RF Filter Theory — RF Theory

// 01 — Fundamentals

Filter Fundamentals

An RF filter is a two-port network that passes signals in a desired passband while attenuating signals in the stopband. Filters appear everywhere in RF systems: after the antenna to reject out-of-band blockers, between stages to suppress harmonics, in diplexers to separate TX and RX bands, and as channel-select filters in receivers.

Filter Specification Parameters

f_c — passband edge / cutoff frequency · f_s — stopband edge
A_p — max passband ripple (dB) · A_s — min stopband attenuation (dB)
Ω_s = f_s/f_c — normalised stopband frequency (selectivity factor)

Steeper rolloff or larger A_s−A_p requires higher order n — more components.

All filter types (HP, BP, BS) are derived from a lowpass prototype by frequency transformation — LP is the fundamental building block.

// 02 — Filter Approximations

Filter Approximations

Butterworth — Maximally Flat

Butterworth Filter

|H(jΩ)|² = 1/(1 + Ω²ⁿ) · At Ω=1: |H| = 1/√2 = −3.01 dB always
Rolloff: −20n dB/decade · No passband ripple — maximally flat magnitude
Poles: equally spaced on circle of radius ω_c in left half-plane

Chebyshev — Equiripple Passband

Chebyshev Type I Filter

|H(jΩ)|² = 1/(1 + ε²·T²_n(Ω)) · T_n = Chebyshev polynomial
ε² = 10^(A_p/10)−1 · A_p=0.5 dB: ε=0.3493 · A_p=3 dB: ε=0.9976
Advantage: Steeper rolloff than Butterworth for same n · Trade-off: Passband ripple

Bessel — Maximally Flat Group Delay

Bessel / Thomson Filter

τ(ω) = −dφ/dω ≈ constant for ω < ω_c (linear phase)
Step response: zero overshoot · Rolloff: slower than Butterworth
Use for: radar pulse shaping, digital IF filters, wideband modulations

Comparison — n=5, same cutoff

Parameter	Butterworth n=5	Chebyshev 0.5 dB n=5	Bessel n=5
Passband ripple	0 dB (flat)	0.5 dB	0 dB (flat)
Atten. at 2×f_c	−31 dB	−43 dB	−14 dB
Atten. at 3×f_c	−50 dB	−67 dB	−25 dB
Group delay	Good	Poor	Excellent
Best for	General purpose	Sharp selectivity	Linear phase / pulse

Butterworth n=5 at 2f_c: 10·log₁₀(1+2¹⁰) = 10·log₁₀(1025) = 30.1 dB ≈ −31 dB ✓

// 03 — Filter Order

Determining Filter Order

Minimum Order Formulas

Butterworth: n ≥ log[(10^(A_s/10)−1)/(10^(A_p/10)−1)] / (2·log Ω_s)
Chebyshev: n ≥ arccosh(√[(10^(A_s/10)−1)/(10^(A_p/10)−1)]) / arccosh(Ω_s)

Worked Example

Example 1 — A_p=0.5 dB, A_s=40 dB, Ω_s=f_s/f_c=2.0

1

10^(40/10)−1 = 9999 · 10^(0.05)−1 = 0.1220

2

Butterworth: log(9999/0.1220)/(2·log2) = 4.9135/0.60206 = 8.16 → n = 9

3

Chebyshev: arccosh(√81959)/arccosh(2.0) = 6.350/1.317 = 4.82 → n = 5

✓ Butterworth needs n=9, Chebyshev achieves same spec with n=5 — major saving. Prefer Chebyshev when passband ripple is acceptable.

// 04 — LP Prototype & Denormalization

LP Prototype & Denormalization

All filter designs start from a normalized lowpass prototype (ω_c=1 rad/s, R=1 Ω). Denormalization scales to any cutoff frequency and impedance.

