Training Electrical Engineering Transfer Functions, Poles & Zeros
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Transfer Functions, Poles & Zeros

24 min Electrical Engineering

Transfer Functions, Poles & Zeros

A transfer function $H(s)$ describes the input-output relationship of a linear time-invariant (LTI) system in the Laplace domain.

Transfer Function

$$H(s) = \frac{Y(s)}{X(s)} = \frac{b_m s^m + \cdots + b_1 s + b_0}{a_n s^n + \cdots + a_1 s + a_0}$$

The zeros are the roots of the numerator (where $H(s) = 0$). The poles are the roots of the denominator (where $H(s) \to \infty$).

Stability Criterion

A system is BIBO stable if and only if all poles have negative real parts (lie in the left half of the $s$-plane). A pole on the imaginary axis gives sustained oscillation; a pole in the right half-plane gives unbounded growth.

Pole-Zero Plot

Poles are marked with × and zeros with ○ on the complex $s$-plane. The plot reveals stability, natural frequencies, and damping at a glance.

Example 1

Find the poles and zeros of $H(s) = \dfrac{s + 3}{s^2 + 5s + 6}$.

Zero: $s + 3 = 0 \implies s = -3$

Poles: $s^2 + 5s + 6 = (s+2)(s+3) = 0 \implies s = -2, \; s = -3$

The pole at $s = -3$ cancels the zero at $s = -3$ (pole-zero cancellation). The system behaves as $H(s) \approx 1/(s+2)$.

All poles have negative real parts → stable.

Example 2

Determine the stability of $H(s) = \dfrac{10}{s^2 + 2s + 5}$.

Poles: $s = \frac{-2 \pm \sqrt{4 - 20}}{2} = -1 \pm 2j$

Real part $= -1 < 0$ → stable.

Natural frequency $\omega_n = \sqrt{5} \approx 2.24$ rad/s, damping ratio $\zeta = 1/\sqrt{5} \approx 0.447$.

Example 3

An RC low-pass filter has $H(s) = \dfrac{1}{RCs + 1}$. For $R = 1$ kΩ, $C = 1\,\mu$F, find the pole and cutoff frequency.

$\tau = RC = 10^3 \times 10^{-6} = 10^{-3}$ s

Pole: $s = -1/\tau = -1{,}000$

Cutoff frequency: $f_c = 1/(2\pi\tau) = 159.2$ Hz

Practice Problems

1. Find the poles and zeros of $H(s) = \dfrac{2s}{s^2 + 4s + 3}$.
2. Is the system $H(s) = \dfrac{1}{s^2 - 4}$ stable?
3. Find the poles of $H(s) = \dfrac{5}{s^2 + 6s + 25}$ and identify the natural frequency and damping ratio.
4. A second-order system has poles at $s = -3 \pm j4$. Find $\omega_n$ and $\zeta$.
5. Find $H(s)$ for an RLC series circuit ($V_{\text{out}}$ across $C$): show it equals $\dfrac{1/LC}{s^2 + (R/L)s + 1/LC}$.
6. If $H(s) = \dfrac{s+1}{(s+2)(s+5)}$, find the DC gain ($s \to 0$).
7. A system has poles at $s = 0$ and $s = -5$. Is it BIBO stable?
8. Find the transfer function with poles at $-1 \pm j$ and a zero at $s = 0$, DC gain $= 1$.
9. Sketch the pole-zero plot for $H(s) = \dfrac{s^2 + 1}{s^2 + 2s + 1}$.
10. For a first-order system $H(s) = K/(\tau s + 1)$, the step response is $y(t) = K(1 − e^{−t/\tau})$. Find the time to reach 63.2% of the final value.
11. The Routh-Hurwitz criterion: for $s^3 + 3s^2 + 5s + 2 = 0$, are all roots in the left half-plane?
12. A system has $H(s) = \dfrac{10(s+2)}{(s+1)(s+3)(s+10)}$. Find all poles, zeros, and the DC gain.
Show Answer Key

1. Zeros: $s = 0$. Poles: $(s+1)(s+3)=0$ → $s=-1, -3$.

2. $s^2 - 4 = (s-2)(s+2) = 0$ → poles at $s = 2$ (RHP) and $s = -2$. Unstable.

3. $s = -3 \pm 4j$; $\omega_n = 5$, $\zeta = 3/5 = 0.6$.

4. $\omega_n = \sqrt{9+16} = 5$; $\zeta = 3/5 = 0.6$.

5. Voltage divider: $H(s) = Z_C/(Z_R + Z_L + Z_C) = (1/Cs)/(R + Ls + 1/Cs) = 1/(LCs^2 + RCs + 1) \cdot (1/LC)/($ top & bottom by $1/LC)$.

6. $H(0) = 1/(2 \times 5) = 0.1$

7. Pole at $s = 0$ (on imaginary axis) → marginally stable / not BIBO stable (integrator).

8. $H(s) = K \cdot s/((s+1-j)(s+1+j)) = Ks/(s^2+2s+2)$. DC gain: $H(0) = 0$, so DC gain condition is ambiguous; typically $K$ is set by other criteria. With $K=2$: $H(s) = 2s/(s^2+2s+2)$.

9. Zeros at $s = \pm j$ (imaginary axis). Poles at $s=-1$ (double). Plot: ○ at $\pm j$, ×× at $-1$.

10. At $t = \tau$: $y(\tau) = K(1-e^{-1}) = 0.632K$. So $t = \tau$ (one time constant).

11. Routh array: row 1: [1, 5], row 2: [3, 2], row 3: [$5 - 2/3 = 13/3$, 0], row 4: [2]. All first-column entries positive → all roots in LHP → stable.

12. Poles: $s = -1, -3, -10$. Zero: $s = -2$. DC gain: $H(0) = 10(2)/(1 \cdot 3 \cdot 10) = 0.667$.

AC Circuits — Phasors & Impedance The Laplace Transform in Circuits