2nd year electronics: Lecture notes


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1 Todd Huffman
2 2nd year electronics
3 Lecture notes
4

ii T O D D H U F F M A N
5

6 Contents

7

Reminder of 1st year material

1

8

Passive sign convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

9

Kirchhoff’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

10

Network replacement theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

11

AC circuit theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

12

Ideal op-amps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

13

More realistic op-amps

4

14

Semiconductors

7

15

Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

16

Transistors

10

17

A first transistor amplifier circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

18

The common-emitter amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

19

The common-collector amplifier or emitter follower . . . . . . . . . . . . . . . . . . . 17

20

The long-tailed pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

21

The current mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

22

Appendices

21

23

A Semiconductor principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

24

B Summary of approximate transistor circuit properties . . . . . . . . . . . . . . . 23

25

C Circuit diagram symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

26

D Open issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

27

Todd: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

28

Tony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

29

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

30 Reminder of 1st year material

31 Passive sign convention
32 The passive sign convention is the standard definition of power in elec33 tric circuits. It defines electric power flowing from the circuit into an 34 electrical component as positive, and power flowing into the circuit out 35 of a component as negative. A passive component which consumes 36 power, such as a resistor, will have positive power dissipation. Active 37 components, sources of power such as electric generators or batteries, 38 can have positive or negative power dissipation if there are more than 39 one of them in a circuit, but if there is only one, it will have negative 40 power dissipation. 41 The practical application of this principle of power for passive circuit 42 elements is that one must label the positive terminal of the passive ele43 ment as the one in which current flows. One can choose whether to first 44 label the positive voltage terminal or to choose the direction of current 45 flow, but once one of these two options is chosen, the other must be set 46 accordingly so that the current flows into the positive terminal.

47 Kirchhoff ’s laws

48 There are two circuit laws first described by the German physicist Gustav 49 Kirchhoff in 1845, which are extremely useful to understand any electri50 cal circuit. The first one deals with the currents in the circuit:

51

At any point in the circuit the sum of currents flowing in is equal to the sum

52

of currents flowing out.

53 This is also called ‘Kirchhoff’s current law’ or KCL. For example see fig54 ure 1. Here the KCL at the node gives

In = I1 + I2 − I3 − I4 = 0.
n
55 For physicists this is an obvious consequence of the charge carrier 56 densities following a continuity equation, and ultimately local conserva57 tion of charge.

58 The second of Kirchhoff’s laws is Kirchhoff’s voltage law (KVL):

59

The directed sum of the electrical potential differences (voltage) around any

60

closed network is zero.

I1 I2
I3 I4
Figure 1: Example for Kirchhoff’s current law.

2 TODD HUFFMAN

61 The direction of the potential drops across resistances is given by the 62 direction of the currents flowing through the resistor. For an example see 63 figure 2. For this circuit
Vn = −V0 + I R1 + I R2 + I R3 = 0.
n
64 The KVL is a consequence of the conservative character of the electro65 magnetic force.

66 Network replacement theorems

67 Any linear electrical network with voltage and current sources and re68 sistances can be replaced by an equivalent source and an equivalent 69 resistor. Two types of equivalences are possible:

70 • Thevenin equivalent: The circuit can be replaced by an equivalent

71 voltage source in series with an equivalent resistance. The equivalent

72 voltage is the voltage obtained at the terminals of the network when

73 they are not connected. The equivalent resistance is the resistance

74 between the terminals if all voltage sources in the circuit are replaced

75 by a short circuit and all current sources are replaced by an open

76

circuit.

77 • Norton equivalent: The circuit can also be replaced by an equivalent 78 current source in parallel with an equivalent resistance. The equiv79 alent current is the current obtained if the terminals of the network 80 are short-circuited. The equivalent resistance is again the resistance 81 obtained between the terminals of the network when all its voltage 82 sources are shorted and all its current sources open circuit.

83 The equivalent resistances for the two cases are the same, and the equiv84 alent voltage and current sources are related as VThevenin = ReqINorton 85 with the equivalent resistance Req. The equivalent resistance can there86 fore be found from the ratio of the voltage between the terminals with 87 no load connected (VThevenin) divided by the current flowing when the 88 terminals are shorted (INorton).

89 AC circuit theory

90 Alternating currents (AC) can be described by a harmonic time depen91 dence V (t ) = V0 cos(ωt ). More complicated time dependencies can be 92 expressed by the superposition of harmonic components with different 93 frequencies using a Fourier series. In general the Fourier series is com94 plex and we therefore describe the voltage by V = V0e j ωt . The complex 95 phase introduced by this generalization is relevant as different parame96 ters in a circuit can have a different complex phase, equivalent to a phase 97 shift of their harmonic development. 98 In passive circuits we can use a generalized form of Ohm’s law

V = ZI,

(1)

99 where the impedance of a pure resistor is Z = R. For a pure inductor 100 Z = j ωL (can be easily seen from the definition of the self-inductance

+IR1
R1

-V + 0–

R2 +IR2

R3
+IR3
Figure 2: Example for Kirchhoff’s voltage law.

