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6.3 Unity Gain buffer .................................................................................................................... 65
6.4 Frequency dependent gain .................................................................................................... 66
6.5
Gain-bandwidth product ........................................................................................................ 67 6.6 Control of gain - the non-inverting amplifier .......................................................................... 68
6.7
The inverting amplifier ........................................................................................................... 69
Electronics Notes-2014 Imperial College London page 4 of 72
1 Introduction to fundamentals
Electronics is all about the motion of electrons.
These tiny particles have a property called charge which gives rise to some of the most dramatic andfar reaching behaviour.
Protons also have a charge, equal and opposite to that of the electron, but, by-and-large, they tend tobe bound up inside materials and are not mobile.
Electrons, however, can often find themselves liberated from their parent atoms and they can thenrespond much more freely to electric and magnetic field.
A current, , is then just the movement of charge, .
In circuit theory the electrons move along wires and the current then has a magnitude equal to the
amount of charge per second passing a given point, , with units of Coulombs/sec or Amp(ere)s.
It also has a direction along the wire.
Historically, unfortunately, the direction has been defined as positive for + charges and negative for –charges. Hence the electrons themselves actually travel in the opposite direction to the current.
The movement of the electrons is caused by the presence of forces acting upon them.
Usually these are electrostatic forces due to electric fields set up by the presence of other charges.
A charge, , in an electric field , will experience a force .
Sometimes they are magnetic forces
A charge, , moving with velocity in a magnetic field with experience a Lorentz force
Note: Faraday induction arises through the combined action of these two forces. The chargeswithin a conductor which is formed into a loop of area, , through which a time dependent magneticfield is present will separate causing a potential difference, (sometimes misleadingly called an
electromotive force), which has units of Volts (and not force!).
2 Basic AC/DC Circuits
2.1 Electrostatic Forces, Field and Potential Energy
Coulomb first established that the electrostatic force, , between two charges and separated by
a distance, , follows an inverse square law relationship
|| 2-1
Where is a fundamental constant called the (electrical) permittivity of free space. In SI units the
charges should be in coulombs (1 electron has a charge of 1.60210 C), and
9.0 10Nm2/C2. The concept of a ‘field’ mediating the force can be expressed by saying that
the charge experiences the electric field, , due to . Alternatively it can be said that the charge
Electronics Notes-2014 Imperial College London page 5 of 72
experiences the electric field, , due to . These must be equivalent statements and it follows
that the electric field, , due to a charge is
|| 2-2
and that the force on another charge, , which sits in this field, is
2-3
Another useful pair of concepts is that of potential energy and potential. The force equation 1-1 can
be integrated from ∞ to to find out the potential energy, , between the two charges to give
∙
2-4
Potential energy exists between two (or more) charges due to their relative positioning.
However one can also talk about the potential due to a single charge. This is similar to the use of
field (equation 2-2) rather than force, and we can say that the potential energy in equation 2-4 is dueto the presence of charge, , siting in the potential of charge , and vice-versa. This implies the
potential, , due to a charge, , is given by
2-5
and the potential energy due to anther charge, , which sits in this potential is
2-6
Note and are vector quantities, whereas and are scalars. The relationships between them
can be written and V . The potential, , is often referred to as voltage.
2.2 Definition of current
Current, , is the rate of flow of electric charge, :
dt
dq I Ampères (= Coulombs/s) 2-7
The charges can be ions in an electrolyte (like a battery) or a gas discharge (like a fluorescent strip
light) but for most solids it is the movement of electrons that generates current. Each electron carries
a negative charge of 1.60210-19
coulombs and the unit of current, the ampere, is defined as the
passage of 1 coulomb of charge (about 61018
electrons) per second along a metal wire. Such largenumbers are difficult to comprehend but to give you an idea, this number is greater than the age of
Electronics Notes-2014 Imperial College London page 7 of 72
is the charge on the electron) where /. is a vector and lies along the direction shown in the
diagram from the positive potential (anode) of the battery to the negative potential (cathode). The
convention for the direction of current is that positive charges move along the direction of . Electrons
have a negative charge and will travel in the opposite direction but the direction of current is the
same.
Applying Newton’s law of motion, , the electron should accelerate in the electric field, but just
as an object falling in a gravitational field reaches a terminal velocity (due to a drag force – air
resistance), so an electron attains a drift velocity, . What causes the drag force which limits the
velocity here? If we were able to see inside the metal the movement of the electrons would be rapid
and apparently random. This is due to them having a relatively large thermal energy, ~, which
manifests itself as a large thermal speed, from , of typically 10
6 m/s. Superimposed upon
this motion is the drift of electrons due to the applied electric field. Since the density of electrons is
very high they will collide with each other or with the Cu nuclei/atoms that make up the wire. Thesecollisions randomise the direction of the electrons and limit the net speed in the direction of the
electric field. This is the origin of Resistance. The energy lost in electron collisions ends up as heat in
the wire.
