---
title: "Sensitivity of -7; RLC lowpass ωc and Q"
subject: "Filter Design"
chapter: 7
key: "semester_5/filter_design:sensitivity-rlc-lowpass"
source: https://www.bctengineer.com/q/semester_5/filter_design/sensitivity-rlc-lowpass
site: BCT Engineering
---

# Sensitivity of -7; RLC lowpass ωc and Q

Filter Design · Chapter 7: Sensitivity

- Asked in: 2083 Baisakh, 1+4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/sensitivity-rlc-lowpass
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-7

## Question

What do you understand when someone tells you that sensitivity of x with respect to y is -7? Compute the sensitivity expression for cutoff frequency and quality factor of RLC lowpass filter.

## Short answer

S^x_y = -7 indicates a 1% rise in y produces roughly a 7% fall in x. For RLC lowpass: S^(omega_o)_L = S^(omega_o)_C = -1/2, S^(omega_o)_R = 0; S^Q_R = -1, S^Q_L = 1/2, S^Q_C = -1/2.

## Answer

#### Meaning of $S^x_y = -7$
$$
\frac{\Delta x}{x} \approx -7\frac{\Delta y}{y}
$$
A 1% increase in $y$ causes about a **7% decrease** in $x$ (opposite direction, seven times larger). $x$ is highly sensitive to $y$, so $y$ must be held to a very tight tolerance.

#### Sensitivity of RLC low-pass filter
Take the series RLC low-pass circuit with the output across the capacitor:
```text
Vin o--R--L--+--o Vout
             |
            === C
             |
GND o--------+--o
```
$$
T(s) = \frac{1/(sC)}{R + sL + 1/(sC)} = \frac{\dfrac{1}{LC}}{s^2 + \dfrac{R}{L}s + \dfrac{1}{LC}}
$$
Comparing with $\dfrac{\omega_o^2}{s^2 + (\omega_o/Q)s + \omega_o^2}$:
$$
\omega_o = \frac{1}{\sqrt{LC}} = L^{-1/2}C^{-1/2}, \qquad Q = \frac{\omega_o L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} = R^{-1}L^{1/2}C^{-1/2}
$$
Here $\omega_o$ is the cutoff (pole) frequency; for the Butterworth case $Q = 0.707$ it is also the half-power frequency.

**Cutoff frequency:**
$$
\begin{aligned}
S^{\omega_o}_L &= \frac{L}{\omega_o}\frac{\partial\omega_o}{\partial L} = \frac{L}{\omega_o}\left(-\frac{\omega_o}{2L}\right) = -\frac12 \\
S^{\omega_o}_C &= -\frac12, \qquad S^{\omega_o}_R = 0
\end{aligned}
$$
**Quality factor:**
$$
\begin{aligned}
S^Q_R &= \frac{R}{Q}\frac{\partial Q}{\partial R} = \frac{R}{Q}\left(-\frac{Q}{R}\right) = -1 \\
S^Q_L &= +\frac12, \qquad S^Q_C = -\frac12
\end{aligned}
$$

| Element | $S^{\omega_o}$ | $S^Q$ |
|---|---|---|
| R | 0 | −1 |
| L | −1/2 | 1/2 |
| C | −1/2 | −1/2 |

So a 1% rise in L or C lowers the cutoff by 0.5%, while R affects only Q: a 1% rise in R lowers Q by 1%.

## Check yourself

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

1. If S of x with respect to y is -7, what is the implication?
   A. x is insensitive to y
   B. x rises about 7% for a 1% rise in y
   C. x falls about 7% for a 1% rise in y, so y needs tight tolerance
   D. x and y are unrelated
   Answer: C. dx/x is about -7 dy/y; the large magnitude means y must be held tightly. (Key point: Meaning of S=-7)

2. For the series RLC low-pass (output across C), which pair is correct?
   A. wo = 1/sqrt(LC), Q = (1/R) sqrt(L/C)
   B. wo = 1/(RC), Q = sqrt(LC)
   C. wo = R/L, Q = 1/sqrt(LC)
   D. wo = 1/sqrt(LC), Q = R sqrt(C/L)
   Answer: A. Comparing T(s) with the standard form gives these. (Key point: RLC cutoff and Q)

3. What are S of wo with respect to L, C and R?
   A. -1/2, -1/2, 0
   B. -1/2, -1/2, -1
   C. 1/2, 1/2, 0
   D. -1, -1, 0
   Answer: A. wo = L^(-1/2) C^(-1/2) and does not contain R. (Key point: Sensitivity of wo)

4. What are S of Q with respect to R, L, C?
   A. -1, +1/2, -1/2
   B. +1, +1/2, -1/2
   C. -1, -1/2, +1/2
   D. 0, 1/2, -1/2
   Answer: A. Q = R^(-1) L^(1/2) C^(-1/2). (Key point: Sensitivity of Q)

## Related in Filter Design

- [Sensitivity of 2; centre frequency of Sallen-Key LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-value-2-sallen-key) — Filter Design, chapter 7
- [Sensitivity of -3; Q of Tow-Thomas LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-minus-3-q-tow-thomas) — Filter Design, chapter 7, asked 2×
- [Sensitivity of 10; ωo of Sallen-Key LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-value-10-sallen-key) — Filter Design, chapter 7, asked 3×
- [Sensitivity of ωo and Q in Tow-Thomas LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-wo-q-tow-thomas) — Filter Design, chapter 7, asked 3×
- [Sensitivity of 0.5; single and multi-parameter sensitivity](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-value-0-5-single-multi) — Filter Design, chapter 7
- [Sensitivity of -5; ωo of Sallen-Key LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-value-minus-5-sallen-key) — Filter Design, chapter 7, asked 4×

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