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Semiconductor Detector Systems$

Helmuth Spieler

Print publication date: 2005

Print ISBN-13: 9780198527848

Published to British Academy Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198527848.001.0001

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(p.438) APPENDIX D FEEDBACK AMPLIFIERS

(p.438) APPENDIX D FEEDBACK AMPLIFIERS

Source:
Semiconductor Detector Systems
Publisher:
Oxford University Press

The most basic amplifier is a gain device itself, e.g. a bipolar transistor, JFET, or MOSFET. None of these is perfectly linear, so any amplifier utilizing these devices will also exhibit nonlinearity. Furthermore, the overall gain depends on device parameters that can vary among nominally identical devices.

Feedback amplifiers can provide predictable gain and significantly improve linearity by making the gain dependent primarily on linear components, i.e. resistors or capacitors, whose values are independent of signal amplitude.

D.1 Gain of a feedback amplifier

Figure D.1 shows the principle. A portion of the output is fed back to a summing circuit at the input. The net voltage applied to the amplifier input

(D.1)
v i a = v i + v f b .
The feedback signal
(D.2)
v f b = A f b v o .
Assume an inverting amplifier, so the ratio of output to input voltage
(D.3)
v o v i a = A v .
This relationship always applies, whether feedback is present or not. Thus, the voltage at the amplifier input
(D.4)
v i a = v o A v .
Inserting eqns D.2 and D.4 into eqn D.1 yields
APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.1 In a feedback system a portion of the output signal is fed back to a summing junction at the input.

(p.439)
(D.5)
v o = v i 1 A v + A f b
and the closed loop gain, i.e. the gain with feedback
(D.6)
A C L = v o v i = 1 1 A v + A f b = A v 1 + A f b A v .
When the amplifier gain (the “open loop gain”) is sufficiently large that A v ≫ 1/A fb, the gain of the overall system (“closed loop gain”)
(D.7)
A C L v o v i 1 A f b
is independent of the open loop amplifier gain A v and determined by the feedback network alone. If the feedback network is an attenuator (A fb < 1), e.g. formed by resistors, the overall gain is set by the resistor values and is independent of the amplifier gain.

However, note that when the amplifier gain A v is marginal it cannot be ignored. For example, when A v = 1/A fb (i.e. the amplifier gain equals the nominal closed loop gain)

(D.8)
A C L = 1 2 1 A f b ,
only half the naively expected value.

D.2 Linearity

Feedback linearizes the amplifier response. For a deviation Δ A v in open loop gain

(D.9)
A v + Δ A v 1 + ( A v + Δ A v ) A f b A v + Δ A v 1 + A f b A v = A C L + Δ A v 1 + A f b A v ,
where A CL is the nominal closed loop gain. Thus, deviations from linearity are reduced by the factor
(D.10)
1 1 + A f b A v .

D.3 Bandwidth

Similarly, the bandwidth is also improved. The frequency dependent gain of a single-pole amplifier with a corner frequency f c

(D.11)
A v ( f ) = A v 0 1 + i ( f / f u ) .
Inserting this into eqn D.6 yields (p.440)
APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.2 Feedback can be applied as a voltage in series with the input signal (left) – series feedback – or in parallel as a current (right) – shunt feedback.

(D.12)
A C L = A v ( f ) 1 + A v ( f ) A f b = A v 0 1 + i ( f / f u ) 1 + A v 0 1 + i ( f / f u ) = A v 0 1 + A f b A v 0 + i ( f / f u ) .
By dividing the numerator and denominator by (1 + A fb A v0) this result can be rewritten as
(D.13)
A v f = A v 0 f 1 + i ( f / f u f ) ,
where
(D.14)
A v 0 f A v 0 1 + A f b A v 0 and  f u f f u ( 1 + A f b A v 0 ) .
The gain and phase response of the fedback amplifier are of the same form as for an open loop single-pole amplifier. The midband gain is as predicted by eqn D.6 and the frequency response is extended by the factor (1 + A fb A v0). The gain–bandwidth product and unity gain frequency remain unchanged whether or not feedback is applied. By a similar calculation, a lower corner frequency is reduced by the factor 1/(1 + A fb A v0).

