Transmission Lines

Coupled Microstrip Design

Q: How do I calculate the coupling of an edge-coupled microstrip coupler?

The voltage coupling factor is C = (Z0e - Z0o) / (Z0e + Z0o), where Z0e is the even-mode impedance and Z0o is the odd-mode impedance. For a target coupling and a system impedance Z0, the modal impedances must satisfy Z0 = sqrt(Z0e times Z0o). A 10 dB coupler in a 50 ohm system needs C = 0.3162, which gives Z0e = 69.4 ohm and Z0o = 36.0 ohm. A 3 dB coupler needs Z0e = 120.7 ohm and Z0o = 20.7 ohm, a wide spread that is difficult to realize with simple edge-coupled microstrip; tight couplers usually move to broadside-coupled or Lange geometries instead.

Q: Why does edge-coupled microstrip have poor directivity?

Microstrip is an inhomogeneous medium: the even mode stores more field in the air above the substrate while the odd mode concentrates field in the higher-permittivity dielectric between the lines. The two modes therefore travel at different phase velocities, so the effective permittivities differ (epsilon_eff,o is larger than epsilon_eff,e because the odd mode is concentrated in the dielectric). This velocity mismatch prevents perfect cancellation at the isolated port, limiting practical directivity to roughly 10 to 15 dB. Remedies include adding capacitive compensation at the line ends, serrating or wiggling the coupled edges to equalize the modal velocities, or using a homogeneous medium such as stripline where directivity above 25 dB is routine.

Q: What spacing and substrate give a 10 dB coupler at 10 GHz?

On a 0.254 mm (10 mil) alumina substrate with relative permittivity near 9.9, a 10 dB edge-coupled microstrip coupler at 10 GHz uses line widths of about 0.20 mm and an edge gap of roughly 0.08 to 0.10 mm, with a coupled-section length of one quarter wavelength (about 2.4 mm in the dielectric). On lower-permittivity RT/duroid 5880 (permittivity 2.2, 0.508 mm thick), the same coupling needs wider lines near 1.4 mm and a larger gap close to 0.4 mm. Gaps tighter than about 0.05 mm strain standard photolithography, so very tight coupling pushes the design toward Lange or broadside structures.

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When two microstrip traces run close together over a shared ground plane, their electromagnetic fields interact, and the engineering task is to choose the line width, edge spacing, and substrate so the structure behaves as intended. The coupled pair supports two independent quasi-TEM modes, each with its own characteristic impedance: an even-mode impedance Z_0e when both lines carry equal in-phase voltages, and an odd-mode impedance Z_0o when they carry equal out-of-phase voltages. The ratio of these two impedances sets the coupling, while their geometric mean must equal the system impedance for a matched coupler. Because microstrip is an open, inhomogeneous medium, the even and odd modes travel at slightly different phase velocities, a mismatch that ultimately limits directivity to roughly 10 to 15 dB unless it is compensated.

Category: Transmission Lines

Coupling: 3 to 20 dB typical

Directivity: 10 to 15 dB uncompensated

Designing Two Lines That Share Their Fields

A single microstrip trace has one characteristic impedance set by its width-to-height ratio and the substrate permittivity. Bring a second trace alongside it and the picture changes: the electric and magnetic fields of the two conductors overlap, and the structure can no longer be described by a single impedance. Instead it supports two normal modes. In the even mode both strips are excited with the same voltage and the same sign, so a magnetic wall sits on the symmetry plane and the fields fringe outward into the air. In the odd mode the strips carry opposite voltages, an electric wall appears on the symmetry plane, and the field crowds into the high-permittivity gap between the lines. The designer manipulates trace width W, edge gap s, and substrate height h and permittivity to land Z_0e and Z_0o on their target values.