Figure 2-2 (right) shows a short section of copper wire of cross-sectional area, . When current flows
in the wire, electrons travel along the wire with a drift velocity ms-1
. All the electrons within the
shaded cylinder of length will pass through the wire in one second. The volume of the shaded
cylinder is and the amount of charge contained within it is coulombs where is the number
density of free electrons (per m3) in copper (given previously as 8.510
28 m
-3). Since current is the
rate of flow of charge C/s or amperes (amps). We can use this information to estimate
for a specific case as follows; suppose a bicycle lamp connected to a battery by copper wires 1.5 mm
thick carries a current of 350 mA. Then 350 10 8.5 10 1.6 10 0.75 10
which gives: 1.46 10 ms-1
or 0.0146 mms-1
. This is a surprisingly small value especially when
compared with the electron’s thermal velocity.
Even conductors have some resistance and the value depends on the geometrical shape of the
conductor. The material itself has an intrinsic property called resistivity, .
A wire with a large cross-sectional area has more electrons to carry the current whilst for longer wires
there will be more scattering events as the electron travels through the material. Thus we expect the
resistance of a wire to vary directly with length , and inversely with area , so that /. The
constant of proportionality is the resistivity, , and thus
A
L R
2-8
where the units of are m. The resistivity of some metals, semiconductors and insulators is shown
in the table. The range is huge and reflects the ease with which electrons become delocalised (or how
tightly bound they are to their parent atoms). Copper is the most commonly used highly conducting
Electronics Notes-2014 Imperial College London page 10 of 72
2.6 Resistors in series
R1
V i
0 V
I
R2
Figure 2-4: Simple circuit with two resistors in series
Consider the circuit shown in Figure 2-4. This circuit draws a current from the +ve terminal of thebattery. The same current flows back into the –ve terminal having passed through both resistorsone after the other.
By Ohm’s Law (equation 2-10) the voltage across resistor 1 must be equal to 1.
Similarly the voltage across resistor 2 must be equal to 2.
However the sum of the two voltages across the two resistors must add up to the voltage provided bythe battery as there is no other source of voltage in the circuit; i.e. the battery voltage alone is whatdrives the current.
On the left-hand side is the driving voltage whilst on the right-hand are individual voltages developedacross resistors around a closed loop. In a generalised form to include an arbitrary number of
resistors this is Kirchhoff’s Voltage Law (KVL)
Hence 12 12 2-11
Equation 2-11 just looks like Ohm’s Law again with where is effectively the total resistancepresented by the resistors in series.
We now have 1 2 which can be generalised to any number of resistors in series to give
1 2 ⋯ 2-12
2.7 Resistors in parallel
R1
V i
0 V
I
R2
I
Figure 2-5: Simple circuit with two resistors in parallel
Consider the circuit shown in Figure 2-5. This circuit draws a current from the +ve terminal of thebattery. The same current flows back into the –ve terminal. However on its outward journey at point
Electronics Notes-2014 Imperial College London page 11 of 72
A it has a choice of two paths and so it divides without loss. At point B the two currents mustrecombine. We can express this as
at point A at point B
Where the left-hand side has current flowing into the node and the right-hand side current flowing out.
This is an example of Kirchhoff’s Current Law (KCL) which can be generalised to any number ofconnections to a single node.
By Ohm’s Law (equation 2-9) the current through resistor 1 must be equal to /1.
Similarly the current through resistor 2 must be equal to /2.
However the sum of the two currents through the two resistors must add up to the current drawn fromthe battery as there is no other source of current in the circuit.
Hence
/1/2 2-13
Equation 2-13 can be rearranged to look like Ohm’s Law with
where is effectively the total resistance presented by the resistors in
parallel.