D.4 Series and shunt feedback

Feedback can be applied as either voltage or current, as illustrated in Figure D.2. In series feedback the feedback signal is applied as a voltage in series with the input. The feedback signal is commonly applied to the inverting input of a differential amplifier input. Shunt feedback adds currents, so the feedback signal connects directly to the input to form a current summing node.

D.5 Input and output impedance

Although for simplicity in analysis we often use amplifier models with infinite input resistance (i.e. no current ever flows into the amplifier input), in reality all amplifying devices have a finite input impedance (impedance because the input generally does not appear purely resistive). A MOSFET input, for example, appears capacitive at low frequencies.

(p.441) D.5.1 Series feedback

Consider the configuration in the left panel of Figure D.2. Without feedback the input current i i = v i/Z i0, where Z i0 is the input impedance of the amplifier without feedback. With feedback the voltage applied to the amplifier input is reduced

(D.15)
v i a = v i v f b .
Thus, the input current
(D.16)
i i = v i v f b Z i 0
is reduced relative to the open loop configuration and the input impedance becomes
(D.17)
Z i f = v i i i = Z i 0 ( 1 + A f b A ) ,
where
(D.18)
A f b = R 1 R 1 + R 2 .
Series (voltage) feedback increases the input impedance.

D.5.2 Shunt feedback

Again consider an amplifier with an infinite input impedance and a voltage gain A v, but now the output is fed back directly to the input through an impedance Z f, as shown in the right panel of Figure D.2. Since the amplifier has an infinite input impedance, any input current i i must flow through the feedback impedance Z f. Thus, the input current

(D.19)
i i = v i v o Z f .
The output voltage
(D.20)
v o = A v v i ,
so the input current
(D.21)
i i = v i ( 1 + A v ) Z f .
The input impedance
(D.22)
Z i f = Z f 1 + A v
and for large gains A v ≫ 1
(D.23)
Z i f Z f A v .
Shunt negative feedback reduces the input impedance. Thus, a large amplifier gain can yield a small input impedance, depending on the feedback impedance.

This is the mechanism that leads to the notion of “virtual ground”. For example, an amplifier with a gain of 105 and a feedback resistor of 104 Ω yields (p.442)

APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.3 In a shunt feedback amplifier the input impedance of the non-fedback amplifier Z a is in parallel with the active input impedance.

an input resistance is 0.1 Ω. This is quite practical at low frequencies. However, at high frequencies the amplifier gain may be only 100, and the input resistance is 100 Ω, which may qualify as a low impedance, but not a virtual ground.

It is straightforward to extend the result to an amplifier with a finite input impedance Z a. As indicated in Figure D.3 the input current splits into two components, each corresponding to a component of the input impedance,

(D.24)
i i = i f + i a = v i Z i f + v i Z a v i Z i .
The total input impedance is the parallel combination of the feedback and amplifier input impedance
(D.25)
1 Z i = 1 Z i f + 1 Z a .

D.5.3 Output impedance

The same reasoning can be applied to the output impedance. In this case shunt feedback takes the voltage directly from the output, as in Figure D.2. Any reduction in the output signal with increasing load current due to a finite output

APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.4 An emitter resistor R E introduces local series feedback into a common emitter stage. The same principle applies to other devices.

(p.443) impedance will also reduce the magnitude of the fedback signal, compensating for the output drop. This effectively reduces the output impedance to
(D.26)
Z o f = Z o 1 + A f b A v .
Conversely, series feedback will increase the output impedance. Figure D.4 shows an example of local series feedback. The emitter resistor introduces a voltage drop into the input circuit that depends on the output current. If the output voltage sags because of the inherent output resistance of the device, the current decreases, which in turn reduces the voltage drop across the emitter resistor and increases the base–emitter voltage to counteract the decrease in output current.

For a constant input voltage V B applied to the base, the collector voltage

(D.27)
V C = I C R E + V C E ( V B E , I C ) .
The total differential
(D.28)
d V C = d I C R E + d V B E d V C E d V B E + d I C d V C E d I C .
Since dV BE = − dI C R E and the output resistance of the transistor r o = dV CE/dI C, the change in output voltage
(D.29)
d V C = d I C R E d I C R E d V C E d V B E + d I C r o .
Using dI C/dV BE = g m and dV CE = −dI C R E, the output resistance with feedback becomes
(D.30)
r o f d V C d I C = r o + R E ( 1 + g m R E ) .
The source resistor also introduces local series feedback into the input, increasing the input resistance as discussed for the emitter follower in Chapter 6.

