Frequency scaling relations and filter design steps

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.