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 opamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
13
More realistic opamps
4
14
Semiconductors
7
15
Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
16
Transistors
10
17
A ﬁrst transistor ampliﬁer circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
18
The commonemitter ampliﬁer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
19
The commoncollector ampliﬁer or emitter follower . . . . . . . . . . . . . . . . . . . 17
20
The longtailed 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 shortcircuited. 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 selfinductance
+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 timedependent 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 opamps
111 An ideal opamp is a differential ampliﬁer: it’s output is Vout = A(V+ − V−), 112 with A the openloop gain. Ideally, the openloop 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 opamp is 0. Often opamps 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: Noninverting 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 opamp).
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 − FreeAir Temperature − °C
Figure 4. Maximum Peak Output Voltage vs
FreeAir Temperature
±15
RL = 10 kΩ TA = 25°C
More rea±12l.5istic opamps
±10
VOOMM − Maximum Peak Output Voltage − V
±7.5
120
121
122
123
124
4 7 10
125
kΩ
126
put Voltage
127
A more realistic model of the opamp will encompass a ﬁnite and
±5
frequencydependent openloop gain. Commonly the frequency re
sponse will be similar to a simple lowpass 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 opamps
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 – LargeSignal 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. LargeSignal Differential Voltage Amplification and
Phase Shift
e
128 A simple replacement circuit reprodvuscing the behaviour around the
129 rolloff 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 opamp 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 opamp has a slightly lower gain (∼ 2×105) and cutsoff at a slightly higher frequency ( fcutoff 20 Hz) than described in the text.
Figure 7: Simple replacement circuit to model a real opamp.
131 The ampliﬁer in this circuit is again an ideal opamp with frequency132 independent openloop 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 cutoff 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 opamp 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 noninverting ampliﬁer has a similar
159 behaviour.
160 The next level of opamp imperfection which could be considered
161 for an even more realistic opamp 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
101 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 currentvoltage 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 nonlinear 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.
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 opamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
13
More realistic opamps
4
14
Semiconductors
7
15
Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
16
Transistors
10
17
A ﬁrst transistor ampliﬁer circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
18
The commonemitter ampliﬁer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
19
The commoncollector ampliﬁer or emitter follower . . . . . . . . . . . . . . . . . . . 17
20
The longtailed 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 shortcircuited. 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 selfinductance
+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 timedependent 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 opamps
111 An ideal opamp is a differential ampliﬁer: it’s output is Vout = A(V+ − V−), 112 with A the openloop gain. Ideally, the openloop 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 opamp is 0. Often opamps 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: Noninverting 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 opamp).
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 − FreeAir Temperature − °C
Figure 4. Maximum Peak Output Voltage vs
FreeAir Temperature
±15
RL = 10 kΩ TA = 25°C
More rea±12l.5istic opamps
±10
VOOMM − Maximum Peak Output Voltage − V
±7.5
120
121
122
123
124
4 7 10
125
kΩ
126
put Voltage
127
A more realistic model of the opamp will encompass a ﬁnite and
±5
frequencydependent openloop gain. Commonly the frequency re
sponse will be similar to a simple lowpass 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 opamps
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 – LargeSignal 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. LargeSignal Differential Voltage Amplification and
Phase Shift
e
128 A simple replacement circuit reprodvuscing the behaviour around the
129 rolloff 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 opamp 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 opamp has a slightly lower gain (∼ 2×105) and cutsoff at a slightly higher frequency ( fcutoff 20 Hz) than described in the text.
Figure 7: Simple replacement circuit to model a real opamp.
131 The ampliﬁer in this circuit is again an ideal opamp with frequency132 independent openloop 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 cutoff 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 opamp 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 noninverting ampliﬁer has a similar
159 behaviour.
160 The next level of opamp imperfection which could be considered
161 for an even more realistic opamp 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
101 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 currentvoltage 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 nonlinear 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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