Microstrip Hairpin Bandpass Filter Synthesizer

Microstrip Hairpin Bandpass Filter Synthesis

The hairpin filter is a compact planar microwave bandpass filter synthesized by folding traditional half-wavelength ((\lambda/2)) parallel-coupled microstrip resonators into a "U" shape. Folding drastically conserves PCB real estate without requiring grounding vias.

1. Resonator Guided Wavelength & Effective Permittivity

For a microstrip line of width (W) on substrate height (h) and relative dielectric (\varepsilon_r), the quasi-TEM effective permittivity (\varepsilon_{\text{eff}}) is given by Hammerstad & Jensen:

$$\varepsilon_{\text{eff}} \approx \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \left(1 + 12 \frac{h}{W}\right)^{-0.5}$$

The total unfolded electrical length of each hairpin resonator is half a guide wavelength:

$$L_{\text{total}} \approx \frac{\lambda_g}{2} = \frac{c}{2 f_0 \sqrt{\varepsilon_{\text{eff}}}}$$

2. External Quality Factor ((Q_e)) & Tapped Feed Distance

Rather than using extremely narrow coupling gaps at the I/O ports, hairpin filters employ direct tapped 50Ω microstrip lines. The tapping distance (l_t) measured from the bottom folded bend sets the input match:

$$Q_e = \frac{g_0 g_1}{FBW}, \quad l_t \approx \frac{L_{\text{total}}}{\pi} \arcsin\left(\sqrt{\frac{\pi}{4} \frac{Z_0}{Z_{\text{res}} Q_e}}\right)$$

3. Inter-Resonator Coupling Coefficients & Spacing Gaps

Chebyshev low-pass prototype values (g_0, g_1, \dots, g_{n+1}) dictate the required mutual coupling coefficients between adjacent folded resonators:

$$k_{i,i+1} = \frac{FBW}{\sqrt{g_i \cdot g_{i+1}}}$$

The physical spacing gaps (S_{i,i+1}) between adjacent hairpin arms are determined empirically or via coupled line synthesis such that the fringing field produces the exact target mutual coupling.