Microstrip Rat-Race 180° Coupler Calculator

Microwave 180° Rat-Race Hybrid Ring Couplers

The rat-race hybrid coupler (or 180° hybrid ring) is a 4-port microwave device consisting of a continuous circular microstrip transmission line with a total circumference of $1.5\ \lambda_g$. Four ports are positioned along the ring: three ports separated by quarter-wavelengths ($\lambda_g / 4 = 90^\circ$) and one longer section separated by three quarter-wavelengths ($3\lambda_g / 4 = 270^\circ$).

Ring Characteristic Impedance

To ensure a matched input impedance ($S_{11} = 0$) and equal 3 dB power split when all four ports are terminated in system impedance $Z_0$ (typically $50\ \Omega$), the ring transmission line must have a characteristic impedance of:

Sum ($\Sigma$) vs Difference ($\Delta$) Port Operation

• Port 1 Excitation (Sum $\Sigma$ Port): Waves travel clockwise and counter-clockwise to Ports 2 and 3 with equal path lengths, producing two in-phase (0° phase delta) outputs of equal power ($-3.01\ \text{dB}$). At Port 4, clockwise path ($270^\circ$) and counter-clockwise path ($90^\circ$) cancel destructively ($180^\circ$ out of phase), giving theoretical infinite isolation ($S_{41} = 0$).
• Port 4 Excitation (Difference $\Delta$ Port): The wave traveling to Port 2 travels $90^\circ$ whereas the wave to Port 3 travels $270^\circ$. The outputs at Ports 2 and 3 are equal in magnitude ($-3.01\ \text{dB}$) but exactly $180^\circ$ out of phase. Port 1 is isolated ($S_{14} = 0$).