SC · Laplace

The Transfer Function

The transfer function H(s) = output/input is the system itself, independent of the input. Laplace turns the derivative d/dt into a multiply by s, making a differential equation into algebra. Move a coefficient and watch the differential equation, the transfer function, the pole, and the impulse response move as one.

One system, four languages

The differential equation, its Laplace transform H(s) = a/(s+a), the pole that makes the denominator zero, and the impulse response h(t) = a·e^(−at) are all the same system. Move the coefficient a and all four change together. Match the dashed target response and see one coefficient fix the whole system.

s-plane (× = pole −a)

impulse response h(t) = a·e^(−at)

coefficient a2.40

Four representations

H(s) = 2.40 / (s + 2.40)

dy/dt + 2.40 y = 2.40 x   ·   s = −2.40

The pole position s = −a differs from the target. The coefficient a in the differential equation is the transfer function’s denominator and pole, and it even fixes how fast the response decays. All four point to the same a.

Off

Differentiation becomes multiplication

With zero initial conditions, the Laplace transform turns the derivative d/dt into a multiply by s. So transforming both sides of the linear differential equation a system obeys replaces every derivative by a power of s, and the equation becomes algebra. Rearranged as the ratio of output Y(s) to input X(s), it gives H(s) = N(s)/D(s), a ratio of polynomials in s. A hard differential equation has simplified into a single fraction.

ObserveH(s) = Y(s)X(s)

The transfer function is output over input.

ChooseLaplace: d/dt → ?

Laplace turns d/dt into s.

Fill inY(s) = H(s) ?

The output is H times the input.

On your ownpoles: D(s) = ?

Poles are the roots that make the denominator zero.

The transfer function is the system’s ID

H(s) = Y(s)/X(s) is the same function whatever input you use. Being independent of the input, it is the system itself. Feed an impulse and X(s)=1, so Y(s)=H(s): the transfer function is exactly the Laplace transform of the impulse response, H(s)=L{h(t)}. This carries B1’s claim that the impulse response tells the whole system into the s-domain. And for any input the output is Y(s) = H(s) X(s) — multiplication in the s-domain. C4’s convolution-is-multiplication is completed here, now including transients.

Poles are the roots of the denominator

Building the transfer function from the differential equation makes the denominator D(s) the equation’s characteristic polynomial. Its roots — the solutions of D(s)=0 — are the poles, which, as the last unit showed, set the system’s natural modes and stability. The roots of the numerator N(s) are the zeros. So knowing the differential equation’s coefficients, you factor the denominator and numerator of H(s) to read off poles and zeros at once, and from their positions read how the system behaves. The first-order system you just saw had a single pole (−a), hence simple decay.

Back to the first screen

Moving the coefficient a moved the differential equation, the transfer function H(s) = a/(s+a), the pole s = −a, and the impulse response a·e^(−at) all together, because the four are one system written in four languages. Because Laplace turned the derivative into s and the equation into a single fraction, the system became an H(s) you can handle with one multiplication. The roots of its denominator are the poles, and the output for any input is Y(s) = H(s) X(s).

The transfer function H(s) = Y(s)/X(s) is the ratio of output to input Laplace transforms — the system itself, independent of the input. Transforming the differential equation turns d/dt into s, giving H(s) = N(s)/D(s), a ratio of polynomials in s. H(s) = L{h(t)} (the transform of the impulse response), and for any input the output is Y(s) = H(s) X(s). The roots of D(s)=0 are the poles, those of N(s)=0 the zeros (D2).

On to the next unit

If one system is written as a single H(s), then several systems wired together must also collapse into one H(s). The next unit, block diagrams and feedback, draws systems as boxes and learns the rules: in series the transfer functions multiply, in parallel they add, and with feedback — output fed back to the input — they combine as H/(1+HG). Feedback moves the poles to make an unstable system stable or a slow one fast — the central tool of control. There, signals and systems cross over into control.