---
title: "RC impedance properties; Foster form"
subject: "Filter Design"
chapter: 4
key: "semester_5/filter_design:rc-impedance-properties-foster"
source: https://www.bctengineer.com/q/semester_5/filter_design/rc-impedance-properties-foster
site: BCT Engineering
---

# RC impedance properties; Foster form

Filter Design · Chapter 4: Properties and Synthesis of Passive Networks

- Asked in: 2080 Baisakh, 3+2+3 marks
- Video: [Foster 1 & Foster 2 Forms- LC,RC,LR- KTU Qn #EE201 #CIRCUITS](https://www.youtube.com/watch?v=1tnqYLnfpMw) — Base Your Btech
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/rc-impedance-properties-foster
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-4

## Question

What are the properties of RC driving point impedance function? Which of the following functions are valid RC driving point impedance function and why? Z(s) = (s+3)(s+6)/((s+1)(s+5)); Z(s) = 2(s+1)(s+3)/((s+2)(s+4)). Find Foster form of valid RC driving point impedance function.

## Short answer

RC impedance: poles and zeros on negative real axis, simple, alternating; lowest is a pole, highest a zero; Z(0)>Z(∞). Valid: Z(s)=(s+3)(s+6)/((s+1)(s+5)). Foster I: series 1 Ω, then parallel branches R=2.5 Ω, C=0.4 F and R=0.1 Ω, C=2 F.

## Answer

#### Properties of RC driving-point impedance

An RC driving-point impedance $Z_{RC}(s)$ has these properties:

1. All poles and zeros are simple and lie on the negative real axis of the $s$-plane (including the origin).
2. Poles and zeros alternate along the negative real axis.
3. The critical frequency nearest the origin (lowest) is a **pole**; it may be at $s=0$.
4. The critical frequency farthest from the origin is a **zero**; it may be at $s=\infty$.
5. $Z_{RC}(0) > Z_{RC}(\infty)$, and $Z_{RC}(\infty)\ge 0$ is a constant (no pole at infinity).
6. Residues of $Z_{RC}(s)$ at its poles are real and positive; on the real axis $dZ(\sigma)/d\sigma<0$.
7. The same form describes an RL admittance $Y_{RL}(s)$.

#### Checking the functions

| Function | Critical frequencies | Result |
|---|---|---|
| (a) $\dfrac{(s+3)(s+6)}{(s+1)(s+5)}$ | p 1, z 3, p 5, z 6 | Lowest is a pole, highest a zero, alternate; $Z(0)=3.6 > Z(\infty)=1$ → **valid RC** |
| (b) $\dfrac{2(s+1)(s+3)}{(s+2)(s+4)}$ | **z 1**, p 2, z 3, p 4 | Lowest critical frequency is a zero and $Z(0)=0.75 < Z(\infty)=2$ → **not RC impedance** (it is an RL impedance / RC admittance) |

#### Foster form of (a)

**Foster-I (series) form:** Expand $Z(s)$ in partial fractions; every residue must be positive:

$$
\begin{aligned}
k_\infty &= Z(\infty) = 1 \\
k_{2} &= [(s+1)Z(s)]_{s=-1} = \frac{5}{2} \\
k_{3} &= [(s+5)Z(s)]_{s=-5} = \frac{1}{2}
\end{aligned}
$$

$$
Z(s) = 1 + \frac{5}{2 \left(s + 1\right)} + \frac{1}{2 \left(s + 5\right)}
$$

| Term | Rule | Element value |
|---|---|---|
| $R_1$ (series) | $k_\infty$ | 1 Ω |
| $R_2 \parallel C_2$ (pole at $-1$) | $C=1/k$, $R=k/\sigma$ | $R_2$ = 5/2 Ω (≈ 2.5 Ω), $C_2$ = 2/5 F (≈ 0.4 F) |
| $R_3 \parallel C_3$ (pole at $-5$) | $C=1/k$, $R=k/\sigma$ | $R_3$ = 1/10 Ω (≈ 0.1 Ω), $C_3$ = 2 F |

```text
o--[R1]--[R2||C2]--[R3||C3]--o
```

