Transfer function

A transfer function $H(s)$ is the ratio of output to input in the s-domain for a linear time-invariant (LTI) system with zero initial conditions. It encapsulates the system’s behavior in a single algebraic expression.

For a second-order linear ODE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=g(t)$ with $y(0)=y^{\prime }(0)=0$:

Taking the Laplace transform: $(a{s}^2+bs+c)Y(s)=G(s)$, so

$Y(s)=\frac{G(s)}{a{s}^2+bs+c}=H(s)G(s)$

where

$\boxed{H(s):=\frac{1}{a{s}^2+bs+c}}$

is the transfer function.

What it captures

$H(s)$ encodes everything about how the system responds to inputs:

• The denominator poles (zeros of $a{s}^2+bs+c$) tell you the system’s natural frequencies and damping.
• The numerator zeros (here: none) shape the response.
• For any input $G(s)$, the output is $Y(s)=H(s)G(s)$.

This is the s-domain equivalent of “the system’s response is determined by its dynamics, not by what’s driving it.”

Why it’s useful

You can compute the system’s response to arbitrary input by:

1. Taking the Laplace transform of the input: $G(s)=\mathcal{L}\{g(t)\}$.
2. Multiplying by $H(s)$: $Y(s)=H(s)G(s)$.
3. Inverse-transforming: $y(t)={\mathcal{L}}^{-1}\{Y(s)\}$.

Or in the time domain via Convolution integral: $y(t)=(h\ast g)(t)$ where $h(t)={\mathcal{L}}^{-1}\{H(s)\}$ is the Impulse response.

The key insight: once you know $H(s)$, you can predict the system’s response to any input. The transfer function is a complete description of the system’s input-output behavior.

Poles and stability

The poles of $H(s)$ — values of $s$ where the denominator vanishes — determine the system’s stability:

• All poles strictly in the left half-plane (negative real parts): asymptotically stable. The unforced output decays to zero.
• Simple (non-repeated) poles on the imaginary axis, none in the right half-plane: marginally stable. The unforced response is a bounded sinusoid that neither grows nor decays.
• Repeated poles on the imaginary axis (e.g. a double pole at the origin or a double pole at $\pm i\omega$): unstable. A double pole at $s=0$ inverse-transforms to $t$, which grows without bound; a double pole at $\pm i\omega$ gives terms like $t\cos (\omega t)$, also unbounded. So repeated imaginary poles fail BIBO stability even though simple imaginary poles do not.
• Any pole in the right half-plane (positive real part): unstable. Outputs grow exponentially.

For the second-order example above, the poles are the roots of $a{s}^2+bs+c=0$ — exactly the Characteristic equation roots. So the transfer function’s poles coincide with the natural frequencies of the homogeneous system.

This is no coincidence: the transfer function is the “frequency-domain shadow” of the homogeneous solution.

Example

For the spring-mass-damper $m{y}^{\prime \prime }+\gamma y^{\prime }+ky=F(t)$:

$H(s)=\frac{1}{m{s}^2+\gamma s+k}$

The poles are at $s=(-\gamma \pm \sqrt{{\gamma }^2-4mk})/(2m)$. For positive damping ($\gamma >0$), both poles have negative real parts → system is stable, transient response decays.

In the wider picture

Transfer functions are the foundation of:

• Classical control theory — design of controllers (PID, lead-lag, etc.) is done in the transfer-function domain.
• Signal processing — filters are described by their transfer functions, with poles and zeros placed to shape the frequency response.
• Circuit analysis — RLC circuits’ input-output relationships are transfer functions in $s$.

For the time-domain equivalent (the system’s response to a delta input), see Impulse response. For applying $H$ to general inputs via time-domain integration, see Convolution integral.

General form from an ODE

For an LTI system described by

$a_n{y}^{(n)}+\cdots +a_{0}y=b_m{x}^{(m)}+\cdots +b_{0}x,$

Laplace-transforming both sides with zero initial conditions and applying the differentiation property gives

$H(s)=\frac{Y(s)}{X(s)}=\frac{b_m{s}^m+b_{m-1}{s}^{m-1}+\cdots +b_0}{a_n{s}^n+a_{n-1}{s}^{n-1}+\cdots +a_0}.$

The numerator polynomial encodes the right-hand side of the ODE (input and its derivatives), the denominator encodes the left-hand side. The roots of the numerator are the system’s zeros; the roots of the denominator are the poles — see Pole and zero.

The poles coincide with the characteristic-equation roots of the homogeneous ODE: they are the system’s natural frequencies of free response.

Frequency response from H(s)

For a stable system, the frequency response is just $H(s)$ restricted to the imaginary axis of the s-plane:

$H(j\omega )=H(s){|}_{s=j\omega }.$

This is one of the most-used identities in the course. Compute $H(s)$ algebraically from the ODE, substitute $s=j\omega$, and you have the system’s response to every sinusoidal input — magnitude $|H(j\omega )|$ and phase $\angle H(j\omega )$.

For unstable systems, $H(j\omega )$ doesn’t equal $H(s){|}_{s=j\omega }$ — the imaginary axis isn’t inside the ROC and the Fourier transform of $h$ doesn’t exist.

In microelectronics: RC filter transfer functions (Electronics I)

In Electronics I the transfer function shows up directly as the voltage ratio $H(j\omega )=V_{out}/V_{in}$ of the coupling/bypass RC networks in an amplifier. Treat each capacitor as an impedance $Z_C=1/(j\omega C)$, write the output as a complex Voltage divider, and the result is a first-order $H(j\omega )$ whose corner frequency is ${\omega }_c=1/(RC)$. The two cases are the RC lowpass filter (capacitor across the output, passes DC, rolls off the highs) and the RC highpass filter (capacitor in series, blocks DC, passes the highs — exactly what coupling capacitors do). These are the building blocks of the low-frequency end of an Amplifier frequency response.