We now have
which can be generalised to any number of resistors in parallel to give
⋯ 2-14
The parallel arrangement of resistors in the circuit in Figure 2-5 is sometimes referred to as a currentdivider in that a fraction of the current goes through each resistor. Combining equation 2-13 with theKCL relation at node A (or B) gives
2 1 2
and 2-15 1
1 2
2.8 The voltage divider circuit
In Figure 2-4 we saw a circuit with two dissimilar resistors. This is redrawn in Figure 2-6 to emphasise
its use as a voltage divider in which the voltage across R2 is a fraction of that produced by the driving
Electronics Notes-2014 Imperial College London page 13 of 72
oxide electrodes immersed in H2SO4. These batteries are able to deliver 12V at very large currents to
power the car starter motor, lights and ignition plugs. Such a battery cannot last forever and the
electrodes deteriorate due to the chemical reactions. Fortunately this can be reversed by continuously
recharging using an alternator driven by the fan belt. A modern ‘zinc carbon’ battery or “dry cell”
consists of an anode (carbon), a cathode (zinc) and an acidic electrolyte manganese oxide (chloride)paste. These batteries are gradually being replaced by alkaline types which use potassium hydroxide,
KOH as the electrolyte and deliver 1.5V. Lithium-ion batteries are more sophisticated (they use
organic electrolytes) and are used to power laptops, mobile telephones, kitchen scales, watches and
many other devices. None of these is rechargeable.
To maintain a potential difference at the battery terminals requires the movement of ions within the
electrolyte. This comes at a cost: there is opposition to the ion flow which manifests itself as an
internal resistance.
2.9.4 Real voltage and current sources
2.9.4.1 Voltage sourcesIn section 2.9.3 the concept of an internal resistance was introduced. This turns out to be a very good
way of representing the way in which real power sources differ from the ideal. The concept for a
voltage source is illustrated in Figure 2-7. The voltage source is enclosed in the dashed box and has
an internal resistance in series with the voltage, V, and it has a value, RS. The presence of this
internal resistance means that even if the load resistance, RL goes to zero, the current drawn is
limited to /. The load resistor RL represents a generic way of representing any resistor or
circuitry attached to the power supply.
Another, unfortunate effect of the internal resistance is that the circuit now looks like that of a potential
divider in Figure 2-6, with and which means that the voltage which actually is
presented across the load is reduced to
. Not only is this lower than V but it also depends on
the load resistance.
Figure 2-7: A real voltage source with an internal resistance, RS
Electronics Notes-2014 Imperial College London page 21 of 72
Then the voltage across AB, now labelled OC V is given by 1mA 10k =10V. Accordingly the value
of the open circuit voltage including both sources is given by V16 OC OC OC V V V .
To find the source resistance we have two options: find the short circuit current in each case above
and apply (5.2) or adopt the method presented in the previous example of replacing each source by
an open circuit (current source) or a short circuit (voltage source) and calculating the total resistance.
This is the easier method and the resulting circuit is shown above. is the resistance you would
measure by placing an ohmmeter across points A and B. This is the parallel combination of 30k and
15k resistors giving the source resistance as 10k.
2.11 Impedance matching
When connecting different electrical circuits the equivalent circuit concept is very useful. For instance,
consider an audio system for a concert hall or recording studio. A microphone generates a voltage
that is proportional to the sound produced by instruments/voices. The measured voltage is usuallyvery small and must be amplified before connecting the output of the amplifier to a loudspeaker which
then transforms the electrical signal back into one that we can hear. Connecting the
microphone/amplifier/loudspeaker to make a complete audio system is made much easier if each
individual sub-system can be considered as an equivalent circuit.
The main problem is “impedance matching” (we will discover more about impedance later –
essentially impedance takes account of capacitances and inductances which make the “resistance”
change with frequency). This involves designing the input impedance of a subsystem and the output
impedance of the signal source in order to achieve maximum power or maximum voltage transfer. We
have just seen that equivalent circuits are an elegant way of simplifying circuits to voltage (current)
sources and series (parallel) source resistances. The microphone can be considered as a Thévenin
circuit comprising an ideal voltage source in series with a source resistance . A high quality
microphone will have an output impedance (or source resistance) of 600. The output voltage of a
microphone is usually very small (0-100 V) and this has to be amplified by about 70dB before it can
be output to a loudspeaker. In this case the object is to maximise voltage (and not power) transfer
from the microphone to the amplifier. If we consider the amplifier to be a “black box” with input and
output terminals, maximum voltage transfer is achieved by having an amplifier input impedance that is
much larger than the output impedance of the microphone (think voltage divider). At the output sideof the amplifier, loudspeakers typically have input impedances of 8 and in this case it is desirable to
have maximum power transfer from amplifier to loudspeaker. To cover both requirements the audio
amplifier will be designed to have a very large input impedance (M) and a relatively small output
impedance (a few ) to fulfil the criteria specified above. We will return to this later when we look at
operational amplifiers.