**Foster-II (parallel) form:** Take $Y(s)=1/Z(s) = \frac{s^{2} + 6 s + 5}{s^{2} + 9 s + 18}$. Expand $Y(s)/s$ in partial fractions and multiply back by $s$:

$$
\begin{aligned}
k_0 &= Y(0) = \frac{5}{18} \\
k_{2} &= \left[\frac{(s+3)Y(s)}{s}\right]_{s=-3} = \frac{4}{9} \\
k_{3} &= \left[\frac{(s+6)Y(s)}{s}\right]_{s=-6} = \frac{5}{18}
\end{aligned}
$$

$$
Y(s) = \frac{5}{18} + \frac{4 s}{9 \left(s + 3\right)} + \frac{5 s}{18 \left(s + 6\right)}
$$

| Term | Rule | Element value |
|---|---|---|
| $R_1$ (shunt) | $1/k_0$ | 18/5 Ω (≈ 3.6 Ω) |
| $R_2$–$C_2$ in series (pole at $-3$) | $R=1/k$, $C=k/\sigma$ | $R_2$ = 9/4 Ω (≈ 2.25 Ω), $C_2$ = 4/27 F (≈ 0.1481 F) |
| $R_3$–$C_3$ in series (pole at $-6$) | $R=1/k$, $C=k/\sigma$ | $R_3$ = 18/5 Ω (≈ 3.6 Ω), $C_3$ = 5/108 F (≈ 0.0463 F) |

```text
o--+-------+-------+
   |       |       |
  [R1]    [R2]    [R3]
   |      [C2]    [C3]
   |       |       |
o--+-------+-------+
```

## Check yourself

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

1. Which statement about RC impedance critical frequencies is correct?
   A. Simple, alternating on the negative real axis; nearest to origin is a pole, farthest is a zero
   B. On the jw axis, nearest to origin a zero
   C. Poles may be double
   D. Pole at infinity is allowed
   Answer: A. Z_RC has simple alternating poles and zeros on the negative real axis, starting with a pole and ending with a zero. (Key point: Pole/zero location properties)

2. What relation holds between Z_RC(0) and Z_RC(infinity)?
   A. Z(0) < Z(infinity)
   B. Z(0) = Z(infinity)
   C. Z(0) > Z(infinity)
   D. Z(0) = 0 always
   Answer: C. Resistance falls as frequency rises (capacitors short), so Z(0) > Z(infinity), and dZ/dsigma < 0. (Key point: Value relation)

3. Which function is a valid RC impedance?
   A. (s+3)(s+6)/((s+1)(s+5))
   B. 2(s+1)(s+3)/((s+2)(s+4))
   C. Both
   D. Neither
   Answer: A. (a) has p1, z3, p5, z6 with Z(0)=3.6 > Z(infinity)=1; (b) starts with a zero, so it is RL type. (Key point: Valid function)

4. In Foster-I of (a), Z = 1 + 5/(2(s+1)) + 1/(2(s+5)); what does the term 5/(2(s+1)) represent?
   A. R = 5/2 ohm in parallel with C = 2/5 F
   B. R = 2/5 ohm in series with C = 5/2 F
   C. A series inductor
   D. R = 5 ohm only
   Answer: A. For a term k/(s+sigma): C = 1/k = 2/5 F and R = k/sigma = 5/2 ohm, as a parallel RC block in series. (Key point: Foster-I form)

## Close variants set in other exams

- [LC impedance properties; Foster I of valid function](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alc-properties-foster-i-three-functions) — Filter Design, chapter 4, asked 2×
- [LC impedance check; Foster parallel and Cauer I](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alc-impedance-foster-parallel-cauer-i) — Filter Design, chapter 4, asked 5×

## Related in Filter Design

- [Valid RC impedance; Foster-II and Cauer-I](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Arc-impedance-foster-ii-cauer-i) — Filter Design, chapter 4, asked 3×
- [RC impedance properties; Foster I and Cauer I](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Arc-impedance-four-functions-foster-cauer) — Filter Design, chapter 4
- [RC impedance properties; Foster and Cauer synthesis](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Arc-impedance-properties-foster-cauer) — Filter Design, chapter 4
- [RC properties; lossless check; Foster series and Cauer II](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Arc-properties-lossless-foster-series) — Filter Design, chapter 4
- [LC impedance properties; identify valid function](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alc-properties-identify-two-functions) — Filter Design, chapter 4
- [Valid RC admittance; Foster II and RC ladder](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Arc-admittance-foster-ii-ladder) — Filter Design, chapter 4

## Related in other subjects

- [Driving point impedance and voltage ratio, two 0.5 F](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Adriving-point-impedance-voltage-ratio-two) — Electrical Circuit Theory, chapter 5
- [Driving point impedance and voltage ratio, bridged RL network](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Adriving-point-impedance-voltage-ratio-bridged) — Electrical Circuit Theory, chapter 5