---
title: "Poles and zeros of a network function"
subject: "Electrical Circuit Theory"
chapter: 5
key: "semester_3/electrical_circuit_theory:poles-and-zeros-significance"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/poles-and-zeros-significance
site: BCT Engineering
---

# Poles and zeros of a network function

Electrical Circuit Theory · Chapter 5: Frequency Response of Network

- Asked in: 2068 Shrawan, 4 marks
- Video: [Zeros and Poles of a Transfer Function](https://www.youtube.com/watch?v=AZ7_MvANy_Q) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/poles-and-zeros-significance
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-5

## Question

What do you understand by poles and zeros of a network function? State their significance in analyzing the time domain response of a network.

## Short answer

Poles and zeros are the roots of denominator and numerator of the network function H(s). Poles determine the natural response terms (e.g., e^(p t)), stability, and damping; zeros set amplitudes and block specific frequencies. A pole-zero diagram on the s-plane directly reveals time-domain behavior.

## Answer

A network function H(s) (driving point or transfer function) of a lumped linear network is a ratio of polynomials in the complex frequency s = σ + jω:

```text
H(s) = N(s)/D(s) = K (s − z1)(s − z2)...(s − zm) / [(s − p1)(s − p2)...(s − pn)]
```

- **Zeros** z1, z2, ... are the values of s for which H(s) = 0 (roots of N(s)).
- **Poles** p1, p2, ... are the values of s for which H(s) → ∞ (roots of D(s)). They are also the natural frequencies of the network.

They are shown on the s-plane (poles as ×, zeros as ○) in a pole-zero diagram. Poles and zeros are either real or occur in complex-conjugate pairs.

#### Significance in the time domain

1. **Natural response.** Each pole pi contributes a term Ki·e^(pi·t) to the response (partial fractions followed by the inverse Laplace transform). Example: H(s) = (s + 2)/[(s + 1)(s + 3)] = 0.5/(s + 1) + 0.5/(s + 3), so the impulse response is 0.5e^(−t) + 0.5e^(−3t).
2. **Stability.** Poles in the left half plane (σ < 0) give decaying terms, so the network is stable. Poles on the jω axis give sustained oscillation (marginal stability). Poles in the right half plane give growing terms (unstable).
3. **Speed and damping.** A real pole at −a gives e^(−at): the farther from the origin, the faster the decay. A complex pair −α ± jβ gives e^(−αt) cos(βt + θ): α sets the decay rate and β the oscillation frequency.
4. **Zeros** do not create new time functions, but they set the amplitudes (residues Ki) and the phase of each term. A zero at s = z means that an input e^(zt) produces no output; a zero on the jω axis blocks that sinusoidal frequency completely.
5. The forced response depends on the poles of the input as well; the poles of H(s) decide the transient (natural) part.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are poles and zeros of H(s) = N(s)/D(s)?
   A. Zeros are roots of N(s); poles are roots of D(s) and are the natural frequencies
   B. Poles are roots of N(s); zeros are roots of D(s)
   C. Both are roots of D(s)
   D. Poles are values where H = 0
   Answer: A. H = 0 at zeros and H goes to infinity at poles. (Key point: Definition)

2. What does each pole pi contribute to the time response?
   A. A term Ki sin(pi t)
   B. A constant Ki
   C. A term Ki/t^pi
   D. A term Ki e^(pi t)
   Answer: D. Partial fractions and inverse Laplace give exponentials at each pole. (Key point: Natural response)

3. What do poles in the right half plane mean?
   A. Decaying terms and stability
   B. Sustained oscillation
   C. Growing terms, so the network is unstable
   D. No effect on the response
   Answer: C. σ > 0 gives e^(σt) growing without bound. (Key point: Stability)

4. What do zeros of H(s) do in the time domain?
   A. Create extra exponential terms
   B. Set the amplitudes (residues) and phases of the terms without creating new time functions
   C. Decide stability
   D. Determine the decay rate of each term
   Answer: B. Zeros change residues; the pole positions fix the exponential forms. (Key point: Role of zeros)

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- [Frequency response of networks and complex frequency](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afrequency-response-complex-frequency-role) — Electrical Circuit Theory, chapter 5
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## Related in other subjects

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