Transfer Functions
Laplace turns a differential equation into algebra, cascades into products, and stability into the sign of a pole.
Skip to the animationThe transfer function is the Laplace-domain ratio of output to input with zero initial conditions, which converts a differential equation into algebra, makes cascaded systems multiply instead of convolve, and turns stability into a question about which side of the imaginary axis the poles sit on.
The problem it solves
An RC circuit gives RC·dv/dt + v = v_in; a mass on a spring gives m·d²x/dt² + c·dx/dt + kx = F. Same shape, and solving either afresh for every input — step, ramp, sinusoid, disturbance — is intolerable. Worse, it mixes what belongs to the system with what belongs to the input.
Laplace, and what it buys
Under the transform, d/dt becomes × s and ∫dt becomes ÷ s. With zero initial conditions the differential equation becomes algebraic, and rearranging gives G(s) = C(s)/R(s) — a ratio containing no reference to any particular input.
- Any input can now be run through it by multiplication:
C(s) = G(s)·R(s). - Series systems multiply, where in the time domain they convolved.
- Parallel paths add.
- A closed loop reduces to
G/(1 + GH).
Those last three are the whole of block diagram reduction, and they exist only because the transform turned convolution into a product.
Poles and zeros
Factorised, G(s) = K(s+z₁)(s+z₂)/(s+p₁)(s+p₂). Zeros are where the numerator vanishes; poles where the denominator does. Each pole contributes one exponential term to the time response, so poles determine *how* the system behaves while zeros shape *how much* of each mode appears.
A pole at s = −a contributes e^(−at) and decays; one at s = +a contributes e^(+at) and grows. Stability is therefore the question of whether every pole lies in the left half-plane — answered by reading a sign rather than solving anything.
What it assumes
- Linearity — no saturation, backlash or stiction.
- Time invariance — parameters do not drift as it runs.
- Zero initial conditions — or the ratio is not clean.
- Single input, single output.
So it describes the forced response and says nothing about starting state, which is exactly what state-space methods were developed to handle. The usual workaround for a non-linear plant is to linearise about an operating point, which works within a range.
The numbers you will be asked for
- Transfer function
G(s) = C(s) / R(s)
Zero initial conditions.
- Series
G₁ · G₂
Convolution in time becomes a product.
- Closed loop
T(s) = G / (1 + G·H)
Negative feedback.
- Characteristic equation
1 + G(s)H(s) = 0
Its roots are the closed-loop poles.
Advantages and disadvantages
Advantages
- Turns calculus into algebra, once and for all.
- Cascades multiply, so block diagrams reduce mechanically.
- Stability becomes a question about pole position.
- Independent of the input, so one model serves every test signal.
Disadvantages
- Requires linearity and time invariance.
- Assumes zero initial conditions, so it describes only the forced response.
- SISO only — multivariable systems need state space.
- A transport delay's e^(−sT) is not a polynomial and must be approximated for some methods.
Watch it work
Check yourself
question 1 / 4
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