Application Application of of Neg Negative-Feedback ative-Feedback Theor Theory y

4.4 Application Application of of Neg Negative-Feedback ative-Feedback Theor Theory y

The power behind negative feedback resides on its resulting approxi-mations and loop-gain effects. Memorizing circuit topologies and equations help but designing beyond the state of the art demands

(a) (b) V _BIAS v _I

v_FB

R V _DD

v_O

M ₁ M ₂

i_O _

+

v _I

R

v_O &v_FB

M ₁ M ₂ V _DD

i_O V _BIAS

_

+

F

FIGUREIGURE4.154.15 Series-mixed shunt-sampled source-sensor feedback circuits.

Negative Feedback

172 172

Chapter FourChapter Four

deeper understanding. The circuits presented in the previous sections aim to exemplify the fundamentals of negative feedback with the three-terminal transistor as its core, as all circuits ultimately decompose into transistors and their constituent small-signal parameters. The most basic, yet most powerful feature of a negative-feedback circuit is it reg-ulates (i.e., controls) its output (when its loop gain is considerably larger than 1) so that the difference between its mixing inputs is negli-gibly small, virtually short-circuiting or mirroring the input signals.

The closed-loop gain of the circuit at any point is therefore, and simply, the gain translation from the virtual short/mirror point. For instance, output s_Ois approximately the ratio ofs_FB(or s_I ) and feedback factor

β

_FB

(i.e., s_O= s_FB/

β

_FB

≈

s_I /

β

_FBor A_CL

≈

1/

β

_FB) and, similarly, errors_Ethe ratio of s_FBand loop gain A_OL

β

_FB(i.e., s_E= s_FB/ A_OL

β

_FB

≈

s_I / A_OL

β

_FB).

Shunt feedback, whether it is at the input or output, decreases the impedance of the circuit. At the input, for example, the loop responds to ensure a virtual mirror exists between input and feedback currents i_I and i_FB, producing a substantially small error current i_Eand there-fore a negligibly small input voltage v_I (across R_{I .OL}and R_O.FB). Irre-spective of the magnitude of i_I , v_I is small, which is equivalent to say-ing the negative-feedback circuit offers considerably low input impedance. In the case of shunt sampling, the loop ensures sensed output voltage v^O remains unchanged in the face of changing load currents. In other words, shunt sampling regulates v_Oagainst varia-tions in output current i_O, which has the same effects of a change in i_I on v_I in the shunt-mixing example—little to no effect on v_O— so the circuit presents a substantially low output impedance.

Conversely, series mixing and sampling increase the impedance of a circuit. In the case of series mixing, for instance, the loop responds to ensure feedback voltage v_FBequals input voltage v_I , producing a substantially small error voltage v_E whose implied ohmic current through input resistance R_{I .OL}is also considerably small (noting i_I is v_E/R_{I .OL}). The small variation that results in input current i_I when pre-sented with an input-voltage excursion is the manifestation of high input resistance. Similarly, sensing and therefore regulating output current i_O(as in series sampling) against variations in output voltage v^Oresults in small changes in i^Oin spite of considerable changes in v^O, in other words, produces high output impedance.

In applying these generalized conclusions, however, it is important to identify the mixing and sampling functions. Fundamentally, voltage mixers (i.e., series) and samplers (i.e., shunt) reduce to base-emitter and gate-source terminals. Voltage mixers result when input voltagev_I is at one terminal and a loop signal (i.e.,v_FBor v_LOOP) is at the other and voltage samplers when output voltagev_Ois at one terminal and the other is any-thing but a loop signal (i.e.,v_I or ac ground). Current (i.e., shunt) mixers result from star-mixing connections between input currenti_I , feedback current i_FB, and error currenti_Einto base/gate or emitter/source termi-nals. Because collectors/drains and emitters/sources carry the driving

Negative Feedback

Negative Feedback

Negative Feedback

173 173

current of a transistor, collectors/drains and emitter/sources are good current samplers, but the latter only when their respective bases/gates are in the feedback path with a loop voltage (i.e.,v_LOOP) driving them.