These theorems can be extended to capacitances and inductances when they are expressed by their complex impedances (see next section below).
A voltage source is an electrical component which generates and maintains a difference in the electrical potential between its terminals, independent of the load (current).

A current source is a component which provides a defined current, independent of the voltage between its terminals.

Electronics engineers prefer the use of the letter ‘j’ for the imaginary unit, to distinguish it from ‘i’, which is often used to denote a current. We will follow this convention.

2ND YEAR ELECTRONICS 3

101

V

=

L

dI dt

),

and

for

a

pure

capacitance

Z

= ( j ωC )−1

(from Q

= VC

and

102 I = ddQt ). These can be combined, and the overall impedance of a passive

103 network can be written as

Z = |Z |e j φ.

(2)

104

The current is then given by

V V0e j ωt V0 j (ωt−φ)

I

=

Z

=

|Z |e j φ

=

e |Z |

.

105 |Z | gives the ratio of magnitudes of V and I , and φ gives the phase differ106 ence by which the current lags the voltage. 107 Notice that the time-dependent part is a common factor for voltage 108 and current, so e j ωt can be omitted, but it is understood to be present 109 when returning to the time domain.

110 Ideal op-amps
111 An ideal op-amp is a differential amplifier: it’s output is Vout = A(V+ − V−), 112 with A the open-loop gain. Ideally, the open-loop gain is very large 113 (A → ∞), and the inputs have infinite input impedance (no current is 114 flowing into the ‘+’ and ‘-’ inputs). The output impedance of the ideal 115 op-amp is 0. Often op-amps are used in circuits with negative feedback 116 (for examples see figures 3 and 4). In circuits with negative feedback the 117 voltages at the two inputs adjust until they are equal, so that V+ = V−. 118 This equality is sometimes referred to as a ‘virtual’ short.

+

V

in

R

Vout

1

R2

Warning: this is only true for circuits with linear behaviour.
Negative feedback circuits are circuits where the output is connected (through a resistor) back to the input, reducing the input (here connecting back to the ‘-’ input).
Figure 3: Non-inverting amplifier. The ideal gain of this circuit is Vout/Vin = (R1 + R2)/R2 (You can see that from V+ = V− and the voltage divider R1 and R2).

I R2 R1 Vin I + V
out

Figure 4: Inverting amplifier. The ideal gain of this circuit is Vout/Vin = −R2/R1 (You can see that from V− = 0 and the currents through the resistors must be equal as there is no current flowing into the op-amp).

VOOMM − Maximum

±5

4 M 10 M
put Voltage
119

±2.5 VCC± = ±15 V
See Figure 2
0 −75 −50 −25 0

25 50 75 100 125

TA − Free-Air Temperature − °C
Figure 4. Maximum Peak Output Voltage vs
Free-Air Temperature

±15
RL = 10 kΩ TA = 25°C
More rea±12l.5istic op-amps

±10

VOOMM − Maximum Peak Output Voltage − V

±7.5

120
121
122
123
124
4 7 10
125

126
put Voltage
127

A more realistic model of the op-amp will encompass a finite and
±5
frequency-dependent open-loop gain. Commonly the frequency re-

sponse will be similar to a simple low-pass RC filter. Stray capacitances
±2.5
within the circuit would8cause such a behaviour, but often it is achieved

by design and the d0eliberate use of capacitances in the circuit, to pre-

vent instabilities at h0igh f2requ4encie6s. Ty8pica1l0DC1g2ain1s4for1r6eal op-amps

are

about

106

,

and

the

gain

|VCC±| − Supply Voltage − V
starts to decrease above

a

frequency

around

Figure 6. Maximum Peak Output Voltage

ω = 1 rad/s (see figure 6 for an examplves).

Supply Voltage

106

VCC± = ±5 V to ±15 V

RL = 2 kΩ

105

TA = 25°C

104 Differential 0°

Voltage

Amplification

103

45°

Phase Shift

AAVD – Large-Signal Differential Voltage Amplification

102 Phase Shift
101

90° 135°

1

180°

75 100 125

1 10 100 1 k 10 k 100 k 1 M 10 M

− °C

f − Frequency − Hz

ltage Amplification

Figure 8. Large-Signal Differential Voltage Amplification and

Phase Shift

e

128 A simple replacement circuit reprodvuscing the behaviour around the

129 roll-off and up to reasonably highFfrreeqquenciyes (∼ 1 MHz) is shown in

130 fig. 7.

V R Copyright © 10977–2015, Texas Instruments Incorporated A0
81 TL081A TL081B TL082 TL082A TL082B TL084 TL084A TL084B
Vin C Vout

+

dV
A(w)dV

-

Vout

Figure 5: More realistic op-amp replacement diagram.

Figure 6: Frequency response of the TL081, a widely used operational amplifier from Texas Instruments (datasheet from http://www.ti.com/lit/ds/symlink/tl081.pdf ).
This specific op-amp has a slightly lower gain (∼ 2×105) and cuts-off at a slightly higher frequency ( fcut-off 20 Hz) than described in the text.

