Transmission Lines

Coupling Factor (Design)

Q: How do I convert a coupling factor in dB to even and odd mode impedances?

Start by converting the dB coupling factor to a linear voltage coupling coefficient: k = 10^(-C[dB]/20). For a 20 dB coupler, k = 0.1; for a 10 dB coupler, k = 0.316; for a 3 dB coupler, k = 0.707. The even-mode impedance is Z0e = Z0 times sqrt((1+k)/(1-k)) and the odd-mode impedance is Z0o = Z0 times sqrt((1-k)/(1+k)), where Z0 is the system impedance, normally 50 ohms. For a 20 dB coupler in a 50 ohm system this gives Z0e of about 55.3 ohms and Z0o of about 45.2 ohms. Note that Z0e times Z0o equals Z0 squared, which is the matched condition for a backward-wave coupled-line coupler.

Q: Why is a tight 3 dB coupling factor so hard to realize in microstrip?

A 3 dB coupling factor requires k = 0.707, which forces Z0e of about 120 ohms and Z0o of about 21 ohms in a 50 ohm system. Achieving an odd-mode impedance near 21 ohms demands an extremely narrow gap between the two coupled lines, on the order of 10 to 30 micrometers on common substrates, which is below the resolution of standard PCB etching and even challenges thin-film processes. This is why tight couplers are usually built as Lange couplers, which interleave multiple fingers to raise the effective coupling without an impractically small gap, or as broadside-coupled stripline where the two conductors overlap on separate layers.

Q: How does coupling factor change with frequency in a quarter-wave coupled section?

A single quarter-wave coupled section has a coupling factor that peaks at the design frequency, where the coupled length equals a quarter wavelength, and falls off symmetrically on either side. The mid-band coupling equals the design value k, and the 3 dB coupling bandwidth of a single section is roughly plus or minus 35 percent. The coupling response follows C = k sin(theta) divided by sqrt(1 minus k squared times cos squared theta), where theta is the electrical length. To widen the bandwidth, designers cascade multiple coupled sections of graded coupling, producing multi-octave couplers with coupling flatness within plus or minus 0.5 dB.

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As the central design target in coupled-line and directional coupler synthesis, the coupling factor sets the fraction of input power, expressed in dB, that appears at the coupled port. Once a target such as 10 dB or 20 dB is fixed, it converts to a linear voltage coupling coefficient k = 10^−C/20, which in turn dictates the required even-mode impedance and odd-mode impedance of the coupled section. For a matched backward-wave coupler in a 50 Ω system, Z_0e × Z_0o = 2500 Ω², so tighter coupling spreads the two modal impedances farther apart, and a 3 dB coupler pushes Z_0o below 21 Ω, which is why tight couplers usually adopt Lange or broadside-coupled geometries.

Category: Transmission Lines

Typical Range: 3 dB to 30 dB

Match Condition: Z_0e·Z_0o = Z₀²

Turning a dB Coupling Target Into Physical Dimensions

Coupling factor is the quantity that anchors the entire coupled-line design flow. A system requirement, for example a 20 dB sample for a power-monitor coupler or a 3 dB split for a balanced mixer, arrives as a number in dB. The first design step is to translate that into the linear voltage coupling coefficient k, because k is what maps directly onto the geometry of the two coupled conductors. The coupling factor is defined as C = −20 log₁₀(k), and k represents the ratio of coupled-port voltage to incident voltage at the design frequency, where the coupled length is exactly a quarter wavelength.

From k the synthesis is deterministic. The matched coupled-line coupler must present the system impedance Z₀ at all four ports, which forces the geometric-mean condition Z_0e × Z_0o = Z₀². Combined with the coupling relationship, this yields a unique even-mode and odd-mode impedance pair for every coupling factor. Those two modal impedances then set the line widths and, critically, the gap between the lines: tighter coupling means a smaller gap and a wider impedance spread. The conductor gap is usually the limiting fabrication parameter, since odd-mode impedance falls quickly as coupling tightens.

The same coupling factor target also drives the directivity budget. Coupling factor specifies the wanted signal at the coupled port, but the isolated port leakage determines directivity, and the difference between coupling and isolation is what a system actually uses. In ideal coupled-line theory directivity is infinite, yet real microstrip couplers suffer from unequal even-mode and odd-mode phase velocities, which limits practical directivity to 10 dB to 25 dB unless compensation such as wiggly lines or lumped capacitors is added.

Coupling Factor Synthesis Equations

Coupling Factor to Voltage Coupling Coefficient:
C(dB) = −20 log₁₀(k) → k = 10^−C/20

Modal Impedance Synthesis (matched coupler):
Z_0e = Z₀ × √((1 + k) / (1 − k))
Z_0o = Z₀ × √((1 − k) / (1 + k))
Z_0e × Z_0o = Z₀²

Coupling vs. Electrical Length θ:
C(θ) = k sinθ / √(1 − k² cos²θ)

Where C = coupling factor in dB, k = voltage coupling coefficient, Z₀ = system impedance (typically 50 Ω), θ = electrical length (90° at the design frequency). Example: C = 20 dB → k ≈ 0.1 → Z_0e ≈ 55.3 Ω, Z_0o ≈ 45.2 Ω.

Coupling Factor Design Points (50 Ω System)

Coupling C (dB)	Coeff. k	Z_0e (Ω)	Z_0o (Ω)	Through-Port Loss	Typical Realization
3 dB	0.707	120.7	20.7	3.0 dB	Lange / broadside stripline
6 dB	0.501	86.6	28.9	1.26 dB	Tight edge-coupled / Lange
10 dB	0.316	69.4	36.0	0.46 dB	Edge-coupled microstrip
20 dB	0.100	55.3	45.2	0.044 dB	Power monitor / sample tap
30 dB	0.0316	51.6	48.4	0.004 dB	Wide-gap reflectometer tap

Common Questions

Frequently Asked Questions

How do I convert a coupling factor in dB to even and odd mode impedances?

Convert the dB coupling factor to the linear voltage coupling coefficient k = 10^−C/20; a 20 dB coupler gives k = 0.1, a 10 dB coupler 0.316, and a 3 dB coupler 0.707. Then Z_0e = Z₀√((1+k)/(1−k)) and Z_0o = Z₀√((1−k)/(1+k)). For a 20 dB coupler in 50 Ω that is Z_0e ≈ 55.3 Ω and Z_0o ≈ 45.2 Ω, satisfying Z_0e·Z_0o = Z₀².

Why is a tight 3 dB coupling factor so hard to realize in microstrip?

A 3 dB coupling factor needs k = 0.707, forcing Z_0e ≈ 121 Ω and Z_0o ≈ 21 Ω in a 50 Ω system. That low odd-mode impedance demands a 10 to 30 μm gap between the coupled lines, below standard PCB etch resolution. Tight couplers therefore use Lange geometries that interleave fingers to raise effective coupling, or broadside-coupled stripline where the conductors overlap on separate layers.

How does coupling factor change with frequency in a quarter-wave coupled section?

A single quarter-wave section peaks at the design frequency, where the coupled length equals λ/4, and rolls off symmetrically on either side. The response follows C(θ) = k sinθ / √(1 − k² cos²θ), giving a 3 dB coupling bandwidth near plus or minus 35 percent. Cascading multiple graded-coupling sections widens this to multi-octave couplers with coupling flatness inside plus or minus 0.5 dB.

Related Terms

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