2.12 Alternating Current
So far we have looked at direct current (DC) circuits where the driving voltage is produced by
chemical means in a battery or a mains-operated DC voltage generator using rectification. Alternating
Electronics Notes-2014 Imperial College London page 24 of 72
3 RC Filters At the end of the last section we noted that for resistors AC signals can be treated more or less like
DC signals and that there is no frequency dependent behaviour involved. However, there are two
other electronic components that do exhibit a frequency dependent behaviour and one of the mostuseful applications of this is in filter circuits. These can be used to enhance otherwise weak signals
whose frequency is known but would remain buried in noise without the use of filters. In this sectin we
will look at the first of these components, the capacitor
3.1 Capacitors
Capacitors are components that can be charged by connecting them to a voltage source such as a
battery and they are able to store electrostatic energy. This energy can be released rapidly as
happens in a camera flash or pulsed laser applications. The time taken to fully charge and discharge
depends on the size of the capacitor and any resistance in the circuit. This chapter will describe theconstruction of capacitors and their transient response during charging and discharging. Later we will
describe the response of capacitors to alternating current, AC waveforms and show that they can be
described with a complex impedance as the voltage and current are not in phase.
3.1.1 Capacitors: design and construction
A capacitor consists of two conducting (metal or metal foil) electrodes with air or a dielectric
(insulating) material separating the two.
Figure 3-1: Parallel plate capacitors with air (left) and with dielectric (middle)
The left part of Figure 3-1 shows a schematic of an air filled capacitor indicating the electric field
which is generated by the presence of charge or – on the electrodes (plates). It can be shown
that the charge on the plates is proportional to the voltage applied and hence:
3-1
The proportionality constant is called the capacitance and is measured in Farads (or Coulomb2/
Joule) in honour of Michael Faraday. In the E&M course you will see how to apply Gauss’s law to
show that the electric field between the plates is given by
Electronics Notes-2014 Imperial College London page 26 of 72
electric field of the light wave produces a time varying dipole moment. The magnitude of the dipole
moment will depend on the detailed nature of the bond between the electron and its parent atom and
also the frequency of the light, since the bonds will have a resonant frequency; for instance at very
high light frequencies the electron may not be able to follow the rapidly changing electric field. When
the frequency of the light matches the resonant frequency of the atom/electron bond the displacementis large; when an electron oscillates with large amplitude it emits light and so certain light frequencies
are able to travel through the glass. We know that this happens in the visible part of the
electromagnetic spectrum. This is the origin of the refractive index of a material and for a material
such as glass the refractive index √ . At non-resonant frequencies there will be absorption of the
light and the refractive index can be represented as a complex quantity.
3.2 Capacitors in series and parallel
Figure 3-2: Capacitors in series (left) and parallel (right)
Figure 3-2 (left) shows two capacitors in series. The voltage across is
1
11
C
QV whilst that across
is
2
22
C
QV . Now consider the wire that joins and . At one end there is a charge and at the
other end a charge . Since the wire connecting and has a high conductivity, charge will flow
freely until . Kirchhoff’s voltage law tells us that the total voltage across the two capactors is:
Assuming that the two capacitors can be replaced by a single capacitor of value , where
total
totaltotal
C
QV and substituting for the voltages gives:
2
2
1
1
C
Q
C
Q
C
Q
total
total
but giving:
21
111
C C C total
3-4
Figure 3-2 (right) shows a parallel combination of the capacitors. In this case the voltage across both
components is the same which means that the charge on each must be different; so
Electronics Notes-2014 Imperial College London page 34 of 72
would correspond to a voltage which leads by /2. On the Argand diagram this rotation is
equivalent to multiplication by where 1. Then /2 or
–. Alternatively we can use cosV a and sinV b and we
now have a third way of representing the vector: )sin(cos jV ; this is the trigonometric
form. Finally, invoking Euler’s theorem jeV we have the exponential form which completes
the different ways of representing the voltage . In the following rectangular representation will be
used.
3.7 Resistance becomes impedance
Figure 3-9: Phase diagrams showing the relationship between voltage and current for varioustypes of component. The left shows resistance. The right-hand diagram is for a capacitor.The centre diagram will be discussed in detail in the next section.
For an AC voltage, resistance is given by the instantaneous value of I
V . For instance, if
t V V cos0 and t I I cos0 then / where the phasors 0 and / 0 both
lie along the real axis as shown in Figure 3-9 (left). Thus / is a fixed value since and
are always in phase.
