DM/CM equivalent circuit analysis

Under the excitation of DM signals, a virtual short appears along the symmetric line as in Fig. 2. Then, the odd-/even- mode equivalent circuits for the DM are obtained in Fig. 3.

Figure 2.

The DM equivalent circuit of the proposed balanced reflectionless filter.

Figure 3.

The (a) odd-mode, and (b) even-mode equivalent circuits for DM circuit.

According to the voltage/current relationship, the input impedances Z_ino,DM, Z_ine,DM are derived as follow:

(1)\begin{equation}{Z_{ino,DM}} = \frac{{4 + 2{Z_1}({Z_e} + {Z_o})}}{{ - ({Z_e} + {Z_o})\left[ {{Z_{in1}}({Z_e} + {Z_o}) + j2\cot \theta } \right]}}\end{equation}

(2)\begin{equation}{Z_{ine,DM}} = \frac{\begin{array}{l}{a^2}({a^2} - {b^2} - {c^2} - {d^2}) + 2abcd + (2{a^3} - a{b^2} \\[4pt] \qquad - 2a{c^2} - a{d^2} + 2bcd){Z_{in1}} + ({a^2} - {c^2}){Z_{in1}^2}\end{array}}{{(a + {Z_{in1}})({a^2} + a{Z_{in1}} - {b^2})}}\end{equation}

where

(3a)\begin{equation}a = - j\frac{{{Z_{e1}} + {Z_{o1}}}}{2}\cot \theta \end{equation}

(3b)\begin{equation}b = - j\frac{{{Z_{e1}} - {Z_{o1}}}}{2}\cot \theta \end{equation}

(3c)\begin{equation}c = - j\frac{{{Z_{e1}} - {Z_{o1}}}}{2}\csc \theta \end{equation}

(3d)\begin{equation}d = - j\frac{{{Z_{e1}} + {Z_{o1}}}}{2}\csc \theta \end{equation}

(4)\begin{equation}{Z_{in1}} = j{Z_1}\tan \theta \end{equation}

To realize the passband of DM signals, S _dd11 = 0 needs to be met. Therefore, one transmission pole can be obtained by imposing Z_ino,DM × Z_ine,DM = Z ₀² [Reference Matthaei, Young and Jones23]:

(5)\begin{equation}{\theta _{TP1}} = \frac{\pi }{2}\end{equation}

Under the excitation of CM signals, a virtual open appears along the symmetric line as in Fig. 4. Then, the odd-/even- mode equivalent circuits for the CM are obtained in Fig. 5. The input impedances Z_ino,CM, Z_ine,CM are derived as

(6)\begin{equation}{Z_{ino,CM}} = \frac{{4{Z_e}^2{Z_o}^2 - j2{Z_{in2}}{Z_e}{Z_o}({Z_e} + {Z_o})\cot \theta }}{{{Z_{in2}}{{({Z_e} + {Z_o})}^2} - j2{Z_e}{Z_o}({Z_e} + {Z_o})\cot \theta }}\end{equation}

(7)\begin{equation}{Z_{ine,CM}} = \frac{{{{({Z_e} + {Z_o})}^2} - j2{Z_{in2}}({Z_e} + {Z_o})\cot \theta }}{{4{Z_{in2}} - j2({Z_e} + {Z_o})\cot \theta }}\end{equation}

Figure 4.

The CM equivalent circuit of the proposed balanced reflectionless filter.

Figure 5.

The (a) odd-mode, and (b) even-mode equivalent circuits for CM circuit.

where

(8)\begin{equation}{Z_{in2}} = \frac{{(2R + j{Z_1}\tan \theta ){Z_1}}}{{{Z_1} + j2R\tan \theta }}\end{equation}

To meet S _cc21 = 0, Z_ino,CM = Z_ine,CM should be satisfied [Reference Matthaei, Young and Jones23]. Two transmission zeros are obtained as

(9)\begin{equation}{\theta _{TZ1}} = 0,{\theta _{TZ2}} = \pi \end{equation}

To enhance CM absorption, S _cc11 = 0 should be satisfied. By imposing Z_ino,CM × Z_ine,CM = Z ₀² [Reference Hsiao, Cheng and Wu9], the value R of resistor is extracted as

(10)\begin{equation}R = \frac{{{Z_1}^2{Z_0}}}{{2{Z_e}{Z_o}}}\end{equation}

Proposed balanced reflectionless filter

Figure 6 illustrates the influences of the coupling coefficient k (k = [(Z_e-Z_o)/(Z_e + Z_o)]) on the proposed filter. As k increases, the bandwidth and the selectivity for DM improves, whereas the out-of-band rejection decreases. The two transmission zeros aside the passband slightly moves as k changes. It indicates that k affects more on the DM bandwidth. On the other hand, as k increases, the CM reflectionless bandwidth remains relatively stable, while the absorption around center frequency gets enhanced. For S _cc21, the suppression level is gradually decreased. The two transmission zeros fixedly locate at DC and 2f ₀, which only depends on the center frequency and hardly influence the CM bandwidth. Therefore, the value of k for the half-wavelength ring resonator should be carefully designed to achieve initial response at first. Then, Z ₁ needs to be adjusted to further realize satisfied DM passband and CM suppression performances. Ultimately, the R value can be determined using equation (10) to effectively absorb CM noise.

Figure 6.

The (a) DM, (b) CM responses of the proposed filter with k (k ₁ = 0.29: z_e = 100 Ω, z_o = 55 Ω; k ₂ = 0.38: z_e = 107 Ω, z_o = 48 Ω; k ₃ = 0.47: z_e = 110 Ω, z_o = 40 Ω).

To provide specific guidelines, designing the proposed reflectionless filter requires following procedures: (1) Select center frequency f ₀ and determine the demanded bandwidth; (2) Choose proper impedance (Z_e, Z_o, Z ₁) of the transmission lines and adjust to obtain fundamental property; (3) Derive the resistor value R to achieve CM suppression and absorption properties; (4) Obtain layout dimensions and realize physically; (5) Optimize the layout parameters. Moreover, tight coupling is needed for this design that the spacing between coupled lines is relatively small, which requires higher processing capability. Through optimization, the final circuit parameters are: Z_e = 107 Ω, Z_o = 48 Ω, Z ₁ = 84 Ω, Z ₀ = 50 Ω, R = 50 Ω, θ = 90°.