Copyright © 2011, 2022 by Wayne Stegall
Updated September 26, 2022. See Document History at end
for details.
Local Feedback
Is Local Feedback the same
concept as Global Feedback?
Introduction
The operation of a transistor with respect to the linearizing
effect of a source or emitter resistor is so straightforward that
it is often difficult to regard its operation as feedback in the
same sense as global feedback. Is local feedback the same
mechanism as global feedback? It the purpose of this article
to examine this question by solving for the closedloop feedback
gain equations of circuits using global feedback and local
feedback.
Feedback around a voltage gain block
The first circuit to examine applies global feedback to a
differential input voltage gain block. In all these circuits
β represents a numerical value which when multiplied to the output
value gives the voltage applied as feedback. Usually, global
feedback uses a voltage divider and local feedback uses an
impedance to implement β, both usually resistive.
Figure 1: Circuit
illustrating global feedback applied to a differential
input voltage gain block 

Beginning with the open loop gain equation, solve for the
closedloop feedback gain equation.
(1)

v_{o} = A_{OL}(v_{i}
 v_{f})

(2)

v_{o} = A_{OL}(v_{i}
 βv_{o}) 
(3) 
v_{o} = A_{OL}v_{i}
 A_{OL}βv_{o} 
(4) 
v_{o} + A_{OL}βv_{o}
= A_{OL}v_{i} 
(5) 
v_{o}(1 + A_{OL}β)
= A_{OL}v_{i} 
(6)

A_{CL} =

v_{o}
v_{i} 
=

A_{OL}
1 + A_{OL}β 
This result is the standard equation defining closedloop gain for
global feedback. This is the reference to which the other
equations are compared. All feedback equations of this form
(equations (6), (13), and (19)) reduce to
if A
_{OL}β (or other equivalent term, such as g
_{OL}β)
is
much greater than one.
Feedback around a transconductance gain block
The second circuit to examine applies global feedback to a
differential input transconductance gain block. This is
useful as an intermediate step because of the similarity to the
transconductance gain of a transistor when feedback is applied.
Figure 2: Circuit
illustrating global feedback applied to a differential
input transconductance gain block 

Beginning with the open loop gain equation, solve for the
closedloop feedback gain equation. If β is implemented as a
voltage divider, its contribution should be added to Z
_{L}.
(8)

v_{o} = g_{OL}Z_{L}(v_{i}
 v_{f})

(9)

v_{o} = g_{OL}Z_{L}(v_{i}
 βv_{o}) 
(10) 
v_{o} = g_{OL}Z_{L}v_{i}
 g_{OL}Z_{L}βv_{o} 
(11) 
v_{o} + g_{OL}Z_{L}βv_{o}
= g_{OL}Z_{L}v_{i} 
(12) 
v_{o}(1 + g_{OL}Z_{L}β)
= g_{OL}Z_{L}v_{i} 
(13)

A_{CL} =

v_{o}
v_{i} 
=

g_{OL}Z_{L}
1 + g_{OL}Z_{L}β 
This result is comparable to the to the standard equation above,
except that the output impedance is in the equation.
Therefore loading can somewhat affect the gain result if the
feedback factor is low.
Feedback around a transistor gain block
The last circuit to examine is a transistor amplifier with local
feedback. The circuit is meant to represent a generic
transistor although it is illustrated as a JFET, and could
represent a tube circuit as well. The ac equivalent circuit
makes the gate and source or base and emitter effectively
differential inputs. Because the output current also passes
through the inverting input, the feedback impedance Z
_{F}
by converting the output current to a feedback voltage at the
feedback terminal becomes the feedback factor β.
Figure
3: Transistor amplifier with local feedback
impedance.

Figure
4: AC equivalent circuit of same transistor
amplifier



Beginning with the open loop gain equation and substituting β for
Z
_{F}, solve for the closedloop feedback gain equation.
(14)

i_{o} = g_{OL}(v_{i}
 v_{f})

(15)

i_{o} = g_{OL}(v_{i}
 βi_{o}) 
(16) 
i_{o} = g_{OL}v_{i}
 g_{OL}βi_{o} 
(17) 
i_{o} + g_{OL}βi_{o}
= g_{OL}v_{i} 
(18) 
i_{o}(1 + g_{OL}β)
= g_{OL}v_{i} 
The resulting closedloop transconductance gain equation is in the
same mathematical form as to standard feedback gain equation of
equation (6) above.
(19)

g_{OL} =

i_{o}
v_{i} 
=

g_{OL}
1 + g_{OL}β 
where v
_{o} = Z
_{L}i
_{o}, multiplying the
closedloop transconductance gain equation by Z
_{L} gives
the closedloop voltage gain.
(20)

A_{OL} =

v_{o}
v_{i} 
=

g_{OL}Z_{L}
1 + g_{OL}β 
dividing numerator and denominator by g
_{OL} and
substituting back Z
_{F} for β gives the generic form of
the most recognized commonemitter/commonsource gain equation:
(21)

A_{OL} =

v_{o}
v_{i} 
=

Z_{L}
Z_{F }+ 1/g_{OL} 
Local feedback calculations take the same form as global ones and
even return to the equation most associated with local feedback!
Deductions
Having determined that the mechanism of local feedback is the same
as that of global feedback, the differences attributed between
them must pertain to differences in their openloop gain
blocks. Individual transistors have comparatively low gain,
a onebend transfer curve, a very high frequency lowpass pole,
minimal phase shift of about 90º, and a stable phase margin of
about 90º. Multiple transistor gain blocks have high gain, a
more complicated transfer curve, and a lowpass pole greatly
lowered by the necessity to compensate for 90º or more of phase
shift per stage. The following table summarizes these
differences and others and their significance.
Gain Block Comparison
Characteristic

Multiple
Transistor
Gain Block

Single
Transistor Gain Block

Winner

Comments

Openloop Gain (A_{OL}
and g_{OL})

high

low

multiple transistor

Many of the multiple
transistor gain block's faults are mitigated here by the
high feedback factor (1 + A_{OL}β) that results
from high openloop gain.

Transfer Curve^{1}

possibly complicated

one bend

single transistor

See first comment.

Distortion

low to vanishing

high to low

?

There may be virtue in
multiple transistor gain block distortion when vanishingly
low, however anything even slightly audible favors the
euphony characteristic of a single transistor gain
block. Still a matter of debate.

Output Impedance

low to vanishing 
high to low 
multiple transistor? 
Low output impedance is
usually a virtue, but not always.

Lowpass Pole

lower, compromised by phase
compensation

very high

single transistor

The multiple transistor
gain block's disadvantage here may be mitigated by
increasing gain bandwidth and possibly decreasing gain
until the dominant pole is 10kHz or higher.^{2}

Compensation Required

yes

no

single transistor


Maximum Phase Shift before
Compensation

≈ number_of_stages × 90º 
≈ 90º 
single transistor


Compensated Phase Margin
(180º  Phase Shift at unity openloop gain)

≈ 45º 
≈ 90º 
single transistor

This factor relates to
stability: 90º is associated with highest stability, 45º
is associated with a small amount of ultrasonic overshoot
and ringing on transients.

Direct Coupling

yes

no

multiple transistor 

Final Comment
I still hold out hope that operational amplifiers could gain
virtue by attaining some charactistics of a single transistor,
especially those of wide bandwidth and of a high frequency lowpass
pole.
^{1}See article Transfer Curve
Shape and Distortion.
^{2}See article Musical Feedback
Amplifiers.
Document
History
August 27, 2011 Created.
September 26, 2022 Correct Compensated Phase Margin in
chart where values transposed.