Figure 7: Simple replacement circuit to model a real op-amp.

131 The amplifier in this circuit is again an ideal op-amp with frequency132 independent open-loop gain A0, so that V0 = A0Vin. Here we assume 133 that the load which we will connect to this circuit has infinite input 134 impedance, so that due to the KCL the current the resistor equals the

2ND YEAR ELECTRONICS 5

135 current through the capacitor

136 Substituting V0 yields

Vout − V0 + j ωC Vout = 0. R

A(ω) = Vout = A0 = A0 ,

(3)

Vin

1 + j ωC R

1

+

j

ω ω

0

137 with the cut-off frequency ω0.

138 What effect does this frequency characteristics have on the gain in a

139 negative feedback circuit? We will study this here for the example of an

140 inverting amplifier (figure 4). We start with the KVL for the input and the

141 feedback legs:

VR1 − δV − Vin = 0

VR2 + δV + Vout = 0.

142 We still assume that no current is flowing into the op-amp and therefore,

143 using the KCL, VR1 = VR2 . R1 R2
144 Combining these equations yields

Vin + δV = − Vout + δV ,

R1

R2

145 or

Vin = − Vout − δV

11 +.

R1

R2

R2 R1

146 We can now use the gain characteristics described before

Vout = A(ω)δV ,

147 and we get

Vin = − R1 Vout − Vout R1 1 + 1 ,

R2

A(ω) R2 R1

148 and

Vout = −

R2 + R1+R2 .

Vin

R1 A(ω)

149 With the parametrization in eq. (3) A(ω) = A0(1 + j ωRC )−1 this yields

Vout

R2

Vin = − R1 + (R1+R2)(A10+ j ω/ω0) .

150 A further approximation is that to achieve some gain R2 R1 and

151 therefore

Vout Vin

R2



R2(1+ j ω/ω0) .

R1 +

A0

152 This equation is demonstrated in figure 8. Note that despite the strong

153 frequency dependency of the open loop gain the gain of the feedback 154 circuit is constant up to much higher frequencies (104 rad/s compared to

155 ω0 = 1 rad/s in this example). This is because the amplifier only needs to

156 provide the gain to uphold the feedback, which is much lower than the

157 open loop gain it can provide at low frequency.

158 The frequency response for the non-inverting amplifier has a similar

159 behaviour.

160 The next level of op-amp imperfection which could be considered

161 for an even more realistic op-amp model would be the finite input and

162 output impedances.

Note that we do not assume δV = 0 anymore.
You will see this in the EL16 practical.

6 TODD HUFFMAN

|Vout/Vin| [dB] 120 100 80 60 40 20
10-1 1

open loop gain with feedback
101 102 103 104 105 106ω [rad/s]

Figure 8: Frequency response of inverting amplifier (A0 = 106, ω0 = 1 rad/s, R2 = 10 kΩ and R1 = 100 Ω).
dB is a unit often used in electronics. It is defined as 20 log10(Vout/Vin).

163 Semiconductors

164 Let’s now investigate the components which allow us to build active 165 circuits like operational amplifiers. Today almost all these components 166 are made from semiconductors, typically silicon or germanium. 167 In this course we will use very simple models for the behaviour of 168 semiconductor devices like the diode or the transistor in electrical cir169 cuits which do not require a detailed understanding of the quantum 170 mechanics and the solid state physics in the semiconductor. If you are 171 interested in these topics there is a slightly more detailed discussion in 172 the appendix of this document and further material can be found in the 173 Practical Course Electronics Manual.
174 Diodes
175 A circuit diagram representation of a diode is shown in figure 9.

I

176 In a diode there is one direction in which a current will easily flow 177 (indicated by the arrow in figure 9). A current in the opposite direction 178 will be blocked. 179 The number of charge carriers which are capable of entering the 180 valence band at a temperature T is given by the Boltzmann distribution 181 and therefore the current-voltage relation for a diode is given by

eVD

I = I0 e kBT − 1 ,

(4)

182 where I is the current through the diode and VD the voltage across the 183 diode. At room temperature kBT /e 25 mV. I0 is the reverse bias sat184 uration current. Its exact value depends on the doping concentrations 185 and the temperature, but is typically of the order I0 10−10 mA for sili186 con diodes. The resulting characteristics is shown in figure 10. Note that 187 the second term in the bracket quickly becomes negligible against the 188 exponential for positive bias voltages, and can be omitted in that case. 189 It should be obvious that a diode is a highly non-linear device. Ohm’s 190 law does not apply. Other tricks that don’t work are
191 1. the Norton equivalent, 192 2. the Thevenin equivalent, and

Due to the limited time available we will not discuss diodes in the lectures, but this section is part of the notes to give useful background material. Figure 9: Circuit diagram representation of a diode.
This equation is also known as the Shockley diode equation.
A more general version of this equation includes an ideality factor n (typically varies from 1 to 2) in the denominator of the exponential, to account for imperfections of junctions in real devices.

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2nd year electronics: Lecture notes