Prototype g-values

Butterworth (−3 dB at ω=1)

n	g₁	g₂	g₃	g₄	g₅	g_n+1
1	2.0000	—	—	—	—	1.0000
2	1.4142	1.4142	—	—	—	1.0000
3	1.0000	2.0000	1.0000	—	—	1.0000
4	0.7654	1.8478	1.8478	0.7654	—	1.0000
5	0.6180	1.6180	2.0000	1.6180	0.6180	1.0000

Chebyshev — 0.5 dB ripple

n	g₁	g₂	g₃	g₄	g₅	g_n+1
1	0.6986	—	—	—	—	1.0000
2	1.4029	0.7071	—	—	—	1.9841
3	1.5963	1.0967	1.5963	—	—	1.0000
4	1.6703	1.1926	2.3661	0.8419	—	1.9841
5	1.7058	1.2296	2.5408	1.2296	1.7058	1.0000

Denormalization to Z₀ and f_c

Series L: L_k = g_k·Z₀/ω_c · Shunt C: C_k = g_k/(Z₀·ω_c)
Topology: C–L–C–L… (shunt-first) or L–C–L–C… (series-first, dual network, same response)

Worked Example — 3rd-order Butterworth LP at 100 MHz, Z₀=50 Ω

Example 2 — Butterworth n=3, f_c=100 MHz, Z₀=50 Ω

g-values: g₁=1.0, g₂=2.0, g₃=1.0. Topology: C–L–C shunt-first. ω_c=2π×10⁸=6.283×10⁸ rad/s

1

C₁ = 1.0/(50×6.283×10⁸) = 31.83 pF

2

L₂ = 2.0×50/(6.283×10⁸) = 159.2 nH

3

C₃ = 1.0/(50×6.283×10⁸) = 31.83 pF

4

Verify at f=200 MHz (Ω=2): 10·log₁₀(1+2⁶) = 10·log₁₀(65) = 18.1 dB attenuation ✓

✓ C₁=C₃=31.83 pF, L₂=159.2 nH. E24 standard: C=33 pF, L=150 nH. Attenuation doubles every octave above f_c.

50 Ω

Source

C₁ ↓

31.83 pF

GND

L₂ →

159.2 nH

C₃ ↓

31.83 pF

GND

50 Ω

Load

LC Filter Design Calculator — automates all of the above →

// 05 — Frequency Transformations

LP → HP / BP / BS Transformations

Lowpass → Highpass

LP → HP (substitute s → ω_c/s)

LP series L → HP series C: C = 1/(g_k·Z₀·ω_c)
LP shunt C → HP shunt L: L = Z₀/(g_k·ω_c)

Example n=3 Butterworth HP at 100 MHz, Z₀=50 Ω:
C₁=C₃ = 1/(1.0×50×6.283×10⁸) = 31.83 pF · L₂ = 50/(2.0×6.283×10⁸) = 39.79 nH

Lowpass → Bandpass

LP → BP (substitute s → Q·(s/ω₀ + ω₀/s))

ω₀=√(ω₁ω₂), Q=ω₀/BW. Each LP element → resonator pair (doubles component count).
LP series L_k → series LC: L_s=g_k·Z₀/BW · C_s=BW/(g_k·Z₀·ω₀²)
LP shunt C_k → parallel LC: C_p=g_k/(Z₀·BW) · L_p=Z₀·BW/(g_k·ω₀²)

Bandpass Design Example

Example 3 — Chebyshev 0.5 dB BP, f₀=900 MHz, BW=50 MHz, Z₀=50 Ω, n=3

g₁=1.5963, g₂=1.0967, g₃=1.5963. ω₀=5.655×10⁹ rad/s, BW=3.142×10⁸ rad/s, Q=18.0

1

Choose resonator L_r=10 nH. C_r=1/(ω₀²·L_r)=1/(3.198×10¹¹) = 3.128 pF

2

Verify: f₀=1/(2π√(10n×3.128p))=1/(2π×1.769×10⁻¹⁰) = 900 MHz ✓

3

FBW=50/900=5.56% — narrowband, validates approximations

✓ 3 resonators each = 10 nH + 3.128 pF at 900 MHz. Q=18, BW=50 MHz, ripple ≤0.5 dB, stopband >40 dB at band edges.

Lowpass → Bandstop

LP → BS (substitute s → BW·s/(s²+ω₀²))

LP series L → shunt parallel-LC (resonates at ω₀, high Z at notch frequency)
LP shunt C → series LC (resonates at ω₀, short at notch frequency)

Single-section notch at 2.4 GHz: L=1 nH, C=1/((2π×2.4G)²×1n) = 4.39 pF
Notch depth ≈ 20–40 dB in practice (limited by component Q)

// 06 — Microwave Realisation

Microwave Filter Realisation

Above ~1 GHz, lumped LC components become impractical due to parasitics. Distributed transmission line structures replace them. Same prototype g-values — only the physical realisation changes.