Ultimately, determining the circuit’s loop gain is important to quantify the effects of feedback on resistance. Determining forward open-loop gain A_OLand feedback factor

β

_FBindependently, however, may not be necessary, not even for stability, which is entirely a func-tion of loop gain A

OL

β

^FB. The only motivation for evaluating their val-ues independently is to ascertain the closed-loop gain of the circuit, and with the simplifying statements already presented in this regard, not even then. In practice, when inspecting a circuit, it is often easier to derive loop gain A_OL

β

_FBthan its constituent A_OLand

β

_FBparameters because the latter involves decomposing the circuit into its two-port equivalent models. The absolute value of this loop gain is the gain across the loop when mixed input i_I or v_I in the mixing transconduc-tor is zero, irrespective of where the starting point is relative to the input or output terminals of the circuit. A shunt resistance would then be the loop-gain-reduced version of its open-loop resistance or approximately R_OL/( A_OL

β

_FB) and the series resistance the loop-gain-increased version of its open-loop resistance or roughly R_OL( A_OL

β

_FB).

The easiest means of quantifying the loop gain is to start with a voltage at a high-resistance node like a gate, base, or op-amp input terminal and traverse through the loop back to that point. In doing so, as already mentioned, the mixed input must be zero. This is not always easy to do at the transistor level because the mixer is inside the transis-tor and zeroing a transistransis-tor’s terminal may, and often does, camou-flage the effects of transistor impedancesr_oor r_ds, r_π, C_πor C_GS, and/or C_μor C_GD. The best way to zero a voltage-mixed inputv_I is to decom-pose the mixing transistor or op amp into its small-signal equivalent and zero the mixed-input component of the dependent source that is mixing s_I and s_FB(e.g., zerov_I in (v_I − v_FB)g_mor (v_I − v_FB) A_V to yield −v_FB g_m or −v_FB A_V ). In shunt-mixed cases, decomposing the incoming signal into its Norton-equivalent circuit and zeroing or discarding only the dependent current source, which is the one that carries current input i_I , is usually the most straightforward means of opening the loop. Note the mixer inverts the signal and the resulting gain across the loop is actually an inverted translation of the loop gain: − A_OL

β

_FB.

In practice, negative-feedback analysis produces, for the most part, reasonable approximations, not exact relationships. To start, the two-port equivalent models used to decipher negative-feedback effects only estimate and mimic the response of a circuit to first order.

To make matters worse, more approximations result when applying feedback theory to intertwined loops, such as emitter- and source-degenerated transistors that also double as mixers or samplers. In such cases, it is prudent to assume the higher loop gain overwhelms the others, but only if the ratio between the two loop gains is suffi-ciently high, like greater than 10.

Negative Feedback

174 174

Chapter FourChapter Four

Below is a succinct and concise, though brief, summary of impor-tant conclusions drawn from the foregoing discussion:

1. Feedback signal s_FBis the virtual short/mirror of s_I : s_FB

≈

s_I . 2. Output signal s_O(and A_CL) is the

β

_FBtranslation of the virtual

short/mirror: s_O= s_FB/

β

_FB

≈

s_I /

β

_FB.

3. Shunt feedback reduces the open-loop resistance by a factor of 1 + A^OL

β

^FB.

4. Series feedback increases the open-loop resistance by a factor of 1 + A_OL

β

_FB.

5. Transistor’s transconductance g_mis both a good voltage mixer and sampler.

6. Current mixers are star connections of i_I , loop signal i_FB, and error current i_E.

7. Collector/drain and emitter/source terminals are good current samplers, but the latter only when their respective base/gate terminals carry loop signals.

8. In opening a loop to determine loop gain A_OL

β

_FB, zero s_I only in the mixing current/voltage source.

9. Feedback approximations include: (a) the use of two-port models, (b) the assumption that the effects of the higher loop gain of multiple intertwined loops overwhelm the others’, and (c) small-signal impedance approximations.

