Directional Coupler Calculator — Coupled-Line Even & Odd Mode Impedance

Design a single-section quarter-wave coupled-line directional coupler: the even- and odd-mode impedances for the coupling you need, the coupled length, the through-path loss and the isolation, with a warning when the coupling is too tight for a single edge-coupled section.

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Formula

c = 10^{-C/20},\quad Z_{0e} = Z_0\sqrt{\frac{1+c}{1-c}},\quad Z_{0o} = Z_0\sqrt{\frac{1-c}{1+c}},\quad L = -10\log_{10}\left(1-c^2\right),\quad I = C + D

Reference: Pozar, Microwave Engineering, 4th ed. (2012), §7.1, Eqs. (7.20)–(7.21), and §7.6, Eqs. (7.85)–(7.87)

C— Coupling (dB)

c— Voltage coupling factor at the centre frequency

Z_0— System impedance (Ω)

Z_{0e}, Z_{0o}— Even- and odd-mode characteristic impedances (Ω)

L— Through-path loss from the power coupled out (dB)

D, I— Directivity and isolation (dB)

How It Works

A coupled-line directional coupler is two parallel lines, a quarter wavelength long, close enough that part of a wave on one appears on the other. A wave entering port 1 passes to port 2, the through port; a fraction couples backward to port 3, the coupled port at the same end as the input; and ideally nothing reaches port 4, the isolated port. How strongly the lines interact is described by two characteristic impedances of the pair: the even-mode impedance $Z_{0e}$ , with both lines driven in phase, and the odd-mode impedance $Z_{0o}$ , with them driven in antiphase.

Design equations

For a coupling of $C$ dB the voltage coupling factor at the centre frequency is $c = 10^{-C/20}$ . The coupler is matched at every port and perfectly isolated when $Z_{0e}Z_{0o} = Z_0^2$ , which gives (Pozar, Microwave Engineering, 4th ed., equations 7.87a and 7.87b):

Z_{0e} = Z_0\sqrt{\frac{1+c}{1-c}}, \qquad Z_{0o} = Z_0\sqrt{\frac{1-c}{1+c}}

A 20 dB coupler in 50 Ω needs $Z_{0e} = 55.28$ Ω and $Z_{0o} = 45.23$ Ω, Pozar's Example 7.7. The coupled section is a quarter wavelength long at the centre frequency, where the coupling peaks:

\ell = \frac{c_0}{4 f_0\sqrt{\varepsilon_{eff}}}

The through path loses the power coupled out, $L = -10\log_{10}(1 - c^2)$ dB, and the isolation is the coupling plus the directivity, $I = C + D$ .

Equal mode velocities

The design assumes that the even and odd modes travel at the same speed, which holds in stripline and other homogeneous TEM structures. In microstrip the even mode keeps more of its field in the substrate than the odd mode, so it travels more slowly, the two modes see different electrical lengths, and the directivity is degraded. Dielectric overlays, compensating capacitors or a stripline build restore it.

Validity

The calculator designs a single lossless quarter-wave section with equal mode velocities, so the coupling is exact only at the centre frequency. Tight coupling needs the lines very close together: Pozar notes that a single edge-coupled section is too loose to reach 3 or 6 dB, where a Lange coupler or a broadside-coupled pair is used instead. Couplings tighter than the limit you enter, 8 dB by default, are shown with a warning.

Worked Example

Problem: A transmitter at 2.45 GHz needs a 10 dB coupler to sample its output, built in 50 Ω stripline on a substrate with εr = 3.0. The layout is expected to reach 25 dB of directivity.

