Coupling Coefficient (TL)
From Line Spacing to Coupling Factor
Coupled transmission lines support two orthogonal modes. In the even mode both lines carry equal in-phase voltages, the field pattern is symmetric, and each line sees the even-mode characteristic impedance Z0e. In the odd mode the lines carry equal but opposite voltages, a virtual ground sits on the symmetry plane, and each line sees the lower odd-mode impedance Z0o. The coupling coefficient is just the normalized difference between these two impedances. Bringing the conductors closer strengthens the mutual capacitance and inductance, which spreads Z0e and Z0o apart and raises k; pulling them apart drives both modal impedances toward the isolated-line value and sends k toward zero.
For the structure to present a matched port impedance Z0 to the rest of the system, the two modes must satisfy Z0 = √(Z0e × Z0o). Combining that match condition with the coupling definition lets a designer solve directly for the required modal impedances from a target coupling value: Z0e = Z0 √((1 + k)/(1 − k)) and Z0o = Z0 √((1 − k)/(1 + k)). For a 50-ohm 10 dB coupler this gives 69.4 ohms even and 36.0 ohms odd, a comfortable edge-coupled microstrip gap. A 3 dB coupler instead demands 120.7 ohms even and 20.7 ohms odd, an impedance spread so wide that planar designs usually switch to a Lange or broadside-coupled geometry.
The coupling coefficient of transmission lines is distinct from the resonator coupling coefficient used in filter theory, even though both share the symbol. Here k is a traveling-wave voltage ratio between two physical conductors, not a normalized energy exchange rate between two resonant modes. Confusing the two leads to incorrect bandwidth and matching predictions, so the parenthetical "(TL)" flags the transmission-line definition.
Governing Equations
k = (Z0e − Z0o) / (Z0e + Z0o)
Match Condition and Modal Impedances:
Z0 = √(Z0e × Z0o) ; Z0e = Z0 √((1 + k)/(1 − k)), Z0o = Z0 √((1 − k)/(1 + k))
Coupling Factor in dB:
C(dB) = −20 log10(k)
Frequency Response of a Quarter-Wave Section:
|S31(θ)| ≈ (k sinθ) / √(1 − k2 cos2θ), peak at θ = 90° (quarter wave)
Where Z0e = even-mode impedance, Z0o = odd-mode impedance, Z0 = system impedance, θ = electrical length of the coupled section. Example: k = 0.316 → 10 dB coupling, Z0e ≈ 69.4 Ω, Z0o ≈ 36.0 Ω in a 50 Ω system.
Coupling Coefficient vs. Coupler Specifications
| Coupling Factor | Coefficient k | Z0e (50 Ω) | Z0o (50 Ω) | Z0e/Z0o | Typical Geometry |
|---|---|---|---|---|---|
| 3 dB | 0.707 | 120.7 Ω | 20.7 Ω | 5.83 | Lange / broadside |
| 6 dB | 0.501 | 86.7 Ω | 28.9 Ω | 3.00 | Tight edge / Lange |
| 10 dB | 0.316 | 69.4 Ω | 36.0 Ω | 1.93 | Edge-coupled microstrip |
| 20 dB | 0.100 | 55.3 Ω | 45.2 Ω | 1.22 | Loose edge-coupled |
| 30 dB | 0.0316 | 51.6 Ω | 48.4 Ω | 1.07 | Wide-gap stripline |
Frequently Asked Questions
How do even-mode and odd-mode impedances determine the coupling coefficient?
For a symmetric coupled pair, k = (Z0e − Z0o) / (Z0e + Z0o), and the match condition Z0 = √(Z0e × Z0o) must also hold. A tighter gap raises Z0e and lowers Z0o, increasing k. A 10 dB coupler (k = 0.316) needs 69.4 Ω even and 36.0 Ω odd; a 3 dB coupler (k = 0.707) needs 120.7 Ω and 20.7 Ω, often forcing a Lange or broadside geometry.
How is the coupling coefficient converted to a coupler's coupling factor in dB?
Coupling factor in decibels is C(dB) = −20 log10(k), so the two are the same quantity in different units. k = 0.316 is 10 dB, k = 0.1 is 20 dB, and k = 0.707 is 3 dB. This holds at the center frequency where the coupled section is one quarter wavelength; off center the coupled output tracks sinθ, rolling off symmetrically around f0.
Why does the coupling coefficient set coupler bandwidth and how is it widened?
A single quarter-wave section follows one sinusoidal coupling hump centered at f0, giving roughly an octave of usable bandwidth. To go wider, designers cascade multiple quarter-wave sections with per-section coefficients tapered in a binomial or Chebyshev profile, small k on the outer sections and larger k in the center. A three-section Chebyshev design can hold ±0.5 dB coupling flatness over a 3-to-1 band.