4.5 Stability 4.5 Stability

A positive-feedback loop, instead of working against the conditions that force variations across the input terminals of its mixer, helps external forces increase their differentiating impact on what would have otherwise been a virtual short or mirror. This positive-feedback effect is mathematically apparent when loop gain A_OL

β

_FB reaches unity-gain frequency f

0dB

with a total shift in phase of 180° (i.e., A

OL

β

at f _0dB is −1, as shown in Fig. 4.16a). At this point and under theseFB

conditions, there is positive feedback at f _0dBand A_CLexplodes to infin-ity at f _0dB(as illustrated in Fig. 4.16b) because A_CL’s denominator (i.e., 1 + A_OL

β

_FB) approaches zero:

A A

A

A

A

CL OL

OL FB OL FB

= + =

− → ∞

1 = ∠ 1 1

1 180

β

_β ο

OL (4.48)

The criterion for a feedback circuit to remain stable is therefore to ensure its loop gain LG or A_OL

β

_FB(not forward gain A_OLalone) has less than 180° of phase shift at the loop gain’s f _0dB. Because of this unstable

Negative Feedback

Negative Feedback

Negative Feedback

175 175

point, phase margin (PM) refers to how much margin in phase exists at f _0dB before reaching 180°. Similarly, gain margin(GM) is how much margin there is in gain below the 0 dB axis before reaching the fre-quency where the loop gain incurs 180° of phase shift. As a result, the closed-loop circuit’s proneness to instability (i.e., peaking effects) increases with decreasing phase and gain margins PM and GM, as illustrated in Fig. 4.16b.

Each pole shunts incoming signals to ac ground at a rate of 20 dB per decade (i.e., linearly) past the pole’s location as frequency increases and phase-shifts (i.e., delays) signals by approximately −90°

a decade past it, −45° at its location, and 0° a decade before it, as illus-trated in Fig. 4.16a for poles p₁and p₂. A zero, on the other hand, feeds forward a signal and increases its magnitude by 20 dB per decade past the zero’s location and phase-shifts it by +90°, +45°, and 0° a decade past, at, and a decade before it. Similarly, a right-half-plane (RHP) zero also feeds forward a signal and increases it by 20 dB per decade, like a left-half-plane (LHP) zero, but like a pole, phase-shifts it by −90°, −45°, and 0° a decade past, at, and a decade before it.

Figure 4.16a graphically illustrates the Bode-plot response of a two-pole system whose pole locations precede f _0dB by at least one decade so PM is zero, which constitutes the makings of an unstable system. Because each pole phase-shifts a signal −90°, there is already 180° of phase shift by the time the loop gain crosses the 0 dB axis, resulting in zero phase and gain margins. Had the second pole (i.e., p₂) been within a decade of f _0dB, there would have been less than 180°

of phase shift at f _0dBand more phase and gain margins, attenuating the closed-loop gain’s peaking effect in Fig. 4.16b, which is a manifes-tation of instability or growing oscillations at peaking frequency f _0dB.

( f _0dB, 0 dB) ( f _0dB, 180°) L o

o p g a i A n ^O

β L F B [ d B ]

Frequency [Hz] (log scale) –

4 0

d B

/ d e c

– 2 0 d B / d e c _p

2

p₁

0°

P h a s e h s i f t [

°

]

0

dB dB 0

1/ βFB

Phase A ^C

L [ d B ]

Frequency [Hz] (log scale)

A_OLβFB

–90° –180°

PM₁ PM₁> PM₂

PM₂

f _0dB

(a) (b)

F

FIGUREIGURE4.164.16 (a ) Open-loop gain and phase (i.e., Bode plot) and (b ) resulting closed-loop gain responses of a negative-feedback circuit as phase margin (PM) decreases to its unstable state of 0°.

Negative Feedback

176 176

Chapter FourChapter Four

In document Analog IC Design With Low-Dropout Regulators (Page 172-177)