Step 1 - Coupling factor: c = 10^(−10/20) = 0.3162

Step 2 - Mode impedances (Pozar equations 7.87a and 7.87b): Z₀e = 50 × √(1.3162 / 0.6838) = 69.37 Ω Z₀o = 50 × √(0.6838 / 1.3162) = 36.04 Ω Check: 69.37 × 36.04 = 2500 = 50²

Step 3 - Through-path loss: c² = 0.1, so L = −10 log₁₀(1 − 0.1) = 0.46 dB

Step 4 - Isolation: I = 10 + 25 = 35 dB

Step 5 - Coupled length: ℓ = 299792458 / (4 × 2.45e9 × √3.0) = 17.66 mm

The sampled port receives a tenth of the power, the main path loses 0.46 dB, and the next step is to find the strip width and gap that give 69.37 Ω and 36.04 Ω in the stack-up. Asked for 6 dB instead, the coupler needs 86.74 Ω and 28.82 Ω, and the calculator flags it as tighter than the 8 dB a single edge-coupled section can practically reach.

Practical Tips

✓Turn the even- and odd-mode impedances into a strip width and gap with the edge-coupled calculators for your stack-up.
✓For 3 dB, use a Lange coupler or a branch-line hybrid rather than a single edge-coupled section.
✓Terminate the isolated port in a good 50 Ω load; its match limits the directivity you actually measure.
✓For a wider band, cascade several quarter-wave sections with graded coupling, a multisection coupler.

Common Mistakes

✗Treating the even- and odd-mode impedances as the impedances of the two individual lines. They describe the coupled pair as a whole, one for each way of driving it.
✗Building a coupler designed for stripline in microstrip and expecting the same directivity. The unequal mode velocities of microstrip leak power to the isolated port.
✗Confusing coupling with loss on the main line. A 10 dB coupler takes a tenth of the power from the through path, which costs it about 0.46 dB, not 10 dB.
✗Looking for the coupled signal at the far end of the coupled line. A quarter-wave coupled-line coupler is a backward-wave coupler: the coupled port is at the same end as the input.

Frequently Asked Questions

What are even- and odd-mode impedances?

They are the characteristic impedances of a pair of coupled lines for its two basic excitations. In the even mode both lines carry equal in-phase voltages; in the odd mode equal and opposite voltages. Any signal on the pair is a sum of the two, and a coupler's coupling is set by how far apart the two impedances are.

Why must the product of the even- and odd-mode impedances equal Z0 squared?

That condition makes the coupler matched at all four ports and puts no power on the isolated port at any frequency. Pozar derives it from the even- and odd-mode analysis of the four-port; the design equations for Z0e and Z0o follow from it and the coupling factor.

How long is the coupled section?

A quarter wavelength at the centre frequency in the line's medium: the speed of light divided by four times the frequency times the square root of the effective permittivity. That is where the coupling reaches its first maximum.

What is the difference between coupling, directivity and isolation?

Coupling is the input power over the coupled-port power, in dB. Directivity is the coupled-port power over the isolated-port power, a measure of how well the coupler separates forward and reflected waves. Isolation is the input power over the isolated-port power, and it equals coupling plus directivity.

Why can't a single coupled-line section make a 3 dB coupler?

At 3 dB the odd-mode impedance must fall to about 20.7 Ω in a 50 Ω system, which needs the two lines so close together that the gap cannot be etched. Lange couplers interleave several fingers to get tight coupling, and broadside-coupled lines stack the strips on two layers.

Methodology & References

References

Microwave Engineering, 4th ed. — David M. Pozar (2012), §7.1, Eqs. (7.20)–(7.21), pp. 322–323 — coupling, directivity, isolation; §7.6, Eqs. (7.81)–(7.87) and Example 7.7, pp. 351–355 — coupled-line couplers; §7.7, p. 359 — the limit of edge coupling
A Method of Analysis of Symmetrical Four-Port Networks — J. Reed and G. J. Wheeler, IRE Transactions on Microwave Theory and Techniques, vol. MTT-4, pp. 246–252, October 1956 — the even- and odd-mode analysis the coupler design rests on

Reproduces Pozar's Example 7.7 (55.28 Ω and 45.23 Ω at 20 dB); the product Z₀e·Z₀o equals Z₀² for every coupling, and as the coupling weakens both impedances tend to Z₀.

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