Coupling is governed entirely by how far apart the two modal impedances are pushed. A tight gap raises Z_0e and lowers Z_0o, increasing the spread and therefore the coupling; a wide gap brings the two impedances together and the lines decouple. For a backward-wave coupled-line coupler, the coupled and through ports appear on the same physical end, the coupling peaks at a quarter-wave coupled length, and the response is inherently broadband over an octave or more around the design frequency. The same coupled-line section is also the building block of parallel-coupled bandpass filters, where successive quarter-wave sections realize the immittance inverters of the filter prototype.

The catch unique to microstrip is its inhomogeneity. Unlike stripline, where the conductor is buried in uniform dielectric and both modes see the same effective permittivity, microstrip leaves part of the field in air. The even mode, with its outward fringing, sees a lower effective permittivity than the odd mode, which is concentrated in the substrate. Unequal effective permittivities mean unequal phase velocities, and that velocity mismatch is what blunts directivity and skews coupler response. Compensation techniques, lumped capacitors at the line ends, dielectric overlays, or wiggly serrated edges that lengthen the odd-mode path, are how practical microstrip couplers recover their isolation.

Coupled-Line Design Equations

Matched-coupler constraint:
Z₀ = √(Z_0e × Z_0o)

Voltage coupling factor:
C = (Z_0e − Z_0o) / (Z_0e + Z_0o)

Modal impedances from C:
Z_0e = Z₀ × √((1 + C) / (1 − C)) ; Z_0o = Z₀ × √((1 − C) / (1 + C))

Mid-band coupled port (quarter-wave length):
|S₃₁| = C at θ = 90° (ℓ ≈ λ/4)

Example: a 10 dB coupler in a Z₀ = 50 Ω system needs C = 0.3162, giving Z_0e ≈ 69.4 Ω and Z_0o ≈ 36.0 Ω. Directivity is set by the difference in even and odd-mode effective permittivities, ε_eff,e ≠ ε_eff,o.

Coupling Targets and Modal Impedances (50 Ω System)

Coupling (dB)	C (linear)	Z_0e (Ω)	Z_0o (Ω)	Z_0e/Z_0o	Practical Geometry
3 dB	0.707	120.7	20.7	5.83	Lange or broadside
6 dB	0.501	86.6	28.9	3.00	Lange / tight edge gap
10 dB	0.316	69.4	36.0	1.93	Edge-coupled microstrip
15 dB	0.178	59.7	41.9	1.42	Edge-coupled microstrip
20 dB	0.100	55.3	45.2	1.22	Loose edge-coupled

Common Questions

Frequently Asked Questions

How do I calculate the coupling of an edge-coupled microstrip coupler?

Use C = (Z_0e − Z_0o) / (Z_0e + Z_0o), with the matching constraint Z₀ = √(Z_0e × Z_0o). A 10 dB coupler in a 50 Ω system gives C = 0.3162, so Z_0e ≈ 69.4 Ω and Z_0o ≈ 36.0 Ω. A 3 dB coupler demands Z_0e ≈ 120.7 Ω and Z_0o ≈ 20.7 Ω, a spread too wide for simple edge coupling, which is why tight couplers move to Lange or broadside geometries.

Why does edge-coupled microstrip have poor directivity?

Microstrip is inhomogeneous: the even mode fringes into the air while the odd mode concentrates in the substrate, so the two modes have different effective permittivities and different phase velocities. That velocity mismatch prevents full cancellation at the isolated port, capping uncompensated directivity near 10 to 15 dB. Capacitive end compensation, dielectric overlays, or serrated coupled edges equalize the modal velocities; moving to stripline reaches 25 dB or better because the medium is homogeneous.

What spacing and substrate give a 10 dB coupler at 10 GHz?

On 0.254 mm alumina (ε_r ≈ 9.9), a 10 dB edge-coupled section uses line widths near 0.20 mm and an edge gap of about 0.08 to 0.10 mm, with a quarter-wave coupled length near 2.4 mm. On RT/duroid 5880 (ε_r = 2.2, 0.508 mm thick) the same coupling needs wider lines near 1.4 mm and a gap close to 0.4 mm. Gaps below roughly 0.05 mm strain standard photolithography, pushing tight designs toward Lange structures.

Related Terms

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