The divider is the most widely used passive component in the wireless industry. In many cases, the same splitting divider is used [1, 2]. However, in special applications, cases that require unequal distribution [3, 4] or various termination impedances [5, 6] exist. To achieve a divider with unequal or various terminated impedances, a high-impedance transmission line must be implemented. Such a high-impedance transmission line is difficult to implement because of its very narrow line width that must be implemented through microstrip technology. Moreover, to satisfy the characteristics of the high-impedance line, a bulky capacitor or coupled lines with narrow gaps are used.

Recently, a new design method has been proposed to adjust only the electrical length of the transmission lines between the input and output ports, or between the output ports and the isolation resistor, to achieve the operation of an unequal divider. Many dividers using a uniform transmission line have been introduced, such as a divider with uniform transmission lines of various electrical lengths [7–9], a structure using an isolated resistor with open-stub connection to obtain enhanced bandwidth [10], an isolated resistor configuration connected to the same impedance lines between the output ports with long stub length [11], a structure connected to various impedance lines at both sides of an isolation resistor [12], a Gysel divider with a phase shifter instead of a 180° transmission line [13], a ring hybrid coupler with a 50-Ω transmission line [14], and a half mode substrate integrated waveguide (HMSIW) divider with composite right and left-handed transmission line (CRLH-TL) [15]. This type of splitter uses a transmission line with uniform impedance, and it does not use an impedance transformer to match each port, even though it operates as an unequal divider. Such a divider consists of circuits with an arbitrary power split ratio and a termination impedance of 50 Ω.

In this paper, we propose a modified Wilkinson divider using a uniform transmission line with various termination impedances and an arbitrary dividing ratio as well as the previously used 50 Ω termination impedance. This type of device eliminates the requirement for a matching circuit between the divider and the peripheral device with various termination impedances, and allows a small circuit size to be achieved.

Fig. 1 shows the schematic of the proposed power divider with power splitting ratio k² (= P₂/P₃). This proposed divider consists of three transmission lines with uniform impedance Z_uL and various electrical lengths of θ_1L, θ_2L, and θ_3L, an isolation resistor R_iso, and various terminated impedances of R_aT, R_bT, and R_cT.

In addition, this divider must satisfy the S-parameter characteristics of the unequal Wilkinson power divider:

where β is the phase shift of the transmission coefficient.

Because the proposed power divider with various terminated impedances is asymmetrical, we analyzed it using scattering parameters rather than the conventional even-odd method. When port 1 is excited, all power is transmitted to the output ports, P₂ and P₃, and the voltage from the branch of P₁ – P₂ to ground is the same as that from the branch of P₁ – P₃ to ground at any distance from P₁, and no current flows in the isolation resistor. Because the isolation resistor operates as an open circuit, we can design an equivalent circuit between ports 2 and 3, as shown in Fig. 2(a). The ABCD parameters between port 1 and ports 2 and 3 can be expressed as

The ABCD parameters of Eqs. (2) and (3) can be converted into the S-parameters of S₂₁ and S₃₁. Using the relation of S₂₁ =k·S₃₁, the following related equations are then obtained.

In Fig. 2(a), under the input matching condition (S₁₁ = 0), we have

where Z_2e and Z_3e are the input impedances of the upper and lower branches, respectively.

Based on the principle of conservation of energy and ideal transmission lines, the real parts of Z_2e and Z_3e are expressed as follows:

In Fig. 2(b), when port 2 is excited, the networks of Net1 and Net2 are connected in parallel, in which Net1 consists of the isolation resistor R_iso and the transmission line of electrical length θ_3L; and Net2 consists of a termination resistor R_aT and transmission lines of electrical lengths θ_1L, θ_2L. The ABCD parameters of Net1 and Net2 can be expressed as:

The ABCD parameters of Eqs. (11) and (12) are converted into the Y-parameters of Net1 and Net2, respectively, and the admittance parameters of the entire network can be obtained as follows:

where

The admittance parameters of Eq. (13) can be converted to the S-parameters of S₂₂, S₃₂, and S₃₃ with terminated impedances of R_bT and R_cT. The S-parameters of the entire network between ports 2 and 3 can be expressed as

Based on Eqs. (4) to (5), (9) to (10), and (14) to (16), the electrical lengths θ_1L, θ_2L, θ_3L, and isolation resistance R_iso that satisfy the power divider characteristic conditions with |S₂₂| < −20 dB, |S₃₃| < −20 dB, and |S₃₂| < −20 dB at center frequency can be obtained by using MATLAB.

