A conventional branch-line coupler consists of four quarter-wavelength transmission lines as shown in Figure 1, at a center frequency. Characteristic impedances of the ports are 50 Ω, Power entering port 1 is evenly divided between ports 2 and 3, power 4 is isolated port, and there is a 90 degree phase shift between port 2 and 3.

Figure 1: 
Conventional branch-line coupler circuit

The branch-line coupler consists of four λ/4 transmission lines, so in order to miniaturize the branch-line circuit, the length of the λ/4 transmission lines need to be reduced. The size-reduction of quarter-wavelength transmission line is done here through a series of equivalent circuit transformation.

As is well known, the quarter-wavelength transmission line also can be made equivalent to a lumped circuit, as given in Figure 2 (b), and the value of C_A1 is given by

Figure 2: 
Quarter-wavelength transmission line (a), capacitively loaded transmission line (b), reduced capacitive load line with shunt resonant circuits (c), parallel end shorted coupled line (d)

By replacing the lumped inductor, the lumped circuit can be transformed into an equivalent circuit with artificial resonance circuits inserted as shown in Figure 2 (c), furthermore the dotted network can be made equivalent to the parallel end shorted coupled line section with electrical length of θ_A, and the final equivalent circuit is shown in Figure 2 (c), The related formulas are expressed by [3]

Where Z_A and Z₀ are the characteristic impedances of the shorted coupled lines and characteristic impedance of the quarter transmission line. θ_A and ω are the electrical length of the shorted line and angular frequency, respectively. C_A is the combination of C_A0 and C_A1. Z_oe and Z_oo are the even and odd mode characteristic impedance of coupled line. When the electrical length θ_A is very small for compact size, Z_A becomes very large. This large Z_A can be easily achieved by making Z_Boe and Z_Boo nearly the same.

Figure 3 (a) is a miniaturized branch-line coupler employing the capacitively loaded circuit. The equivalent circuit consisting of parallel end shorted coupled lines is shown in Figure 3 (b).

Figure 3: 
Reduced size ring hybrid circuit (a), equivalent circuit of ring hybrid with coupled line (b)

Because of the symmetry of the branch-line coupler, we use even-odd mode method to analyse the amplitudes of the emerging wave at each port of the miniaturized branch-line coupler as shown in Figure 4.

Figure4 
Even- and odd-mode decomposition of the ring hybrid when port 1 is excited with a unit amplitude incident wave. (a) Even mode. (b) Odd mode.

When the amplitudes of the incident waves are 1/2(even-mode excitation), as shown in Figure 4 (a), the magnetic wall is formed at A ㅡ A¹. In this case, the circuit is equivalent to an open circuit, as indicated in Figure 4 (a). We multiply the ABCD matrices of each cascade component in the circuit, to give

Similarly, when the amplitudes of the incident waves are 1/2 (odd-mode excitation), as shown in Figure 4 (b), the electrical wall is created at A ㅡ A¹. In this case, the circuit is equivalent to an short circuit, as indicated in Figure 4 (b),

The reflection and transmission coefficients for the even-and odd-mode of the ring hybrids are given by:

The amplitudes of the scattered waves from the ring hybrid will be:

Then with the aid of (9), (10), (11), and (12), according [4] we have:

The input port 1 should be matched, so we let B₁ = 0, then we can get , now the above results turn to be

Which shows that the input port is matched, port 4 is isolated, and the input power is evenly divided in phase between ports 2 and 3.

The miniaturized branch-line coupler circuit is designed and simulated on Agilent ADS and Ansoft HFSS before fabrication until we get an ideal simulation result. We designed the miniaturized branch-line coupler circuit by replacing quarter transmission line. Figure 5 shows the structure of the miniaturized branch-line coupler in HFSS. The electrical length θ is 15°. Table 1 gives the design parameters of the HFSS simulation. The parameter values were appropriately tuned for proper simulation results. Figure 6 shows the simulated insertion and return loss responses of the branch-line coupler.

Figure 5: 
HFSS design

Figure 6: 
The simulation results of branch-line coupler frequency response and phase shift between port2 and 3

The photograph in Figure 8 shows a fabricated branch-line couper with 4 ports, with total die area (including the capsulated ground plane) of 20 mm × 20 mm. It was manufactured on the FR4 epoxy glass cloth copper-clad plat (CCL) PCB substrate having thickness 0.8mm and dielectric constant εr=4.4.

Figure 7: 
Simulated and measured frequency performance

Figure 8: 
A photograph of the fabricated branch-line coupler

The measured performances of the coupler at 1GHz is compared as shown in Figure 7. It shows equal power splitting performance and good phase response. The insertion loss of simulation results by HFSS is –3.62 dB and measurement result is –4.39 dB. The measured insertion loss is too lossy because of surface and edge roughness of the metal, inferior metal conductivity, the inexactness of fabrication technology and so on, which can be improved by better craftsmanship. The branch-line coupler operating at 1 GHz exhibits a band width of 130MHz and 90^o ± 6^o degree phase difference.