Figure 3-9 (middle) shows the current and voltage phases for an inductor. If the current flowing
through the inductor is t I I cos0 then the voltage leads the current through the inductor by /2
and is given by
2
cos0
t L I V
Las will be shown later. A phase lead of
/2 means the
voltage lies along the positive imaginary axis and has an amplitude . Taking the
complex ratio of the amplitudes of the / we obtain . This has the magnitude of the
inductive reactance but the presence of the term tells us that the voltage and current are not in
phase. is a phasor quantity and is called impedance.
Finally Figure 3-9 (right) shows the phase corresponding to the voltage and current across a
capacitor. Following the earlier reasoning if a current t I I cos0 flows to and from the capacitor
terminals then the voltage across the capacitor is
Electronics Notes-2014 Imperial College London page 42 of 72
where represents the area of a surface intersected by the magnetic field lines. Inside the coil or
close to the end of the coil, where is the radius of the coil.
If another short coil having turns (coil 2), which is not initially carrying current, is moved into place
next to the first coil (coil 1) then every turn of coil 2 becomes intersected by the flux 1 from the coil 1
and a voltage:
dt
dI M
dt
dI
l
r N N
dt
d N 12
2
21012
4-3
is induced in coil 2. This is Faraday’s law of induction. is called the mutual inductance between
the coils. The induced emf will drive current around coil 2 which generates a magnetic field from coil
2 as well. The minus sign shows that this magnetic field is in the opposite direction to that produced
by coil 1 and opposes the motion towards coil 1. This behaviour is referred to as Lenz’s law which
states:
"An induced current is always in such a direction as to oppose the motion or change causing it"
Moving coil 2 away from coil 1 also generates an induced voltage but now in the opposite direction so
that the coils attract. This experiment can also be performed by moving coil 2 towards or away from a
bar magnet. The magnet supplies the field and the motion generates a changing magnetic flux.
A changing magnetic flux can also be produced by passing a time varying current (AC) through coil 1.
This would induce a voltage in coil 2 which will depend on the rate of change of the current (the AC
frequency). This is the basis of a transformer which enables alternating voltages to be amplified
(stepped up or stepped down). Transformers are crucial for efficient electrical power distribution and
in combination with diodes are used to produce DC voltages from a mains source.
4.1.2 Self Induction: Inductors
Now imagine coil 1 to be part of an electric circuit. If the current in the circuit is the coil will generate
a magnetic field and magnetic flux according to equations 4-1 and 4-2. If the solenoid current changes
with time then:
dt
dI
l
N r
dt
d 1
2
0
4-4
Following the argument in the previous section the changing magnetic flux will induce an emf in each
turn of the coil according to:
dt
dI L
dt
dI
l
N r
dt
d N 1
2
1
2
01
4-5
Wherel
N r L
2
1
2
01
4-6
is known as the self-inductance and depends only on the geometry of the coil. Comparing equation 4-
6 with equation 4-3 we see that the changing magnetic flux associated with coil 1 causes the coil toact upon itself. Lenz’s law requires that any change in the current in the circuit will be opposed by the
Electronics Notes-2014 Imperial College London page 47 of 72
t
L
R
R
V i exp1 4-16
which is the form of the curves shown in Figure 4-5. Comparing this with the result we obtained for
the circuit we identify the time constant in the circuit as /. Thus the current rises to0.630 after a time ; 0.861 after 2 and 0.99 after 5 . After a suitably long time → , / = 0 and there will be no induced voltage at the inductor
terminals. However there will be a constant magnetic field within the coil. Now imagine replacing the
battery with a short circuit; the energy associated with the magnetic field will be converted into a
current which will flow through the resistor and be dissipated as heat. The rate at which the current
decays is given by solving the differential equation a-1 given in the appendix to this chapter, but
reducing to zero:
t
L
R
e I I
max 4-17
This is shown in Figure 4-6. At 0, and at long times 0. After one time constant R
L ,
will have dropped to 0.37
Figure 4-6: Current decaying after switch is opened
Electronics Notes-2014 Imperial College London page 51 of 72
4.9 Resonant circuits
Simple pendula, water waves and bungee jumpers are examples of oscillatory motion. In the absence
of any frictional effects the motion would continue without loss of amplitude, but in practice the
oscillations diminish and the mass will eventually come to rest unless it is given a periodic push.
