---
title: "Insertion gain/loss and element scaling equations"
subject: "Filter Design"
chapter: 1
key: "semester_5/filter_design:filter-insertion-gain-scaling"
source: https://www.bctengineer.com/q/semester_5/filter_design/filter-insertion-gain-scaling
site: BCT Engineering
---

# Insertion gain/loss and element scaling equations

Filter Design · Chapter 1: Introduction

- Asked in: 2080 Bhadra, 1+2+4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/filter-insertion-gain-scaling
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-1

## Question

What is a filter? Define the terms: Insertion gain and Insertion loss with neat diagram. Derive the element scaling equations.

## Short answer

A filter is a frequency-selective two-port. Insertion gain = 20 log|V2/V20|, insertion loss = 20 log|V20/V2|, where V20 is load voltage without filter, V2 with filter. Magnitude scaling multiplies impedances by km, frequency scaling by kf: R'=kmR, L'=(km/kf)L, C'=C/(km kf).

## Answer

#### Filter
A **filter** is a frequency-selective two-port network that passes signals in a chosen band of frequencies (passband) with little attenuation and attenuates signals at other frequencies (stopband).

#### Insertion gain and insertion loss
**Insertion gain and insertion loss** compare the load voltage with and without the filter between a source $V_1$ (resistance $R_s$) and load $R_L$.

```text
 Without filter:  Rs                 With filter:  Rs   +--------+
  V1 o--/\/\--+-- RL  (V20)          V1 o--/\/\--| filter |-- RL (V2)
                                                 +--------+
```
$$
V_{20} = \frac{R_L}{R_s + R_L}V_1, \qquad
\text{Insertion gain} = 20\log_{10}\left|\frac{V_2}{V_{20}}\right|\ \text{dB}
$$
$$
\text{Insertion loss} = 20\log_{10}\left|\frac{V_{20}}{V_2}\right| = -(\text{insertion gain})\ \text{dB}
$$
A passive filter normally has a positive insertion loss; an active filter can have insertion gain.

#### Magnitude (impedance) scaling
Every impedance in the network is multiplied by $k_m$, at the same frequency. A voltage transfer function is a ratio of impedances, so it does not change.
$$
\begin{aligned}
R' &= k_m R \\
sL' = k_m (sL) &\Rightarrow L' = k_m L \\
\frac{1}{sC'} = k_m \frac{1}{sC} &\Rightarrow C' = \frac{C}{k_m}
\end{aligned}
$$

#### Frequency scaling
The response at the old frequency $\omega$ must appear at the new frequency $k_f\omega$, where $k_f = \omega_{new}/\omega_{old}$. So each element must have at $k_f\omega$ the same impedance that the old element had at $\omega$:
$$
\begin{aligned}
j(k_f\omega)L' = j\omega L &\Rightarrow L' = \frac{L}{k_f} \\
\frac{1}{j(k_f\omega)C'} = \frac{1}{j\omega C} &\Rightarrow C' = \frac{C}{k_f} \\
R' &= R \quad (\text{frequency independent})
\end{aligned}
$$
The new transfer function is $T'(s) = T(s/k_f)$: the response keeps its shape but is shifted along the frequency axis.

#### Combined scaling
Applying both:
$$
R' = k_m R, \qquad L' = \frac{k_m}{k_f}L, \qquad C' = \frac{C}{k_m k_f}
$$

## Check yourself

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

1. What is V20 in the insertion gain definition?
   A. The load voltage without the filter, V20 = RL V1/(Rs + RL)
   B. The load voltage with the filter in place
   C. The source voltage V1
   D. The filter's output voltage at the stopband edge
   Answer: A. V20 is the reference voltage across RL when the source drives RL directly, with no filter. (Key point: Insertion gain)

2. How is insertion loss defined?
   A. 20 log10|V2/V20| dB
   B. 20 log10|V1/V2| dB, with V1 the source voltage
   C. 20 log10|V20/V2| dB, where V20 is the load voltage without the filter and V2 with it
   D. 10 log10|V2/V20| dB
   Answer: C. Insertion loss compares load voltage without the filter (V20) to that with it (V2); it equals minus the insertion gain. (Key point: Insertion loss)

3. Under magnitude scaling by km (same frequency), how do R, L and C change?
   A. R' = km R, L' = L/km, C' = km C
   B. R' = km R, L' = km L, C' = C/km
   C. R' = R, L' = km L, C' = km C
   D. R' = R/km, L' = km L, C' = C/km
   Answer: B. Every impedance is multiplied by km: R and sL scale by km, while 1/(sC) scaling by km means C is divided by km. (Key point: Magnitude scaling)

4. For frequency scaling by kf = w_new/w_old, how do R, L and C change?
   A. R' = kf R, L' = L/kf, C' = C/kf
   B. R' = R, L' = kf L, C' = kf C
   C. R' = R, L' = L/kf, C' = C kf
   D. R' = R, L' = L/kf, C' = C/kf
   Answer: D. Equal impedance at kf*w requires kf*L' = L and kf*C' = C; R does not depend on frequency. (Key point: Frequency scaling)

## Related in Filter Design

- [Normalization, denormalization and element scaling equations](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anormalization-element-scaling-equations) — Filter Design, chapter 1, asked 5×
- [Analog filter applications and element scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aanalog-filter-applications-scaling) — Filter Design, chapter 1
- [Define αmax, αmin, insertion loss and gain](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Adefine-amax-amin-insertion-loss) — Filter Design, chapter 1, asked 2×
- [Insertion gain/loss, design steps, active vs passive](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainsertion-loss-steps-active-passive) — Filter Design, chapter 1
- [αp, αs, ωp, ωs and element scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Adefine-attenuation-terms-scaling) — Filter Design, chapter 1
- [Ideal filter and magnitude/frequency scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aideal-filter-magnitude-frequency-scaling) — Filter Design, chapter 1

## Related in other subjects

- [Insertion loss method for filter design](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Afilter-design-insertion-loss) — RF and Microwave Engineering, chapter 6, asked 2×
- [Insertion loss design of given LC ladder filter](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Afilter-ladder-insertion-loss-microstrip) — RF and Microwave Engineering, chapter 6
- [Justify insertion loss method for filter design](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Afilter-insertion-loss-justify) — RF and Microwave Engineering, chapter 6
- [Insertion loss method and microstrip LPF](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Ainsertion-loss-lpf-microstrip) — RF and Microwave Engineering, chapter 6