D.6 Loop gain

The quantity A fb A v that improves linearity, extends bandwidth, and affects the input or output impedance is called the loop gain. It can be measured by breaking the feedback loop at any convenient point and measuring the total gain between the break, as illustrated in Figure D.5. The original input signal source should remain connected, but with zero signal.

Since the benefits that accrue from negative feedback depend on the loop gain, as the loop gain decreases with A v (f) beyond the either the lower or upper corner frequency, the effects of negative feedback decrease. For small loop gains the overall response follows the open loop response.

(p.444)

APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.5 The loop gain can be determined by splitting the feedback loop, injecting a signal v il, and measuring the output v o. The loop can be split at any convenient point.

D.7 Stability

The amount of feedback is limited by stability considerations. In reality, amplifiers have multiple corner frequencies, so eventually the additional phase shift attains 180° and negative feedback turns into positive feedback, leading to self-oscillation. Figure D.6 illustrates the frequency response of an amplifier in open loop and closed loop operation. Two values of closed loop gain are shown. For A CL1 the corner frequency is in the regime where the open loop phase shift is 90°, so the total phase shift of the inverting amplifier is 180° + 90° = 270°. At frequencies beyond the second corner frequency the additional phase shift is 180°, so the amplifier is noninverting and potentially unstable if too much feedback is applied. The stability criterion is that the loop gain may not exceed unity at the frequency where the additional phase shift is 180°. However, to ensure a safety margin, a phase margin of 45°, i.e. an additional phase shift of 135° is generally accepted to be the minimum. This is shown for closed loop gain A CL2.

APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.6 Open loop gain (solid line) and closed loop gains (dashed) for two values of closed loop gain A CL1 and A CL2.

(p.445)

APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.7 Open loop gain and phase vs. frequency (left) of a commercial operational amplifier. The phase wraps at values exceeding 180°. The right panel shows the gain vs. frequency for various closed loop gains with the open loop gain for comparison. Pronounced peaking for closed loop gains A CL of 3 and 5 is due to reduced phase margin.

Figure D.7 shows the open loop gain and phase vs. frequency (left) of a commercial operational amplifier. Two corner frequencies with an extended 90° phase shift regime are apparent. The right panel shows the gain vs. frequency for various closed loop gains with the open loop gain for comparison. Beyond their respective corner frequencies the closed loop gain curves follow the open loop response. Reduced phase margin at closed loop gains of 3 and 5 results in pronounced

APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.8 Pulse response of the amplifier for closed loop gains A CL = 3, 5, 10, and 20. The peaking in the frequency response for A CL = 3 and 5 translates into ringing in the pulse response. The output pulse for A CL = 10 shows a very slight overshoot and the pulse for A CL = 20 exhibits a fully monotonic response.

(p.446)
APPENDIX D FEEDBACK AMPLIFIERS

Fig. D.9 Plotting the imaginary part of the loop gain vs. the real part shows whether a feedback amplifier is stable. If the resulting curve encloses the point (1,0), the system will oscillate, as shown in the right panel for A CL = 2. A closed loop gain of 3 fulfills the stability criterion, but exhibits severe peaking in the frequency response and ringing in the time response.

peaking near the corner frequency. Figure D.8 shows the pulse response for closed loop gains A CL = 3, 5, 10, and 20. The peaking in the frequency response for A CL = 3 and 5 translates into ringing in the time response.

The stability of a feedback circuit can be assessed either by inspection of the open loop gain and phase (Bode 1940) or by plotting the imaginary part of the loop gain vs. the real part (Nyquist 1932). If the resulting curve encloses the point (1,0) – i.e. one on the real axis – the circuit is unstable. This is illustrated in Figure D.9. A closed loop gain A CL = 2 leads to self-oscillation, whereas reducing the feedback to obtain A CL = 3 achieves stability, but the small phase margin leads to pronounced peaking in the frequency response (Figure D.7) and ringing in the pulse response (Figure D.8).