When the push is applied at the appropriate time (frequency) the amplitude of the motion can bemaintained with the minimum of force. In some cases the amplitude of motion can increase
dramatically such as happened in the Tacoma Narrows Bridge:
see http://www.youtube.com/watch?v=j-zczJXSxnw
This effect is known as resonance. An example on a microscopic scale is the excitation of atoms in
solids by light which creates oscillating dipoles. Electrons bound to atoms have “natural” or resonant
frequencies which, when excited by a light wave of the correct frequency, will induce large amplitudes
of the electrons. Subsequent collisions with adjacent atoms results in absorption of the light. At
frequencies away from resonance there are phase changes between the electron motion and the light
wave which leads to changes in the speed of light in the solid given by the refractive index, as
described in chapter 6. In this lecture we’ll see how the frequency dependent impedances of
capacitors and inductors combine to produce resonance in electrical circuits. Such a selective
frequency response makes these “tuned” circuits useful in radio and TV transmission.
4.10 The circuit
Figure 4-10: circuit
Figure 4-10 shows an circuit driven with an AC voltage source cos. We can analyse
this circuit by applying Faraday’s law:
dt
dI LV V IRV C Lin
where is the current in the circuit. Since the inductor is only a wire there will be a negligible voltage
drop between the ends of the wire and 0; / and the
Electronics Notes-2014 Imperial College London page 57 of 72
5 Digital CircuitsDigital circuit are used in two main situations; control and computing. In control applications they are
often referred to as ‘logic’ circuits which are used to make true-false type decisions. A typicalsituation would be in the control of industrial processes which require particular conditions to be
fulfilled before a particular action is taken. In computing the current preference is still for arithmetic
using binary representations. In both cases the key point is that only two ‘values’ are used which are
referred to as high-low, true-false, 0-1. The electronics which is used to implement this is thus
designed to only have two possible states for inputs and outputs. Although this may seem restrictive
it gives rise to an enormous richness in possible applications and implementation.
5.1 Binary Arithmetic
Our everyday arithmetic is based on a decimal system (base 10) and numbers are represented bystrings of digits with values between 0 and 9. Each digit then represents a multiplier of 10 where
is position of the digit counting from the right-hand end of the string.
For example the decimal number 1234 can be written as 1 1 0 2 10 3 10 4 10
A binary number (base 2) can only use digits with one of two values (0 and 1) and each digit is then a
multiplier of 2 where is position of the digit again counting from the right-hand end of the string.
For example the binary number 1011 is 1 2 0 2 1 2 1 2 (= decimal 11)
Although binary is amenable to easy electronic implementation it is immediately apparent that a major
disadvantage is that numbers begin to need a large number of digits to represent even modest
decimal numbers. For this reason it is common to combine single digits (or bits) into larger units.
Historically the first level bunching was to put 4 digits (bits) together into ‘nibbles’. Each nibble can
then have a value between decimal 0 and 15 and can be though of as a hexadecimal octal (base 16)
value. It is usual to express hexadecimal values with digits between 0 and 9 followed by A to F cover
decimal 10 to 15.
Hexadecimal notation is still commonly used. Less common nowadays is the use of octal numbers
(base 8). These require 3 binary bits to cover the decimal numbers between 0 and 7
Two nibbles together provide a ‘byte’ (covering decimal numbers 0 to 255). Next is a ‘word’ with 16
bits. Beyond that the units become more standardised in terms of ‘bytes’ as shown in Table 5-1.
Electronics Notes-2014 Imperial College London page 58 of 72
Table 5-1: Commonly used notation for collections of bits and bytes
Number of bits Common terminology Decimal value range
1 (2 Bit 0 or 1 only
4 (2 Nibble 0 to 15
8 (2 Byte 0 to 255
16 (2 Word 0 to 65535
1024 (2 Kbit
8*1024 (2 Kbyte
1024*8*1024 (2 Mbyte
1024*1024*8*1024 (2 Gbyte
1024*1024*1024*8*1024 (2 Tbyte
5.2 Boolean Algebra
In decimal arithmetic we are familiar with the mathematical operations + - / and *. Each operatorrequires two input numbers and produces a third as the result.
In binary arithmetic we need to introduce a different set of basic operators. These are NOT, AND andOR.
The NOT operation only requires one input number and produces its result by changing the state ofeach bit in the number to its other value; i.e. it changes 1s to 0s and 0s to 1s.
The AND operation requires two input numbers and produces its result by comparing each number bitby bit. If either of the two corresponding bits are 0 then the resulting number has a 0 at that location.If both bits are 1s then the result is also a 1 at that location.
The OR operation requires two input numbers and produces its result by comparing each number bitby bit. If either (or both) of the two corresponding bits are 1 then the resulting number has a 1 at thatlocation. If both bits are 0s then the result is also a 0 at that location.
