Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

A directional coupler is a four-port that separates the power travelling forwards from the power travelling backwards (Figure 7.8.20). Port 1 is the input, port 2 the through port, port 3 the coupled port and port 4 the isolated port. It is used in reflectometers, power meters and standing-wave measurements, and it is specified by two numbers:

C (dB) D (dB) = 10 lo g_{10} (\frac{P _{in}}{P _{coupled}}), = 10 lo g_{10} (\frac{P _{coupled, fwd}}{P _{coupled, rev}}), (7.8.41)

the coupling and the directivity. The directivity compares the power that reaches the coupled port from a forward wave with the power that reaches it from an equal backward wave. The isolation, $I = 10 lo g_{10} (P_{in} / P_{4}) = C + D$ , is how far below the input the isolated port stays.

Power flow in a directional coupler. A wave entering port 1 continues to port 2, sends the sample P_{\rm in}10^{-C/10} to the coupled port 3, and almost nothing to the isolated port 4. — **Figure 7.8.20.** Power flow in a directional coupler. A wave entering port 1 continues to port 2, sends the sample $P_{in} 1 0^{- C /10}$ to the coupled port 3, and almost nothing to the isolated port 4.

The ideal matched directional coupler

[S] = 0 α j γ 0 α 00 j γ j γ 00 α 0 j γ α 0, α^{2} + γ^{2} = 1, C = - 20 lo g_{10} γ . (7.8.42)

Ports 1 and 4, and ports 2 and 3, are decoupled.

Two forms of

D

Definition Eq. (7.8.41) compares the coupled-port power for equal forward and backward waves. The datasheet form $10 lo g_{10} (P_{3} / P_{4})$ for a forward wave alone is the same number for a symmetric coupler.

How a two-hole coupler works

Each hole in the common wall couples a small wave into the upper guide, travelling both ways (Figure 7.8.21). With holes A and B a quarter guide wavelength apart:

towards port 3 both coupled waves have travelled the same total path, and they add,
towards port 4 the wave from B has travelled $2 \times λ_{g} /4 = λ_{g} /2$ further than the wave from A, and they cancel.

The cancellation is perfect only at the design frequency. With the spacing $ℓ = λ_{g 0} /4$ and the phase constant $τ = 2 π / λ_{g}$ at another frequency,

\frac{∣ b _{4} ∣}{∣ b _{3} ∣} = ∣ cos τ ℓ ∣ = cos (\frac{π}{2} \frac{λ _{g 0}}{λ _{g}}) . (7.8.43)

If $λ_{g}$ is off by $10 %$ , $D$ drops to about $16 dB$ . Many holes with binomial strengths give broadband couplers (Exercise 7.8.9).

The two-hole directional coupler. The waves coupled through holes A and B towards port 3 (solid) have travelled the same path and add. Towards port 4 (dashed) their paths differ by \lambda_g/2 and they cancel. — **Figure 7.8.21.** The two-hole directional coupler. The waves coupled through holes A and B towards port 3 (solid) have travelled the same path and add. Towards port 4 (dashed) their paths differ by $λ_{g} /2$ and they cancel.

Waveguide couplers have $C = 3$ to $40 dB$ and $D = 30$ to $40 dB$ over a waveguide band. Their through loss is $- 10 lo g_{10} (1 - 1 0^{- C /10})$ , only $0.04 dB$ for a $20 dB$ coupler. Problem 7.8.5 shows what a finite $D$ does to a measurement.

Common mistake · Coupling and directivity mixed up

Using $D$ as the level of the coupled signal. The coupled port sits $C$ below the input. A wave travelling the wrong way reaches it $C + D$ below the input, and the isolated port receives the same $C + D$ from the forward wave. A $20 dB$ coupler with $30 dB$ directivity therefore delivers $1 %$ of the forward power to port 3, and the leakage that spoils a reflection measurement is $50 dB$ , not $30 dB$ , below the line power.

7.8.16 The directional coupler

How a two-hole coupler works