Chapter 1

EEN1046 ELECTRONICS III

Chapter 1Operational Amplifier Fundamentals

1.0 INTRODUCTIONThe term operational amplifier, or op-amp, was originally applied to high-performance DC differential amplifiers that used vacuum tubes. These amplifiers formed the basis of the analog computer, which was capable of solving differential equations. Early operational amplifiers (op-amps) were used primarily to perform mathematical operations such as addition, subtraction, integration, and differentiation, hence the term operational. These devices were constructed with vacuum tubes and worked with high voltages. The vacuum tubes soon gave way to the transistor, and eventually to integrated circuit. In the present day the term operational amplifier is used to refer to very high gain DC coupled differential amplifiers with single-ended outputs. Most of these amplifiers appear in integrated circuit form. Today’s op-amps are linear integrated circuits that use relatively low supply voltages. Except for the reduction in size and cost, the function of today’s op-amp has changed very little from the original version.

The first series of commercially available op-amps was uA702, introduced by Fairchild Semiconductor (later bought over by National Semiconductor) in 1963. In 1965, National Semiconductor had introduced the LM101, while Fairchild unveiled the ever-popular uA741 in 1967. The 741 series op-amp remains until today.

Figure 1: An actual size image of an 8-pin IC low-power op-amp (LM358)

1.1 OP-AMP SYMBOL AND EQUIVALENT CIRCUIT

(a) Basic symbol (b) Symbol with dc supply

Figure 2: Symbol for op-amp

The circuit symbol for the op-amp is a triangle with two inputs and one output, as shown in Fig. 2. The minus (-) and plus (+) at the input specifies the inverting and non-inverting inputs respectively. The former will produce an inverted output, while the latter producing an output of the same polarity as that of the applied input.

Chapter 1: Operational Amplifier Fundamentals

1


The op-amp, being an active element, must also be powered by a voltage supply.

Figure 3: Dual, or split voltage power supply used with op-amps

The equivalent circuit of an op-amp is shown in Fig. 4.

Figure 4: Approximate equivalent circuit of a non-ideal op-amp

The op-amp amplifies the voltage difference between non-inverting and inverting input. It senses the difference between the two inputs, and produces an output which forms the product of both the difference vD, and the gain AOL.

vO == AOL(V+ - V-) = AOLvD [1]Where

AOL = open-loop voltage gainZout = output impedanceZin = input impedance

AOL is called the open-loop voltage gain because it is the gain of the op-amp without any external feedback from the output to the input.

1.2 IDEAL OPERATIONAL AMPLIFIERThe ideal op-amp would be expected to have the following important characteristics: Infinite open-loop voltage gain, . Infinite input impedance, . Zero output impedance, . Infinite bandwidth, AOL remains unchanged from DC to very high frequency. Zero offset voltage, zero input (V+=V-) produces zero output. Infinite common-mode rejection only amplifies voltage difference between non-

inverting and inverting input.

The above characteristics in turn form the basis for two fundamental rules of an ideal op-amp:


2

Ground

+VCC

-VEE


1. no current flows into either of the input terminal2. there is no voltage difference between the two input terminals

While in practice, no commercial op-amp can meet these 6 ideal characteristics, it is still possible to achieve high-performance circuits despite this fact. We will consider practical op-amp in later chapters. For the time being, let us concentrate on circuits using ideal op-amp.

1.3 IDEAL INVERTING AMPLIFIERThe schematic of an inverting amplifier using op-amp with negative feedback is shown in 5. The feedback network consists of a single resistor RF while R1 is usually known as the input resistor. A small signal at the input will be amplified, and its polarity inverted, hence the name inverting amplifier.

Figure 5: Ideal inverting amplifier.

1.3.2 Analysis of Ideal Inverting AmplifierAssuming ideal op-amp, the input impedance for inverting input will be infinite. There will be no current flowing into the inverting input (V -). Applying Kirchoff Current Law (KCL) at the inverting input:

Figure 6: Current flow convention

Since V+ is at ground potential, voltage V- must also be approximately zero volts. This is what we term as ‘virtual ground’, which essentially means that the negative terminal is at zero volts, but it does not provide a current path to ground i.e. it is not directly connected to ground potential.

Since V- = 0 (virtual ground),

And we have the closed-loop gain

[2]


3

RF

-

+

Vi

Vo

R1

V-

V+

I=0

I1

IF

RF

-

+

Vi

Vo

R1

V-

V+

The feedback network


Note the negative in Eqn. (4) which account for the reversal of polarity of the output signal.