Additional Boolean operators include
SHIFT L(eft) which moves all bits one place to the left. A 0 is inserted in the right-most bit and theoriginal left-most bit is discarded.
SHIFT R(ight) which moves all bits one place to the right. A 0 is inserted in the left-most bit and theoriginal right-most bit is discarded.
XOR is exclusive OR. The XOR operation requires two input numbers and produces its result bycomparing each number bit by bit. If either one of the two corresponding bits are 1 then the resultingnumber has a 1 at that location. If both bits are 1s or 0s then the result is also a 0 at that location.
The following examples using 5-bit numbers illustrate how these operators work.
Electronics Notes-2014 Imperial College London page 62 of 72
All of these gates so far discussed are available as chips. In some cases the chips contain more than
one gate. For example the 7404 chip contains six NOT gates. It is normally referred to as an hex
‘inverter’. Each inverter is fully independent with its own input and output. The chip is an active chip
and requires a power supply to operate. Two pins on the chip are provided for this. Pin 7 is labelled
GND for ground and pin 14 is labelled Vcc and this is where the +5V power should be connected.Most chips have the same basic layout with the ground and V cc in the same corner locations.
However different chips may have different numbers of pins; 14 and 16 are most common.
Many chips provide both a normal output and an inverted output for convenience.
Figure 5-6: Pin configuration for the 7404 hex inverter gate.
General considerations when using logic chips are:-1. Use chips from the same family throughout
2. Keep your layout neat and tidy3. If you do not need to use all inputs on a multi-gate chip it is good practice to connect unusedgates to ground.
4. Outputs can be used to feed multiple inputs but check the data sheets to make sure you donot overdo it
5.6 Higher Level Functions
A bewildering range of higher level functions integrated onto single chips is now available. In additionTo the basic gates this includes
Registers and latches – for temporary storage of bit values
Counters
Shift registers – for sequentially transferring bitsMultiplexers – for selecting particular inputs
Clocks
Many circuits require the use of a clock to perform sequential operations. For example latches andregisters require a clock to make sure the information they need to store is accepted at the correcttime.
6 Operational AmplifiersYou may already have encountered operational amplifiers (op-amps) in the first year laboratory.
These consist of a rectangular piece of black plastic with dual-in-line pins for connection (see Figure
6-1). The microcircuit inside is complicated and typically contains over 25 transistors. Since an
understanding of one transistor is well beyond the scope of this course, trying to understand how so
Electronics Notes-2014 Imperial College London page 63 of 72
many work in tandem would seem to be an impossible task. However the details need not concern us
and the approach here is to treat op-amps as ‘black boxes’ which are governed by a single equation.
The full extent of the usefulness of op-amps only becomes apparent when we apply the techniques of
feedback and show how they can perform mathematical operations. In fact the prospect of making
analogue computers was the driving force in the development of op-amps.
Figure 6-1: A 741 operational amplifier; left is a schematic functional diagram, right is the pin
assignment diagram
6.1 Differential Amplifiers
An op-amp is a differential amplifier. It has two inputs and one output. It amplifies the difference
between the inputs. Figure 6-1 shows the circuit symbol for an op-amp which we are going to treat as
a “black box” which obeys the following equation:
6-1
The two input voltages and are referred to as the inverting (-) and non-inverting (+) terminals
respectively. Note that the voltages may be AC or DC. The amplifier gain is large, typically 105.
Note that the gain is written in bold type since it is a complex quantity (i.e. the gain may be frequencydependent and may show a phase shift). Op-amps are voltage driven sources: they generate an
output voltage which is proportional to the input voltage(s) but is derived from the amplifier
supply voltages (pins 4 and 7 on the schematic of the packaged 741 module in Figure 6-1).
can never exceed . Op-amps are designed for relatively small voltages and a typical value for the
supply voltages is 15V. is often omitted in circuit diagrams. Input voltages are measured relative
to the common or earth line shown at the bottom of Figure 6-1 left hand side.
Since and then /
If 15V and | | 10 then 150μV. Hence
Op-amps therefore have a very limited working range. Thus a tiny difference in the values of and
will drive to or depending on the sign of the tiny difference (Figure 6-2). This condition
is in fact used to advantage in some applications including digital voltmeters, analogue to digital
converters and control systems. In these situations the op-amp behaves as a comparator (i.e.
compares with ). Final important properties of op-amps are their input and output impedance.
