Transmission Lines

Coupling Length

Q: Why is the coupling length set to a quarter wavelength?

The coupled and uncoupled signals on a pair of parallel lines accumulate a relative phase shift proportional to electrical length. When the coupled section is one quarter-wavelength long (90 degrees) at the center frequency, the forward and backward voltage components on the coupled port add constructively at the desired output and cancel at the isolated port. This maximizes coupling and gives the deepest isolation. A coupled section shorter than lambda/4 produces weaker, more frequency-flat coupling, while a section longer than lambda/4 over-couples near band center and the response rolls off. For a backward-wave coupler in a homogeneous medium, the quarter-wave length yields a symmetric coupling response centered at the design frequency with roughly an octave usable bandwidth for tight coupling and wider bandwidth for loose coupling.

Q: How do I calculate the physical coupling length for a coupler?

Start from the guided wavelength at the center frequency: lambda_g = c divided by (f0 times the square root of the effective dielectric constant). The physical coupling length is one quarter of that value, L = lambda_g divided by 4. For a stripline coupler on a homogeneous dielectric with er = 2.2 at 10 GHz, lambda_g is about 20.2 mm, so L is about 5.1 mm. For an inhomogeneous microstrip coupler the even and odd modes see different effective permittivities, so designers average the two effective dielectric constants or use the odd-mode value, then add a small length correction for the open-end fringing fields at the coupled-section terminations.

Q: What happens to coupling and bandwidth if the coupling length is wrong?

If the coupling length is shorter than lambda/4 at the intended center frequency, the coupler under-couples; the coupling factor in dB increases (less power on the coupled port) and the response peak shifts up in frequency. If the length is longer than lambda/4, the coupling peaks below the design frequency and rolls off above it, and at twice the design frequency (half-wavelength coupled section) the coupling theoretically drops to zero. A 10 percent error in coupling length typically shifts the center frequency by about 10 percent and degrades directivity, because the even-mode and odd-mode phase velocities no longer cancel cleanly at the isolated port. Microstrip couplers are especially sensitive because unequal even-mode and odd-mode velocities limit achievable directivity regardless of length.

/KUP-ling lengkth/

In a coupled-line coupler, the distance over which two transmission lines run parallel and exchange energy is the parameter that dominates the device's behavior. This coupled section length determines the coupling factor, the center frequency, and the usable bandwidth of a directional coupler. Coupling reaches its peak when the parallel section is one quarter-wavelength (λ/4) long at the design frequency, because the even-mode and odd-mode voltage components then combine to deliver maximum power to the coupled port and the deepest null at the isolated port. A section shorter than λ/4 yields looser, flatter coupling; a longer section over-couples near band center and the response falls off, dropping to zero when the coupled length reaches a half wavelength at twice the design frequency.

Category: Transmission Lines

Optimum length: λ_g/4 (90°)

First null: λ_g/2 (180°)

How Coupling Length Sets a Coupler's Response

A coupled-line directional coupler is built from two transmission lines placed close enough that the fields of one line excite the other along their shared run. That shared run is the coupling length. The structure supports two propagation modes: the even mode, in which both lines carry equal in-phase voltages, and the odd mode, in which the voltages are equal but out of phase. Each mode sees its own characteristic impedance (Z_0e and Z_0o) and, in an inhomogeneous medium such as microstrip, its own phase velocity. The coupled-port voltage results from the interference of these two modes as they propagate over the coupling length, so the electrical length of the coupled section, θ = βL, is the variable that ties physical geometry to electrical performance.

Maximum coupling occurs at θ = 90°, which corresponds to a physical coupling length of one quarter guided wavelength. At this length the differential phase between the even and odd modes is exactly right for the forward components to cancel at the isolated port and reinforce at the coupled port, producing a backward-wave coupler whose coupled output emerges at the port nearest the input. The mid-band coupling factor is fixed by the impedance ratio Z_0e/Z_0o, while the coupling length sets where in frequency that peak lands. Because the coupling response varies as sin(θ), tight couplers (3 dB to 6 dB) sustain only modest fractional bandwidth before coupling flatness degrades, whereas loose couplers (15 dB to 30 dB) hold their nominal coupling across an octave or more.

Quarter-Wave Design and Length Calculation

To build a coupler at a target center frequency f₀, the designer first finds the guided wavelength from the effective dielectric constant, then takes one quarter of it as the physical coupling length. In microstrip the even and odd modes propagate at different velocities, so a single λ/4 cannot satisfy both modes exactly; this velocity mismatch is the root cause of limited directivity in microstrip couplers and is why stripline, with its homogeneous dielectric, achieves far better directivity for the same coupling length. A small open-end length correction is added at each end of the coupled section to account for fringing capacitance.

Coupled-port response vs. electrical length:
|S₃₁| = C × sin(θ) / √(1 − C² × cos²(θ)) where θ = βL

Maximum coupling (mid-band, θ = 90°):
C = (Z_0e − Z_0o) / (Z_0e + Z_0o) → Coupling (dB) = −20·log₁₀(C)

Physical coupling length (quarter-wave):
L = λ_g/4 = c / (4 × f₀ × √ε_eff)

Where C = mid-band voltage coupling factor, Z_0e/Z_0o = even/odd-mode impedances, θ = electrical length, β = phase constant, ε_eff = effective permittivity. Example: stripline, ε_r = 2.2, f₀ = 10 GHz → λ_g ≈ 20.2 mm, L ≈ 5.1 mm.

Coupling Length Across Frequency

Electrical length θ	Physical length	Frequency vs. f₀	Coupling behavior	Notes
45°	λ_g/8	0.5 f₀	Under-coupled (~3 dB below peak)	Low-frequency band edge
90°	λ_g/4	f₀	Maximum coupling	Design center, deepest isolation
135°	3λ_g/8	1.5 f₀	Coupling falling off	High-frequency band edge
180°	λ_g/2	2 f₀	Coupling null (0 power)	Lines effectively decoupled
270°	3λ_g/4	3 f₀	Coupling peak repeats	Periodic response, spurious passband

Common Questions

Frequently Asked Questions

Why is the coupling length set to a quarter wavelength?

At θ = 90° (one quarter guided wavelength) the relative phase between the even and odd modes is exactly right for the backward components to add at the coupled port and cancel at the isolated port, giving maximum coupling and the deepest isolation. A shorter section couples more weakly but more flatly; a longer section over-couples at band center and rolls off, reaching a coupling null at θ = 180°.

How do I calculate the physical coupling length for a coupler?

Find the guided wavelength λ_g = c / (f₀ × √ε_eff), then set L = λ_g/4. For stripline with ε_r = 2.2 at 10 GHz, λ_g ≈ 20.2 mm so L ≈ 5.1 mm. In microstrip the even and odd modes see different effective permittivities, so average the two ε_eff values (or use the odd-mode value) and add an open-end length correction for fringing at the coupled-section ends.

What happens to coupling and bandwidth if the coupling length is wrong?

Too short shifts the coupling peak up in frequency and reduces coupling (higher dB); too long shifts the peak down and rolls off above f₀, hitting a null at 2 f₀. A 10 percent length error moves the center frequency about 10 percent and degrades directivity, since the even-mode and odd-mode phases no longer null cleanly at the isolated port. Microstrip is most sensitive because unequal mode velocities cap directivity regardless of length.

Related Terms

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