Passive Components

Coupled-Line Coupler

Q: How do you calculate the even and odd mode impedances for a given coupling value?

For a single-section quarter-wave coupled-line coupler matched to a system impedance Z0, the even and odd mode characteristic impedances are derived from the voltage coupling coefficient C (linear, not dB). The even mode impedance is Z0e = Z0 times the square root of (1+C)/(1-C), and the odd mode impedance is Z0o = Z0 times the square root of (1-C)/(1+C). For a 10 dB coupler in a 50 ohm system, C equals 10 raised to the power of -10/20, which is 0.3162, giving Z0e of 69.4 ohms and Z0o of 36.0 ohms. The geometric mean of Z0e and Z0o always equals Z0, which is the matching condition. Tighter coupling such as 3 dB drives Z0e toward 120.7 ohms and Z0o toward 20.7 ohms, line widths and gaps that are difficult to realize in single-layer microstrip.

Q: Why do microstrip coupled-line couplers have poor directivity compared to stripline?

Directivity in a coupled-line coupler depends on the even and odd modes having equal phase velocities along the coupling region. In stripline the medium is homogeneous because both lines are fully embedded in the same dielectric, so the even and odd modes propagate at the same velocity and directivity can exceed 30 to 40 dB. In microstrip the fields of the odd mode concentrate in the substrate while the even mode fields spread more into the air above the board, so the two modes travel at different velocities. This phase velocity mismatch leaks energy into the isolated port and typically limits microstrip directivity to 10 to 20 dB. Designers improve it with dielectric overlays, serrated or wiggly coupled edges, lumped capacitive compensation at the line ends, or by switching to stripline or suspended substrate.

Q: At what frequency does a quarter-wave coupled-line coupler achieve maximum coupling?

A single-section coupled-line coupler reaches its peak coupling when the coupled length equals one quarter wavelength at the center frequency, which corresponds to an electrical length of 90 degrees. At that point the coupled port output is exactly the design coupling value and the through port loss is minimized. Coupling falls off on either side following a sin(theta) shape, so a single section gives roughly an octave of usable bandwidth within plus or minus 1 dB of coupling flatness. The coupler is a backward-wave device, meaning the coupled signal exits the port nearest the input rather than the far end. For wider bandwidth, designers cascade three, five, or seven quarter-wave sections with tapered coupling to flatten the response across multiple octaves at the cost of longer physical length.

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Two closely spaced parallel transmission lines, run side by side over a coupling region one quarter wavelength long, exchange energy and form a directional coupler that taps a controlled fraction of the forward signal into a coupled port. The behavior is fully described by even mode and odd mode propagation: superimposing the symmetric and antisymmetric excitations of the two lines yields the coupling, isolation, and through responses. Coupling is set by the ratio of even to odd mode characteristic impedance (Z_0e and Z_0o), whose geometric mean must equal the system impedance for a match. Because the coupled signal emerges from the port nearest the input, the structure is a backward-wave coupler. It is the workhorse for power monitoring, signal sampling, balanced mixers, and reflectometer bridges from below 1 GHz into the millimeter-wave bands.

Category: Passive Components

Coupled length: λ/4 (90°)

Coupling type: Backward-wave

Even and Odd Mode Behavior in Coupled Lines

When two transmission lines run parallel and close enough that their fringing fields overlap, they support two distinct propagation modes. The even mode occurs when both lines are driven with equal-amplitude, in-phase voltages, producing a virtual magnetic wall in the symmetry plane and a higher characteristic impedance Z_0e. The odd mode occurs when the lines are driven equal-amplitude but out of phase, producing a virtual electric wall and a lower impedance Z_0o. Any real excitation, such as power entering port 1 with the other ports terminated, is decomposed into a superposition of these two modes. Solving for the port voltages by recombining the modal solutions yields the coupled, through, and isolated responses directly. This even and odd mode decomposition is the foundation of all coupled-line circuit synthesis.