The input impedance is typically very large (M) while the output impedance is typically very small
(few ). The advantages of this are:
a voltage source connected to either of the input terminals will “see” a very large
impedance. If the source impedance of is relatively small the voltage divider rule ensures
that the true value of appears at the op-amp input
Electronics Notes-2014 Imperial College London page 65 of 72
Figure 6-3: An op-amp with negative feedback
6.3 Unity Gain buffer
Figure 6-4 shows a negative feedback circuit where all of the output voltage is returned to the
inverting input. Putting = 1 into equation 6-2 means that 1. It may seem odd to use an amplifier
which does not appear to amplify the input voltage. The circuit is called a unity gain buffer. The unity
gain is self-evident, the “buffer” part refers to the input and output impedances. In section 6.2 it was
stated that it is desirable for the op-amp to have large input impedance and small output impedance. Although the details will not be presented here another effect of the feedback is to increase
(decrease) the input (output) impedances compared with the open loop values. Section 2.11
described the requirements for impedance matching when joining circuits. The unity gain buffer
provides a method of achieving this.
In summary the unity gain buffer uses a negative feedback loop to:
Reduce the gain to 1 (0dB)
Significantly increase the bandwidth (see next section)
Produce a very high input impedance and low output impedance
Electronics Notes-2014 Imperial College London page 66 of 72
Figure 6-4: An op-amp with unity negative feedback
6.4 Frequency dependent gain
Real components very rarely have ideal properties or exact values. For example manufacturing
tolerances mean the measured value of a resistor is rarely exactly the value indicated by the colour
code but will fall within the tolerance band (silver or gold). Integrated circuits are fabricated on
semiconductor substrates and are subject to variations arising from the many complicated processes
involved in their manufacture. Hence data sheets normally quote typical values and indicate the likely
variation from device to device. So the open-loop gain of an op-amp will state only a typical value,
say 105. In addition there may be variation in properties of internal components which would introduce
a frequency dependence, such as capacitances, including so called stray capacitances in integratedcircuits which act between unintentional conductors. Semiconductors also have relatively large
dielectric constants (for Si 13) which can enhance the stray capacitance. To avoid such stray
capacitance dominating the frequency dependence of the performance, and thus introducing too large
a variation from device to device, a capacitor is deliberately included in the op-amp output to
dominate all other capacitive effects. The net effect is to make the op-amp output stage resemble the
low pass circuit discussed at length in previous chapters. The result is shown in Figure 6-5.
Figure 6-5: Variation of open-loop gain with frequency
Electronics Notes-2014 Imperial College London page 70 of 72
6.8 Differential feedback amplifier
It is sometimes necessary to amplify the difference between two voltages, and , in a controllable
way and Figure 6-8 shows an example of a difference amplifier with gain. Although an op-amp
without feedback is already a difference amplifier, the high gain and strong frequency dependence
(which can vary from one op-amp to another) are not desirable. Note that and, appear at boththe inverting and non-inverting inputs. The input to the non-inverting terminal is a voltage divider so:
1V R R
RV
niF
F
6-4
Figure 6-8: Op-amp configured as a differential amplifier
For we have need to recall that the input impedances of the op-amp are very large compared with
and and, since no current can flow into the inverting terminal . Equating these currents
gives
F
out
in R
V V
R
V V
2
which leads to:
F in
inout F
R R
RV RV V
2 6-5
The action of the feedback forces ~ and equating 6-4 and 6-5 gives (eventually)
)( 21 V V R
RV
in
F out
So the closed-loop gain is / and the output voltage is proportional to the difference between the
input voltages.
6.9 Other op-amp operations
Op-amps were developed to perform mathematical operations and are used extensively in control
circuitry, for example, servo control units in instrumentation (Scanning Tunnelling Microscopes). This
section covers some basic mathematical applications.
so the output voltage is an integrated version of the input voltage. If is a constant (and negative)
voltage as shown in Figure 6-10 the integrator will generate a ramped voltage with a gradient that
depends on and . The integrator circuit is the basis of a ramp generator . A ramp voltage is used indigital voltmeters or analogue to digital converters.
Interchanging the resistor and capacitor will create a differentiator . Using a combination of integrators,
differentiators and adders it is possible to make an analogue computer which is capable of solving
differential equations.
6.10 Positive feedback
Positive feedback acts to reinforce any change at the input(s) to the amplifier and the circuit becomes
unstable. It might be thought that this would have no useful applications but this is far from the case.
Positive feedback is used to make oscillators (like the signal generators used in the laboratory) ormulti-vibrators which switch rapidly between positive and negative voltages generating a train of
square wave pulses for use in timing of digital circuits. However, these are somewhat beyond the