--- title: "Frequency scaling relations and filter design steps" subject: "Filter Design" chapter: 1 key: "semester_5/filter_design:frequency-scaling-design-steps" source: https://www.bctengineer.com/q/semester_5/filter_design/frequency-scaling-design-steps site: BCT Engineering --- # Frequency scaling relations and filter design steps Filter Design · Chapter 1: Introduction - Asked in: 2081 Bhadra, 4+3 marks - Video: [Designing Passive Butterworth Filters (10 - Passive Filters)](https://www.youtube.com/watch?v=9rWztYL1IKQ) — Aaron Danner - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/frequency-scaling-design-steps - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-1 ## Question Define scaling and derive the relations for frequency scaling. Explain the basic steps to be followed while designing a filter. ## Short answer Scaling shifts frequency or impedance while preserving response shape. Frequency scaling: L'=L/kf, C'=C/kf, R' unchanged with kf=ω_new/ω_old. Basic design steps: specification, normalization, approximation, realization, denormalization (scaling), study non-idealities, build and test. ## Answer #### Scaling **Scaling** is the change of element values of a filter so that its response moves to a new frequency (frequency scaling) or a new impedance level (magnitude scaling) while the shape of the response stays the same. It is used to convert normalized designs (1 rad/s, 1 Ω) into practical ones. #### 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. Example: a 1 F capacitor in a 1 rad/s prototype becomes $1/10^4 = 100\ \mu$F when $k_f = 10^4$; combined with $k_m = 1000$ it becomes 0.1 μF. #### Basic steps of filter design 1. **Specifications:** state the filter type (LP, HP, BP, BS) and $\alpha_{max}, \alpha_{min}, \omega_p, \omega_s$, gain, impedance levels. 2. **Normalization:** convert to a normalized low-pass prototype ($\omega_p$ = 1 rad/s, 1 Ω), using frequency transformation for HP/BP/BS. 3. **Approximation:** choose a response (Butterworth, Chebyshev, inverse Chebyshev, elliptic, Bessel), find the order $n$ and the transfer function $T(s)$ that meets the specifications. 4. **Realization (synthesis):** build a circuit for $T(s)$: a passive LC ladder, or cascaded active biquads (Sallen–Key, MFB, Tow–Thomas), or switched-capacitor sections. 5. **Denormalization (scaling):** apply frequency and magnitude scaling (and the inverse frequency transformation) to get practical element values. 6. **Study of non-idealities:** check sensitivity to element tolerances, op-amp limits and temperature; simulate. 7. **Construction and testing:** build, measure and tune the filter. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does scaling do to a filter? A. It changes the response shape to a new approximation B. It changes the filter type from LP to HP C. It changes element values to move the response in frequency or impedance level without changing its shape D. It changes only the number of elements Answer: C. Scaling preserves the shape of the response. (Key point: Scaling definition) 2. 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' = L/kf, C' = C/kf C. R' = R, L' = kf L, C' = kf C D. R' = R, L' = L/kf, C' = C kf Answer: B. Equal impedance at kf*w requires kf*L' = L and kf*C' = C; R does not depend on frequency. (Key point: Frequency scaling) 3. A 1 F capacitor in a 1 rad/s prototype with kf = 10^4 and km = 1000 becomes? A. 0.1 μF B. 100 μF C. 10 μF D. 0.1 mF Answer: A. C' = 1/(1000 x 10^4) = 10^-7 F = 0.1 μF. (Key point: Capacitor example) 4. In the filter design procedure, which order is correct? A. Specification, realization, approximation, normalization, denormalization, testing B. Approximation, specification, denormalization, realization, normalization, testing C. Specification, approximation, scaling, normalization, realization, testing D. Specification, normalization, approximation, realization, denormalization, non-idealities, construction Answer: D. You normalize to a prototype, find T(s), realize a circuit, and only then denormalize to actual values. 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