--- title: "Normalization, denormalization and element scaling equations" subject: "Filter Design" chapter: 1 key: "semester_5/filter_design:normalization-element-scaling-equations" source: https://www.bctengineer.com/q/semester_5/filter_design/normalization-element-scaling-equations site: BCT Engineering --- # Normalization, denormalization and element scaling equations Filter Design · Chapter 1: Introduction - Asked in: 2080 Bhadra, 2+5 marks; 2080 Baisakh, 2+5 marks; 2075 Asoj, 2+4 marks; 2071 Chaitra, 6 marks; 2069 Chaitra, 2+5 marks (asked 5 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/normalization-element-scaling-equations - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-1 ## Question What is the importance of normalization and denormalization (scaling) in filter design? Derive the element scaling equations for magnitude and frequency scaling. ## Short answer Normalization sets ωo=1 rad/s and R=1 Ω, enabling use of universal filter tables and reducing complexity. Denormalization uses magnitude scaling factor km and frequency scaling factor kf: new resistor R' = km·R, inductor L' = (km/kf)·L, capacitor C' = C/(km·kf). For frequency-only scaling (km=1), L' = L/kf, C' = C/kf, R' = R. ## Answer #### Importance of normalization and denormalization **Normalization** means designing the filter with convenient reference values, usually a cut-off (or half-power) frequency of $\omega_o = 1$ rad/s and a termination of $R = 1\ \Omega$. **Denormalization (scaling)** converts the normalized element values to the actual frequency and impedance level required. Importance: - Filter tables (Butterworth, Chebyshev, Bessel poles and ladder element values) are published only in normalized form; one table serves every frequency and impedance level. - Calculations use simple numbers like 1, 1.414, 2 instead of values like $10^{-9}$ and $10^{4}$, so errors are fewer. - Designs can be compared and reused: a 1 rad/s prototype becomes a 1 kHz or 1 MHz filter by simple scaling. - Scaling lets the designer choose practical element values (e.g. capacitors in nF–μF, resistors in kΩ) without changing the shape of the response. #### 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: $$ R' = k_m R, \qquad L' = \frac{k_m}{k_f}L, \qquad C' = \frac{C}{k_m k_f} $$ **Example:** the normalized third-order Butterworth ladder (1 Ω, 1 H, 2 F, 1 H, 1 Ω, $\omega_o$ = 1 rad/s) scaled to $\omega_o = 10^4$ rad/s ($k_f = 10^4$) and 1 kΩ terminations ($k_m = 1000$) gives $R$ = 1 kΩ, $L = 1\times1000/10^4 = 0.1$ H and $C = 2/(1000\times10^4) = 0.2\ \mu$F. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does a normalized filter design usually assume? A. Cut-off frequency 1 rad/s and termination 1 Ω B. Cut-off 1 kHz and termination 50 Ω C. Cut-off 1 rad/s and termination 1 kΩ D. Cut-off 1 MHz and termination 1 Ω Answer: A. Normalization uses convenient references, wo = 1 rad/s and R = 1 Ω, which denormalization later converts to actual values. (Key point: What is normalization) 2. Why are filter tables published only in normalized form? A. Real element values cannot be tabulated because they are negative B. Normalized filters have a different response shape that is easier to build C. One table serves every frequency and impedance level after scaling D. Denormalization changes the response shape to the required one Answer: C. Scaling keeps the shape of the response, so a single normalized table can be scaled to any requirement. (Key point: Why normalize) 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 - [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 - [Frequency and magnitude scaling to 10⁵ rad/s](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ascaling-ladder-1e5-numerical) — Filter Design, chapter 1 - [Insertion gain/loss and element scaling equations](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afilter-insertion-gain-scaling) — Filter Design, chapter 1 - [Frequency scaling relations and filter design steps](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afrequency-scaling-design-steps) — Filter Design, chapter 1 - [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 - [Normalization and denormalization with example](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anormalization-importance-example) — Filter Design, chapter 1 ## Related in other subjects - [CTFS time reversal, scaling, frequency shift](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-reversal-scaling-shift) — Signals and Systems, chapter 2 - [Derive bar element stiffness matrix](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Abar-element-stiffness-derivation) — Finite Element Method, chapter 3 - [Derive beam element stiffness matrix](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Abeam-element-stiffness-derivation) — Finite Element Method, chapter 3 - [Derive shape functions of CST element](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Acst-shape-functions-derivation) — Finite Element Method, chapter 5