ExampleGiven the op-amp configuration in Fig. 7, determine the value of Rf required to produce a closed loop voltage gain of –100.

Figure 7

SolutionKnowing that Ri = 2.2k and ACL = -100,

1.4 IDEAL NON-INVERTING AMPLIFIERAs opposed to the inverting amplifier, the non-inverting amplifier will amplify a small input signal with no polarity reversal at the output.

Figure 8: Ideal non-inverting amplifier

Again, assuming ideal op-amp, the input impedance for inverting input will be infinite. There will be no current flowing into the inverting input. Using the current convention as in Fig. 6 and applying Kirchoff Current Law (KCL) at the inverting input:

Since V- = V+ = Vi ,


4

-

+Vi

Vo

R1

V-

V+

RF


And we have

[3]

Note: with ideal op-amp there is no restriction on the values of RF and R1 because closed loop gain ACL is only dependent on the ratio of RF and R1. However there are several practical considerations that should be kept in mind when we actually build the circuit using real op-amps. More will be discussed on this later.

1.4.1 Ideal Voltage FollowerFrom Eqn. (5), when RF goes to zero and R1 approaches infinity, the closed loop gain ACL becomes one or unity. In this case the output voltage actually follows the input voltage, hence the name voltage follower.

Figure 9: Ideal voltage follower

The reader might ask what is the purpose of having an amplifier with a voltage gain of one? In many instances, the voltage follower is useful as a buffer. The input impedance of the voltage follower is essentially infinite while the output impedance is zero. As an example, consider the case below.

Figure 10: (a) source with a 100k output resistance driving a 1k load, and (b) source with a 100k output resistance, voltage follower, and 1k load

In (a), the ratio of output voltage to input voltage is:

There is severe loading effect, whereby the output signal undergoes high attenuation.

In (b) however, due to the presence of the buffer, hence the loading effect is

eliminated.


5

-

+Vi

Vo

V-

V+


1.5 IDEAL SUMMING AMPLIFIER

Figure 11: Ideal inverting summing amplifier

The output of the summing amplifier is proportional to the algebraic sum of its separate inputs. It is frequently called a signal mixer as it is used to combine audio signal from several microphones, guitars, tape recorders, etc., to provide a single output. There are two types of summing amplifier, the inverting and non-inverting. We will consider the inverting summing amplifier first.

Similar to the analysis of the inverting amplifier, by applying Kirchoff Current Law at the inverting input of the op-amp, we obtain:

[4]

Taking V- as virtual ground,

[5]

In the special case when R1 = R2 = R3 = RF,[6]

ExampleRefer to Fig. 12. Determine the following:(a) VR1 and VR2

(b) Current through Rf

(c) Vout

Figure 12

Solution(a) VR1 = 1V VR2 = 1.8V


6

RF

-

+

V1

Vo

R1V-

V+R2

R3

V2

V3


(b)

(c)

or

In addition to the inverting summing amplifier, it is possible to have a non-inverting summing amplifier, as shown by the schematic in Fig. 13.

Figure 13: Ideal non-inverting summing amplifier

Derivation of the expression for the output voltage will be left to the reader as an exercise. Superposition theorem is needed to find the total voltage for V+. The expression for Vo is given as:

[7]

1.6 IDEAL DIFFERENCE AMPLIFIERAs shown in Fig. 14, a difference or differential amplifier has input voltages that are applied simultaneously to both the inverting and non-inverting inputs. Its output voltage Vo is proportional to the voltage difference (V2 – V1). An ideal difference amplifier only amplifies the difference between two signals,

Figure 14: Ideal difference amplifier

Applying Kirchoff Current Law at the inverting input:


7

RF

-

+V1

Vo

R1

V-

V+

R

RV2

V3

R

-

+

V1

Vo

R1

V-

V+

R2

V2R4

R3


[8]

R3 and R4 form a voltage divider at the non-inverting input, therefore:

[9]

V+ = V- for an ideal op-amp, and substituting Eqn. (11) into (Eqn. 10), we get:

Now if we make R1 = R4 and R2 = R3,

[10]

Here the ratio R2/R1 is referred to as the differential gain. When all resistors are equal, Eqn. (12) reduces to:

[11]

Such a circuit is called a unity-gain analog subtractor.

