Microstrip Branchline 90° Hybrid Coupler Calculator

Microstrip Branchline 90° Quadrature Hybrid Coupler

The branchline coupler (quadrature hybrid) is a symmetric four-port microwave network that splits an input signal equally into two output ports with an exact 90° phase difference, while delivering high isolation to the fourth port.

1. Characteristic Branch Impedances

For an equal 3 dB power division from a port impedance (Z_0) (normally 50Ω), the four interconnected quarter-wave transmission lines have characteristic impedances:

• Horizontal (Series) Branches: (Z_{\text{series}} = \frac{Z_0}{\sqrt{2}} \approx 35.35\ \Omega) (wider microstrip trace).
• Vertical (Shunt) Branches: (Z_{\text{shunt}} = Z_0 = 50.0\ \Omega) (standard 50Ω trace).

2. Scattering Matrix & S-Parameters

At the exact design frequency (f_0), all four branches are (\lambda_g/4) long ((90^\circ)), yielding the ideal unitary S-matrix:

$$[S] = -\frac{1}{\sqrt{2}} \begin{bmatrix} 0 & j & 1 & 0 \\ j & 0 & 0 & 1 \\ 1 & 0 & 0 & j \\ 0 & 1 & j & 0 \end{bmatrix}$$

Port 2 (Through) is at (-90^\circ) electrical delay, and Port 3 (Coupled) is at (-180^\circ) delay, producing the fundamental 90° quadrature relationship utilized in I/Q modulators, balanced amplifiers, and circular patch antennas.

3. Bandwidth Limitations

Single-stage branchline couplers have an inherently narrow operating bandwidth of approximately 10% to 15%. Outside this window, junction discontinuities, line dispersion, and non-quarter-wave electrical lengths degrade return loss and phase balance.