Chapter 8 Microwave Filters

8.1 Periodic Structures

Unit cell:

k : propagation constant of the unloaded line

Two solutions:

: propagation (pass band)

: attenuation (stop band)

Bloch wave: https://en.wikipedia.org/wiki/Bloch_wave

Bloch wave MoM method

Periodic boundary conditon

[Filter]

- 원하는 주파수 성분만 통과 또는 차단

- 용도: 신호간섭 피해 방지, 간섭신호 발생 방지

- 종류:

저역통과필터(LPF)

고역통과필터(HPF)

대역통과필터(BPF)

대역저지필터(BSF)

- 특수 필터

다중 대역통과필터

멀티플렉서

[Filter types]

Butterworth filter

Chebyshev filter

Elliptic (Cauer) filter

Bessel filter: maximally flat linear phase response, preserves the signal shape

Gaussian filter

Optimum "L" (Legendre) filter

Linkwitz-Riley filter

Image impedanc filter

[Filter response function shape]

[Group delay]

- Derivative of the phase with respect to angular frequency

- A measure of the distortion in the signal

[LC filter]

1. Butterworth filter

g₀: source resistance or conductance

g_N+1: load resistance (if g_N is a shunt capacitor) or load conductance (if g_N is a series capacitor)

Pozar: Table 8.3

2. Equi-ripple filter = Chebyshev filter

Pozar: Table 8.4

3. Linear phase response = maximally flat time-delay = maximally flat group-delay

Pozar: Table 8.5

[Filter Transformation]

1. Impedanc and frequency scaling

2. Filter type transformation

Design example:

0.5-dB equi-ripple

N = 3

f_c = 1GHz

[Microstrip filter]

1. Stub-loaded low-pass filter

Design example: 3-dB equi-ripple, LPF, 3GHz, 50Ω

2. Stepped impedance LPF

Design example: maximally flat, f_c 2.5GHz, 20dB insertion loss at 4GHz, 50Ω

microstrip impedance range: 20-120Ω

er =4.2, tand = 0.01, 0.5-mil substrate

N = 6 from attenuation requirement

3. Coupled-line BPF

Coupled-resonator filters

Design example:

Capacitively-coupled series resonator BPF

Degisn example: 2.0GHz, 0.5dB equi-ripple, 10%, 20dB at 2.2GHz

BPF using capacitively-coupled shunt resonators

[Keywords]

immitance: normalized impedance or admittance, normalized (so it is unitless)

prototype filter:

filter transformation: frequency scaling, impedance scaling, type transformation

[Rational function respresentation of the filter transmission function]

poles

zeros

transfer function

complex frequency:

Bandpass filter poles:

Notch filter: a pole close to a zero makes the gain near unity at zero and infinite frequency.

Butterworth filter:

]

[Filter implementation

1. Digital implementation

- Bilinear transform method

- The matched z-transform method

- For higher orders, digital filters are sensitive to quatization errors

2. Sallen-Key topology: uses active and passive components

Example: second-order Butterworth filter

3. Cauer topology: uses passive components