1.7 IDEAL INTEGRATORThe basic integrator circuit is shown in Fig. Figure15.

Figure 15: Ideal integrator

The current across capacitor CF from Vo to V- is:

[12]

Applying Kirchoff Current Law at the inverting input and using the virtual ground concept:

V- = 0,

Or [13]


8

CF

-

+

Vi

Vo

R1

V-

V+


Extra Notes on the Integrator (non-examinable)The schematic in Fig. Figure15 is essentially similar to the inverting amplifier circuit, except the feedback resistor RF is replaced with feedback capacitor CF. At low frequencies, the reactance of CF is very high and the circuit may become unstable. A feedback resistor RF is usually added in parallel to CF as in Fig. 16. At low frequencies, the impedance will be dominated by RF since XcF is large. The integrator circuit will essentially become an inverting amplifier. At higher frequency the impedance will be dominated by XcF, the reactance of CF, and the circuit functions as an integrator.

Figure 16: A compensated integrator

Adding RF as in Fig. 16 has two advantages: This will stabilize the op-amp so that it would not saturate or oscillate at low

frequency. The effect of input offset voltage in a non-ideal op-amp is reduced. If left

unchecked, the input offset voltage will be integrated, resulting in saturation of the output Vo. More will be discussed about this when we consider non-ideal characteristics of practical op-amp.

However, by having RF we also limit the use of the integrator circuit in Fig. 16 to input frequencies above:

[14]

Above fLow, the feedback capacitor CF dominates and we can ignore the effect of RF. There is also an upper frequency limit for the integrator, because in practical op-amp, AOL decreases with frequency. When AOL approaches unity the concept of virtual ground and hence Eqn. (15) will not be valid. As a rule of thumb, the frequency where closed-loop gain ACL becomes unity is taken as the upper cut-off frequency. Thus,

[15]

ConclusionTherefore, between fLow and fHigh, the integrator circuit of Fig. 16 will function properly, with Vo related to Vi by Eqn. (15).


9

CF

-

+

Vi

Vo

R1

V-

V+

RF


1.8 IDEAL DIFFERENTIATORThe schematic for a basic differentiator is shown if Fig. 17.

Figure 17: Ideal differentiator

The current across capacitor C1 from V- to Vi is:

[16]

Applying Kirchoff Current Law at the inverting input,

Since V- = 0, [17]

ExampleA triangular waveform is applied to the input of the circuit in Fig. 18 as shown. Determine what the output should be and sketch its waveform in relation to the input.

Figure 18

Solution

From time 0s to

5s,


10

C1

-

+

Vi

Vo

RF

V-

V+


From time 5s to 10s,

Extra Notes on the Differentiator (non-examinable)

The schematic of Fig. 17 is essentially similar to the inverting amplifier circuit, except the resistor R1 is replaced with capacitor C1. The serious problem with the ideal differentiator circuit is that it is susceptible to high-frequency electrical noise. This is because the reactance of C1 decreases with frequency causing a corresponding increase in the closed-loop voltage gain. Although the basic differentiator’s closed-loop gain increases with frequency, it is limited at the high frequency by the op-amp’s open-loop response curve. To put a limit on the closed-loop gain at high frequencies before being limited by the op-amp’s open-loop response curve, resistor Rs is added in series to C1 as shown in FigureFig. 19.

Figure 19: A compensated differentiator

This type of compensated differentiator circuit has a much improved noise handling ability, however the maximum usable frequency is now limited to input frequencies below:

[18]

At high frequency, reactance of C1 is smaller than RS. The circuit behaves as an inverting amplifier. At frequency below fHigh, the reactance of C1 is much larger than RS and the circuit behaves as a differentiator. High frequency noise will not be differentiated. If we keep the ratio of RF/RS small enough (say 5 to 10), the effect of high frequency noise will be greatly diminished.


11

C1

-

+

Vi

Vo

RF

V-

V+

RS


1.9 IDEAL CURRENT TO VOLTAGE CONVERTER

Figure 20: Ideal current to voltage converter

In some cases, the output of a device or circuit is a current, and we may need to convert this to an output voltage. This is when we use a current to voltage converter configuration. Consider the circuit shown in Fig. 20. From the usual approach of applying virtual ground and Kirchoff Current Law, it it evident that the output voltage Vo is given as:

[19]

The current source in Figure 20 can be substituted with any other current-generating component such as a photovoltaic cell or a photodiode in reverse bias. An example of the application is shown in Fig. 21. The current source in this case corresponds to the photocurrent generated in the depletion region of the PN junction.

Figure 21: Application of current to voltage converter in photodiode detection circuit

The sensitivity of the circuit increases with the value of RF. Sometimes it is not convenient to use a large resistor, as large resistors usually have associated stray inductance and capacitance. In this instance, a T-network can be used to replace RF as illustrated in Fig. 22.