For the classic single-section coupler the coupling region is one quarter wavelength (90 degrees) long at the design center frequency. At that length the coupled port reaches its peak voltage equal to the coupling coefficient C, while the isolated port ideally sees a null. The coupled signal exits the port adjacent to and on the input end of the coupler, which is why these are called backward-wave couplers, in contrast to forward-wave couplers (which appear only in inhomogeneous media, where the coupled signal builds toward the far end). The interdigitated Lange coupler and the broadside-coupled coupler are both backward-wave variants of this same family. The through port carries the remaining power, reduced by a through loss of roughly −10·log₁₀(1−C²) decibels relative to the input.

The matching condition links the two modal impedances to the system impedance: the geometric mean of Z_0e and Z_0o must equal Z₀. This single constraint plus the target coupling fully determines both modal impedances, after which an electromagnetic solver or closed-form microstrip model maps them onto a physical line width and gap. Tighter coupling demands a smaller gap and a wider impedance spread, which is why 3 dB single-section microstrip couplers are impractical and engineers turn to Lange or broadside-coupled stripline instead.

Coupling and Impedance Synthesis Equations

Voltage coupling coefficient (from coupling in dB):
C = 10^{(−Coupling_dB / 20)}

Modal impedance synthesis (matched, single section):
Z_0e = Z₀ × √[(1 + C) / (1 − C)]
Z_0o = Z₀ × √[(1 − C) / (1 + C)]

Matching condition:
Z₀ = √(Z_0e × Z_0o)

Coupled and through response vs. electrical length θ:
S₃₁ = jC·sinθ / [√(1 − C²)·cosθ + j·sinθ]
S₂₁ = √(1 − C²) / [√(1 − C²)·cosθ + j·sinθ]

Where C = voltage coupling (linear), Z₀ = system impedance, θ = electrical length of the coupled region (90° at center). Example: a 10 dB coupler in 50 Ω gives C ≈ 0.3162, Z_0e ≈ 69.4 Ω, Z_0o ≈ 36.0 Ω.

Modal Impedance and Directivity by Coupling Value

Coupling	C (linear)	Z_0e (50 Ω)	Z_0o (50 Ω)	Through loss	Typical directivity
3 dB	0.7079	120.7 Ω	20.7 Ω	3.0 dB	10 to 18 dB (impractical in single-layer MS)
6 dB	0.5012	86.6 Ω	28.8 Ω	1.26 dB	15 to 20 dB
10 dB	0.3162	69.4 Ω	36.0 Ω	0.46 dB	18 to 25 dB
20 dB	0.1000	55.3 Ω	45.2 Ω	0.044 dB	25 to 35 dB
30 dB	0.0316	51.6 Ω	48.4 Ω	0.004 dB	30 to 40 dB (stripline)

Common Questions

Frequently Asked Questions

How do you calculate the even and odd mode impedances for a given coupling value?

Convert the coupling from dB to a linear voltage coefficient C = 10^{(−Coupling_dB/20)}, then apply Z_0e = Z₀√[(1+C)/(1−C)] and Z_0o = Z₀√[(1−C)/(1+C)]. For a 10 dB coupler in 50 Ω, C ≈ 0.3162, giving Z_0e ≈ 69.4 Ω and Z_0o ≈ 36.0 Ω. The geometric mean √(Z_0e×Z_0o) always returns to Z₀, which is the matching condition.

Why do microstrip coupled-line couplers have poor directivity compared to stripline?

Directivity needs the even and odd modes to share one phase velocity. Stripline is a homogeneous medium, so both modes travel at the same speed and directivity can exceed 30 to 40 dB. In microstrip the odd-mode fields sit in the substrate while the even-mode fields spread into the air, so the modes travel at different velocities; the mismatch leaks energy into the isolated port and limits directivity to roughly 10 to 20 dB. Dielectric overlays, wiggly edges, or end capacitors recover some of the loss.

At what frequency does a quarter-wave coupled-line coupler achieve maximum coupling?

Peak coupling occurs when the coupled region is exactly λ/4 (90° electrical) at the center frequency, where the coupled port equals the design coupling and the through loss is minimum. Coupling rolls off as sinθ on either side, giving about an octave within ±1 dB for a single section. Because it is backward-wave, the coupled signal exits the input-side port. Cascading three, five, or seven tapered sections extends flat coupling across multiple octaves.

Related Terms

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