Feedback

Closed-loop transfer functions, sensitivity and disturbance rejection, gain traded for bandwidth, stabilisation, the root locus, and gain and phase margins

August 1927 — on the Lackawanna ferry

Harold Black of Bell Laboratories is trying to reduce distortion in the amplifiers that boost telephone signals along long lines. One morning in August 1927, crossing from Hoboken to Manhattan on the Lackawanna ferry, he finds the answer, and records the diagram and its equations on a blank space in his copy of The New York Times (Negative-feedback amplifier). The idea runs against instinct: build an amplifier with far more gain than needed, then feed part of its output back to the input in opposite phase, sacrificing the excess gain to make the amplifier more stable, more linear and better matched to the line (Harold Stephen Black). He files the patent on 8 August 1928. It takes the Patent Office more than nine years to grant it (Negative-feedback amplifier).

Derivation

The closed loop

A forward path

G(s)

produces the output from an error signal, and a feedback path

H(s)

returns a measurement of the output, which is subtracted from the reference:

E = R - HY

and

Y = GE

. Eliminating

E

Y = GR - GHY \quad\Longrightarrow\quad T(s) = \frac{Y(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)} .

The product

L(s) = G(s)H(s)

is the loop gain. Where

|L| \gg 1

T \approx 1/H

: the closed loop is set by the feedback path alone. A few precise resistors in

H

replace an amplifier whose gain drifts with temperature and age — Black's insight in one line.

💡Sensitivity and disturbances

The sensitivity of

T

to changes in

G

S = \frac{dT/T}{dG/G} = \frac{1}{1 + L} .

With an open-loop gain of

10^5

and

H = 0.01

S \approx 10^{-3}

: a 50 % fall in the amplifier's gain moves the closed-loop gain from 99.90 to 99.80. A disturbance added at the output is divided by the same

1 + L

. Feedback buys precision and immunity with gain.

📐Gain for bandwidth

Give the amplifier a pole:

A(s) = A_0/(1 + s/\omega_a)

, with

A_0 = 10^5

and

\omega_a = 10

rad/s. Closing the loop with

H = \beta

T(s) = \frac{A_0/(1 + A_0\beta)}{1 + s/\big(\omega_a(1 + A_0\beta)\big)} :

the gain falls by the factor

1 + A_0\beta \approx 1000

, and the bandwidth rises by the same factor — from 10 rad/s to about 10 000 rad/s. The product, gain × bandwidth, stays at about

10^6

: feedback spends gain to buy speed.

💡Moving poles with feedback

The closed-loop poles are the roots of

1 + L(s) = 0

, so feedback moves poles — including unstable ones. The plant

P(s) = 1/(s^2 - 1)

, a stand-in for an inverted pendulum, has poles at

\pm 1

. Proportional control

K

gives

s^2 - 1 + K = 0

: for

K < 1

one pole stays in the right half-plane, for

K > 1

both sit on the imaginary axis and the system oscillates for ever. Adding a derivative term,

C(s) = K(1 + T_d s)

with

K = 5

and

T_d = 0.5

, gives

s^2 + 2.5s + 4

: poles at

-1.25 \pm 1.56j

, and a well-damped response.

The root locus. As a loop gain

K

grows from zero, the closed-loop poles travel from the open-loop poles towards the open-loop zeros, or off towards infinity along straight asymptotes. Walter Evans's graphical rules for sketching these paths — the root locus — made it possible to see at a glance which gains a design could tolerate.

💡Nyquist's criterion and stability margins

Feedback can also destabilise: if the loop's phase lag reaches 180° where its gain is still above 1, the "negative" feedback arrives in phase and reinforces itself. Harry Nyquist's 1932 criterion reads stability from the plot of

L(j\omega)

— for a stable open loop, the closed loop is stable if the plot does not encircle

-1

. Two numbers say how close it comes:

the gain margin, the factor by which the gain could rise before $|L| = 1$ at the frequency where the phase is $-180°$ ;
the phase margin, the extra lag the loop could take at the frequency where $|L| = 1$ .

For

L(s) = K/\big(s(s + 1)(s + 5)\big)

the phase reaches

-180°

\omega = \sqrt5

rad/s, where

|L| = K/30

, so the gain margin is

30/K

: comfortable at

K = 10

(a factor of 3, with 25° of phase margin), exhausted at

K = 30

, and gone at

K = 40

One reason for the delay was that the concept was so contrary to established beliefs that the Patent Office initially did not believe it would work.

– Harold Black on his feedback-amplifier patent, quoted in Negative-feedback amplifier

🕰Not the first, but the one that worked

Paul Voigt had patented a negative-feedback amplifier in January 1924, though his theory lacked detail. Black's 1934 paper "Stabilized Feedback Amplifiers" made the method practical, and its use depended on stability theory — Nyquist's criterion of 1932, found independently by Felix Strecker at Siemens in 1930, and later Hendrik Bode's design methods (Negative-feedback amplifier; Nyquist stability criterion).

⚠Too much gain, too much lag

The screech when a microphone gets too close to its loudspeaker is a feedback loop whose margins have run out: loop gain above 1 at a frequency where the delay makes the feedback positive. Every added delay — a slow sensor, a digital controller's computation time — eats phase margin. Design for margin, not merely for stability.

💡Where you will use this

Operational amplifiers, autopilots, attitude control, cruise control and phase-locked loops are all feedback systems, and the Bode plots of Lesson 4-3 are where margins are read in practice. Digital implementations need the z-transform (Unit 8), and the delay a computer adds is one more phase lag to budget for.

📚Further reading

Harold S. Black, "Stabilized Feedback Amplifiers", Bell System Technical Journal 13 (1934), 1–18.
Harry Nyquist, "Regeneration Theory", Bell System Technical Journal 11 (1932), 126–147.
Karl J. Åström and Richard M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (2nd ed., Princeton University Press, 2021).

↓But there is a problem

Black's feedback ran through vacuum tubes. Today most controllers are programs: they read their sensors as samples, compute, and act a sample at a time. The Laplace transform, built for continuous time, does not apply to them directly. In 1952, two engineers at Columbia University gave sampled-data systems their own transform, and a name that stuck: the z-transform.