally generated holes and electrons being swept across the junction as the minority carriers. With the increase in temperature of the junction more and more hole-electron pairs are created and so the reverse saturation current I0 increases. The same effect can be had by illuminating the junction. When light energy bombards a P-N junction, it dislodges valence electrons. The more light striking the junction the larger the reverse current in a diode. It is due to generation of more and more charge carriers with the increase in level of illumination. This is clearly shown in ‘ figure for different intensity levels. The dark current is the current that exists when no light is incident. It is to be noted here that current becomes zero only with a positive applied bias equals to VQ. The almost equal spacing between the curves for the same increment in luminous flux reveals that the reverse saturation current I0 increases linearly with the luminous flux as shown in figure. Increase in reverse voltage does not increase the reverse current significantly, because all available charge carriers are already being swept across the junction. For reducing the reverse saturation current I0 to zero, it is necessary to forward bias the junction by an amount equal to barrier potential. Thus the photodiode can be used as a photoconductive device.

On removal of reverse bias applied across the photodiode, minority charge carriers continue to be swept across the junction while the diode is illuminated. This has the effect of increasing the concentration of holes in the P-side and that of electrons in the N-side But the barrier potential is negative on the P-side and positive on the N-side, and was created by holes flowing from P to N-side and electrons from N to P-side during fabrication of junction. Thus the flow of minority carriers tends to reduce the barrier potential.

When an external circuit is connected across the diode terminals, the minority carrier; return to the original side via the external circuit. The electrons which crossed the junction from P to N-side now flow out through the N-terminal and into the P-terminal This means that the device is behaving as a voltage cell with the N-side being the negative terminal and the P-side the positive terminal. Thus, the photodiode is & photovoltaic device as well as photoconductive device.

Photodiodes have a far lower light sensitivity than cadmium sulphide LDR s, but giving a fair quicker response in light level. Generally LDRs are ideal for use in slow acting direct coupled light-level sensing applications, while photodiodes are ideal for use in fast acting ac coupled signalling applications. Typical photodiode applications include detection (both visible and invisible), demodulation, switching, logic circuits that need stability and high speed, character recognition, optical communication equipment, IR remote control circuits encoders etc.