The formula
Three rules do almost all the work. Every block carries a transfer function (a
ratio of Laplace-domain polynomials in s), and the rules tell you how to merge
them.
Series (cascade): G_total = G1 · G2 · G3 ...
Blocks in a row multiply. The output of one is the input of the next, so their gains stack.
Parallel (summed): G_total = G1 ± G2
Blocks fed the same input and summed at the output add. The sign follows the summing junction.
Feedback loop: T(s) = Y(s)/R(s) = G / (1 ± G·H)
This is the one that matters. G is everything in the forward path from the
error junction to the output. H is everything in the feedback path from the
output back to that junction. Use + in the denominator for negative feedback,
− for positive feedback. When the sensor is ideal (H = 1), it collapses to
T = G/(1+G).
Nested loops: reduce inner first, then treat it as one block
inner = G2 / (1 + G2·H2)
T = G1·G2·G3 / [ (1 + G2·H2) + G1·G2·G3·H1 ]
Work from the innermost loop outward. Each reduced loop becomes a single block in the next level up.
Mason's gain formula: T = ( Σ Pk·Δk ) / Δ, Δ = 1 − ΣLi + ΣLiLj − ...
When the diagram is a tangle of crossing loops that won't reduce cleanly, Mason's
formula reads the answer straight off the graph. Pk are the forward-path gains,
Δ is the graph determinant built from loop gains Li, and Δk is Δ with the
loops touching path k removed. The higher-order sums are restricted:
ΣLiLj runs only over pairs of loops that do not touch (share no node),
ΣLiLjLk over non-touching triples, and so on — include a product of touching
loops and Δ comes out wrong. It is Cramer's rule wearing a diagram.
Where you meet it
Control loop review board. Somebody drops a diagram with a PID controller, a motor plant, a gearbox, and an encoder on a slide. Before anyone argues about gains, the group reduces it to
T = G/(1+GH)so there is a single transfer function to point at when they talk stability margin.Test-stand actuator tuning. A hydraulic load actuator on a Redstone test stand has an inner pressure loop wrapped inside an outer force loop. You reduce the inner loop first, fold it into the forward path, then close the outer loop. Skip the nesting order and your denominator comes out wrong.
RF and mixed-signal chains. A PLL or an AGC loop is a feedback block: forward gain in the phase detector and loop filter, feedback through the divider. Reduction gives the closed-loop bandwidth you spec against.
Op-amp circuits on the bench. The classic non-inverting amplifier is a feedback loop with huge forward gain
Aand a resistor dividerβin the path:T = A/(1+Aβ) ≈ 1/βwhenAβ ≫ 1. Block algebra is why the closed-loop gain depends on two resistors and not on the op-amp.
How it works
The whole method rests on one assumption: every block is linear and
time-invariant, so signals superpose and the blocks commute as algebra. That
lets you treat G(s) and H(s) as numbers you can multiply and divide, even
though they are functions of frequency. The moment a real block saturates,
rate-limits, or has backlash, the algebra describes only the small-signal
behavior around an operating point — not the clipped reality.
There is a second, quieter assumption: every block is unilateral and
unloaded by whatever comes after it — infinite input impedance, zero output
impedance, signal flowing one way. The series rule G1·G2 only holds when
connecting block 2 leaves block 1's transfer function unchanged. Cascade two
passive RC sections on the bench and the second loads the first; the combined
response is not the product of the two individually-measured sections. Same
story for a transducer driving a long cable. When stages load each other,
derive the combined transfer function from the actual circuit (or put a buffer
between the stages) — multiplying the blocks is silently wrong.
The feedback rule earns its keep. Two things fall out of T = G/(1+GH)
immediately. First, sensitivity: the closed-loop gain's sensitivity to a
change in the forward-path gain is 1/(1+GH). Drive GH large and the output
stops caring what G does — that is why a feedback amplifier with a sloppy,
drifting transistor still holds a precise gain set by passive parts. With
G = 10, H = 0.1, GH = 1, the sensitivity is 1/2 — a ten-percent drift in
G shows up as five percent at the output.
Second, stability lives in the denominator. Set 1 + G(s)H(s) = 0 and you
have the characteristic equation. Its roots are the closed-loop poles. A loop
that was fine at low gain goes unstable when raising the gain pushes those roots
into the right half-plane. The numerator sets where the zeros are; the
denominator decides whether the thing rings, settles, or runs away.
The steady-state payoff: for a unity-feedback loop the step-response error is
e_ss = 1/(1+Kp), where Kp = G(0) is the DC forward gain. With Kp = 100,
the error is about 0.0099 — under one percent. Bigger DC gain, smaller error.
This is the arithmetic behind "crank up the integral gain to kill the offset."
The mistake people make is losing the summing-junction signs when they move
a pickoff point or a summer across a block to make the diagram reducible. The
rule for that is mechanical: move a takeoff point downstream past a block G and
you must divide the branch by G to keep the signal the same; move it upstream
and multiply by G. Skip that compensation and the reduced transfer function is
silently wrong — the diagram still looks tidy, which is what makes the error
expensive. The other classic slip is reducing an outer loop before an inner one,
which scrambles the denominator. Always inside-out.
When loops cross so that no clean inner/outer nesting exists, stop pushing blocks around and switch to Mason's formula, or just write the node equations and solve the linear system directly. Reduction is a convenience, not the physics; the node equations are always the ground truth.
History
The feedback rule came out of a bandwidth problem, not a controls textbook. In
the 1920s Bell was building coast-to-coast telephone lines that needed hundreds
of amplifiers in series, and every amplifier's gain drifted with temperature,
tube aging, and supply voltage. Harold Black, a young engineer at Bell
Laboratories, spent years chasing the drift directly before flipping the problem:
throw away most of the gain on purpose and feed the output back to the input out
of phase. The trade was stability for gain. By his own account the idea arrived
on the morning commute — a ferry across the Hudson from New Jersey into Manhattan
in 1927 — and he sketched the circuit and the feedback equations on a spare page
of his New York Times [1][2]. The negative-feedback amplifier became U.S.
Patent 2,102,671, "Wave Translation System," and Black laid out the theory in
"Stabilized Feedback Amplifiers" in the Bell System Technical Journal in
January 1934 [1][2]. His formula A/(1+Aβ) is the same G/(1+GH) an engineer
writes today.
Black gave the circuit; his Bell Labs colleagues gave it the mathematics that made it safe. Harry Nyquist worked out when a feedback loop stays stable — the encirclement criterion — and Hendrik Bode developed the gain-and-phase plots and the feedback bookkeeping that turned it into a design discipline through the 1930s and 1940s [1]. The diagram-as-algebra viewpoint got its most general tool in 1953, when Samuel Mason at MIT published his gain formula for reading a transfer function straight off a signal-flow graph, expanded in his 1956 Proceedings of the IRE paper "Feedback Theory — Further Properties of Signal Flow Graphs" [3][4]. Mason's rule is really Cramer's rule for the loop equations, which is why it always agrees with grinding out the reduction by hand — it is the same linear algebra, read off a picture instead.
Related tools
- /tools/opamp-gain
- /tools/inamp-gain
- /tools/rc-filter
- /tools/series-rlc-impedance