Microstrip Stepped-Impedance Lowpass Filter Synthesizer
Synthesize compact planar microwave lowpass filters using alternating narrow high-Z inductive lines and wide low-Z capacitive pads on microwave PCB substrates.
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Stepped-Impedance Microstrip Lowpass Filter Design
The stepped-impedance lowpass filter (also called a hi-Z / low-Z filter) approximates lumped inductor-capacitor ((LC)) ladder filters using short lengths of alternating very high and very low characteristic impedance transmission lines.
1. Lumped Equivalent of Short Transmission Lines
When electrical length (\beta l < 45^\circ):
- Short High-Z Line ((Z_{0H} \gg Z_0)): Acts almost purely as a series inductor:
$$\beta l = \frac{L \cdot \omega_c}{Z_{0H}} = \frac{g_i \cdot Z_0}{Z_{0H}}$$
- Short Low-Z Line ((Z_{0L} \ll Z_0)): Acts almost purely as a shunt capacitor:
$$\beta l = C \cdot \omega_c \cdot Z_{0L} = \frac{g_i \cdot Z_{0L}}{Z_0}$$
2. Physical Line Synthesis
From the synthesized electrical angle (\beta l_i) in radians, the physical line length (l_i) is derived from the guide wavelength (\lambda_{g}) of that specific line width:
$$l_i = \frac{\beta l_i}{2\pi} \cdot \lambda_{g,i} = \frac{\beta l_i}{2\pi} \cdot \frac{c}{f_c \sqrt{\varepsilon_{\text{eff},i}}}$$
3. High-Frequency Spurious Passband
Because transmission lines are distributed elements rather than ideal lumped components, stepped-impedance filters exhibit harmonic passbands at frequencies where the individual line segments approach half-wavelength resonance ((\beta l = 180^\circ)).
Frequently Asked Questions
Why can't we make Z_0H even higher (e.g. 180Ω) to make the filter smaller?
Microstrip lines with very high characteristic impedance require trace widths below 0.15 mm (6 mils), which exceed standard PCB manufacturing limits and suffer high conductor ohmic losses.
Why is a stepped-impedance filter preferred over a coupled-line filter for lowpass applications?
Stepped impedance filters require zero grounding vias, zero slots, and no tightly coupled gaps. They can be fabricated effortlessly on standard single-layer or two-layer FR-4 and Rogers boards without laser drilling.