12

-

+I

RF

Vo

-

+I

RF

Vo

R1

R21V1

-

+

RF

Vo


Figure 22: Using a T-network to achieve high feedback resistance

[20]

Applying Kirchoff Current Law at node 1:

[21]

Using Eqn. (22) in Eqn. (23):

[22]

If RF and R2 >> R1, Eqn. (24) can be approximated as:

[23]

1.10 IDEAL VOLTAGE TO CURRENT CONVERTER

Figure 23: Ideal voltage to current converter

This is the complement of the current-to-voltage converter from the previous section. We use the voltage-to-current converter when for example, we have a voltage source but would like to drive a coil in a magnetic circuit with a given current.

From the usual approach of applying virtual ground and Kirchoff Current Law, it it evident that the load current iL is given as:

[24]

Eqn. (26) means that the current iL is directly proportional to the input voltage VS and is independent of the load impedance (in this case RL), although it needs to be noted that the circuit in Fig. 23 is far from ideal as one end of the load might need to be grounded, making it impractical in certain real life applications.


13

RL

-

+Vo

+

-

R

vs

iL


ExampleA voltage to current converter can be used as a dc voltmeter. Fig. 24 shows a dc voltmeter, where a moving coil meter is conntected as the load. The full scale current of the moving coil is IM = 200A. Determine the value of R1 to give a full-scale reading of VS(max) = 300V.

Figure 24Solution


14


1.11 IDEAL INSTRUMENTATION AMPLIFIER

An instrumentation amplifier allows us to change the gain by changing only a single resistance value. The schematic is shown if Fig. 25. In this configuration, two noninverting amplifiers are used as the input stage, and a difference amplifier is the second, or amplifying stage.

R4

R2

V2

R4

-

R3

+

-

+

-

R3R1

+

V1Vo2

R2

Vo

Vo1

Figure 25: Instrumentation Amplifier

The current in resistor R1 is given by: [27]

The output voltages of the first stage op-amps are:

[28a]

[28b]

We have seen previously that the output of a difference amplifier is given as

[29]

Combining Eqn.s (28a), (28b), and (29), the output voltage for the instrumentation amplifier is found to be:

[30]

From Eqn. (30), it can be seen that the differential gain is a function of resistor R1, which can easily be varied by using a potentiometer, thus providing a variable amplifier gain with the adjustment of only one resistance.


15

i1


APPENDIXIn the syllabus, the comparators are under Chapter 6: Nonlinear Circuit Applications. However, the basic comparator circuit is attached here for your convenience to study for the ECT1 lab experiment. Various types of comparator circuits are included in Chapter 6.

COMPARATORComparator circuits are used to compare either:a) Two changing voltages with respect to each other orb) A changing voltage to a set dc level.

Output from a comparator is normally a dc voltage that indicates the polarity relationship between the two input voltages. When a comparator is used to compare a signal amplitude to a fixed dc level, the circuit is referred to as level comparator.

Figure A1: Level comparator

Without the feedback path, the voltage gain of the above circuit is equal to the open loop gain AOL. Therefore, the output clips at positive or negative supply voltage whenever the difference is positive or negative respectively. Note that the output voltage will not reach the supply voltage, it will always be slightly less than the supply. In a practical circuit, the reference voltage is derived from the supply voltage by resistive voltage divider circuit as shown in the Figure A2. In this case, the reference voltage VREF with respect to ground is given by:

[Eqn.A1]

Eqn.A1 is obtained using superposition theory. First get the voltage at VREF with VEE

terminal grounded. Then we determine VREF with VCC grounded. The voltage at VREF

is just the sum of the two voltages. There is a negative in Eqn.A1 because the polarity of VEE is negative. Other alternative form of obtaining the reference voltage is shown in Figure A3.


16

Vo

-

+

+

-VREF

Vin

-VEE

+VCC

VREF

Vin

0V

+VCC

-VEE

Vo


Figure A2: Level comparator using a resistive network to obtain the reference voltage

Figure A3: Alternative form of obtaining the reference voltage for level comparator


17

-

+Vo

VREF

Vin -VEE

+VCC

R1

R2

+VCC

-

+Vo

VREF

Vin -VEE

R1

R221

2

RR

RVV CC

REF

Chapter 1

Documents

loop response

kirchoff current

inverting

loop voltage

inverting

inverting

virtual ground

resistor r1