Edge-Coupled BPF — Most Common Microwave BPF

Edge-Coupled BPF (Matthaei Method)

n resonators → n+1 coupling gaps. Even/odd mode impedances per gap:
Z_0e,k = Z₀·[1 + J_k/Y₀ + (J_k/Y₀)²] · Z_0o,k = Z₀·[1 − J_k/Y₀ + (J_k/Y₀)²]

Example 4 — 3-resonator 2.4 GHz BPF, BW=240 MHz (10%), Chebyshev 0.5 dB on RO4003

1

Resonator length: 50 Ω on RO4003 (ε_eff=2.74): λ/2 = 300/(2×2.4×1.655) = 37.8 mm

2

J-inverters (FBW=10%): J₀₁/Y₀=√(π×0.1/(2×1.5963))=0.3137 · J₁₂/Y₀=π×0.1/(2×1.3226)=0.1188

3

Gap 1 impedances: Z_0e=50×1.412=70.6 Ω · Z_0o=50×0.785=39.2 Ω

4

Gap 2 impedances: Z_0e=50×1.133=56.6 Ω · Z_0o=50×0.895=44.8 Ω

5

Physical gaps from coupled-line synthesis: Gap 1: s₁≈0.10 mm · Gap 2: s₂≈0.35 mm

✓ Three 37.8 mm resonators, outer gaps 0.10 mm, inner gaps 0.35 mm. Filter length ≈115 mm. Centre 2.4 GHz, BW=240 MHz, >40 dB stopband.

Microstrip — resonator W and λ/2 length → Plot filter S21 from Touchstone →

// 07 — Practical Considerations

Practical Considerations

Component Q and Insertion Loss

Filter IL from Component Q

IL_min ≈ (4.343/Q_L)·Σg_k (dB at passband centre)

5th-order Butterworth: Σg_k=6.472. Q_L=50 (cheap SMD): IL=0.562 dB. Q_L=200 (good SMD): IL=0.140 dB

Technology	Q_L	f range	IL (n=3)	Best for
SMD LC	30–150	DC–3 GHz	0.3–2 dB	PCB, cost-sensitive
Microstrip coupled	100–200	0.5–20 GHz	0.5–2 dB	Microwave PCB standard
SAW	500–2000	10 MHz–3 GHz	0.5–3 dB	Mobile band-select
BAW / FBAR	1000–3000	0.5–6 GHz	0.5–2 dB	5G, WiFi duplexers
Cavity / coaxial	2000–10000	100 MHz–10 GHz	0.1–0.5 dB	Base stations, instruments
Waveguide	5000–50000	2–100 GHz	0.05–0.3 dB	Satellite, radar, mmWave

Common design mistakes:
· Using inductor Q at 1 MHz to design a 900 MHz filter — Q drops dramatically with frequency
· Ignoring PCB parasitic capacitance across series inductors — shifts cutoff upward
· Not accounting for end-loading in microstrip edge-coupled filters — resonators appear too long
· Too-narrow BPF bandwidth — insertion loss increases as 1/BW²

LC Filter Design Calculator → Microstrip Dimensions → Plot Filter S-params → Cascade Filter + Amplifier →

// 08 — Try the Tools

Put This Theory Into Practice

Use these RFLab tools to design, compute component values, and verify your filters — linking the prototype tables and transformations on this page directly to physical designs.

Filter Design Calculators

⚡

CALCULATOR

LC Filter Design

Butterworth, Chebyshev and Bessel — LP, HP, BP and BS — fully automated. Enter f_c, Z₀, order and type, get every component value with a schematic and frequency response plot. Directly implements the prototype tables and denormalization from Example 2.

⚡

CALCULATOR

Microstrip Calculator

After designing a microwave BPF, use this to find the resonator trace width and ε_eff for your substrate. Compute λ/2 resonator length — Example 4 gives 37.8 mm on RO4003 at 2.4 GHz. Essential for converting electrical filter dimensions to PCB layout.

⚡

CALCULATOR

Waveguide Calculator