To validate the proposed power divider, we designed two types of circuits at a center frequency of 2 GHz. One has a power dividing ratio of k² = 2 and port impedances of R_aT = 40 Ω, R_bT = 70 Ω, and R_cT = 60 Ω. The second circuit has a power dividing ratio of k² = 4 and port impedances of R_aT = 50 Ω, R_bT = 70 Ω, and R_cT = 60 Ω. For the first circuit, when the transmission line characteristic impedance of Z_uL = 40 Ω was chosen, we calculated the electrical lengths and isolation resistance using the equations above and after optimization as follows: θ_1L = 157^o, θ_2L = 146^o, θ_3L = 47^o, and R_iso = 12 Ω. For the second circuit, when a transmission line characteristic impedance of Z_uL = 40 Ω was chosen, we calculated the electrical lengths and isolation resistance using the equations above and after optimization as follows: θ_1L = 153°, θ_2L = 130°, θ_3L = 66°, and R_iso = 20 Ω. The Teflon substrate of the proposed power divider had a dielectric constant of 2.5, a thickness of 0.787 mm, and a conductor thickness of 0.035 mm.

The simulation was performed using Microwave Office software with version 13 developed by National Instruments.

Fig. 3(a) and (b) show the photographs of the circuits in which k² = 2 and k² = 4, respectively, where power dividers of various port impedances and uniform transmission lines are implemented. For the measurement, the impedance transformers shown in Fig. 3 were used to match the input and output ports to 50 Ω.

Fig. 4(a) and (b) show the measured and simulated S-parameters of the circuit with a power dividing ratio of k² = 2 and port impedances of 40, 70, and 60 Ω; the figures show insertion losses of |S₂₁| = 2.0 dB and |S₃₁| = 5.0 dB, an isolation of |S₃₂| > 25 dB, an input return loss of |S₁₁| that is better than −20 dB, and output return losses of |S₂₂|, |S₃₃| that are better than −18 dB at the center frequency of 2 GHz. In addition, Fig. 5(a) and (b) show the measured and simulated S-parameters of the circuit with a power dividing ratio of k² = 4 and port impedances of 50, 70, and 60 Ω; the figures show insertion losses of |S₂₁| = 1.2 dB and |S₃₁| = 6.8 dB, an isolation of |S₃₂| > 25 dB, an input return loss of |S₁₁| that is better than −25 dB, and output return losses of |S₂₂|, |S₃₃| that are better than −25 dB, −13 dB, respectively, at the center frequency of 2 GHz. In Figs. 4 and 5, the |S₃₃| data can be observed with slight frequency deviation, which is caused by the parallel admittance of electrical length θ_3L. Fig. 6 shows that the phase difference between the output ports of the k² = 2 circuit is +3° at the center frequency of 2 GHz. Table 1 shows a comparison of dividers using the conventional uniform transmission line and the results obtained for the proposed divider. In addition, Table 2 summarizes the experimental results and design parameters of the proposed power divider.

This paper presented a modified Wilkinson divider using uniform transmission lines for various terminated impedances and an arbitrary dividing ratio. With this configuration, the desired splitting ratio can be obtained by adjusting only the electrical length of the transmission lines between the ports. Moreover, it has the advantage that the impedance of the ports is set to various terminated impedances, and is connected to a circuit without a matching circuit. The feasibility of the proposed power divider design concept was demonstrated, and the simulated and measured results were confirmed to be in good agreement.