A Novel Synthetization Approach for Multi Coupled Line Section Impedance Transformers in Wideband Applications

Abstract

In this paper, a novel synthetization approach is proposed for filter-integrated wideband impedance transformers (ITs). The original topology consists of N cascaded coupled line sections (CLSs) with 2N characteristic impedance parameters. By analyzing these characteristic impedances, a Chebyshev response can be derived to consume N + 2 design conditions. To optimize the left N − 2 variable parameters, CLSs were newly substituted by transmission lines (TLs) to consume the remaining variable parameters and simplify the circuit topology. Therefore, there are totally 2N − N − 2 substituting possibilities. To verify the proposed approach, 25 cases are listed under the condition of N = 5, and 7 selected cases are compared and discussed in detail. Finally, a 75–50 Ω IT with 100% fractional bandwidth and 20 dB bandpass return loss (RL) is designed and fabricated. The measured results meet the circuit simulation and the EM simulation accurately.

1. Introduction

Impedance transformers (ITs) have been widely used in microwave circuits and systems as a basic circuit component [1]. A diagram of the typical end-fed antenna system is shown in Figure 1, where the IT is used to match the antenna impedance to the coax line impedance (usually 50 Ω). Many IT and filter structures have been developed in recent years [2,3,4,5,6,7,8,9,10,11,12,13], and multi-function ITs which are integrated by filters [14,15,16,17,18,19,20], phase shifters [21], power dividers [22], and antennas [23] have also been widely researched.

In [2,3], a quarter-wave IT is presented using transmission lines (TLs) for single-band applications with a limited operating range. Several IT structures have been designed to meet the requirements of multi-band communication systems [4,5,6,7,8,9], in which multi-section structures are an effective way to design multi-band systems and match impedance [4,5,6,7]. Inserting open/short stubs is also available to realize impedance matching [8,9].

In wideband IT applications, filter-integrated ITs have been proposed and developed in recent years to enhance the out-band suppression and reduce the circuit size [14,16,17,18,19,20]. In [14], the lumped elements are used to realize wideband response. However, lumped elements are limited in high-frequency applications, and the soldering will also involve undesirable parasitic effects. Instead of using lumped components, open/short stubs can be used to provide better high-frequency performance for the filtering ITs [16,17,18,19,20]. The equal-ripple response can be realized by lumped elements and J-inverters [24,25,26,27]. A Chebyshev polynomial is also used to constrain the equal-ripple response in filter and IT designs [28,29,30]. To realize DC block and reduce the circuit size, coupled line sections (CLSs) are widely used in wideband ITs and filters [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. For example, two CLSs are added to each terminal of the stepped-impedance TLs symmetrically to integrate the filtering function [30].

To the best of the author’s knowledge, there still are some relationships among these transformers and filters that are not mentioned. For example, in [26,28,29,30,33], filters and ITs with a Chebyshev response are constituted by cascaded CLSs and TLs. Although the similar design synthesis was reported, the numbers and positions of CLSs and TLs were not clear.

In this paper, a novel synthetization approach is presented for filter-integrated multi-section wideband ITs. From the point of view of the number of variable parameters, the wideband IT, which consists of N cascaded CLSs, is provided with a Chebyshev performance, where N + 2 design conditions are consumed, and N − 2 variable parameters remain. To further optimization, CLSs are newly substituted by TLs to consume the remaining variable parameters and simplify the circuit topology. Therefore, there are 2^N − N − 2 substituting possibilities in total for N-section ITs, and the design of the IT can be more flexible for both symmetric and asymmetric structures with the same wideband filtering performance.

2. The Principle of the Synthetization Approach

2.1. The Number of Variable Parameters of Conventional N-Section IT

Figure 2 shows the topology of N-section IT. It consists of N cascaded CLSs with the electrical length θ, where Z_evi and Z_odi are the even-and odd-mode characteristic impedances of the ith CLS, and i = 1,2, …, N. Z_S and Z_L are two different terminal loads, and its impedance ratio k is defined as k = Z_S/Z_L, Z_L = 1 is defined in this paper; thus, k > 1.

Based on [28], the ABCD matrix of the ith CLS can be summarized as

$$\left[\begin{array}{cc}{A}_i& B_i\\ C_i& D_i\end{array}\right]=\frac{\sin \theta }{T_i}\left[\begin{array}{cc}q{S}_i& \frac{j}{2}\begin{array}{c}[T_i^2+q^2(T_i^2-S_i^2)]\end{array}\\ 2j& q{S}_i\end{array}\right]$$

(1)

where q = cotθ, and

$$S_i=Z_{evi}+Z_{odi}$$

(2)

$$T_i=Z_{evi}-Z_{odi}$$

(3)

Then, the ABCD matrix of N-section filter-integrated IT can be written as

$$\left[\begin{array}{cc}{A}_T& B_T\\ C_T& D_T\end{array}\right]=\left[\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right]\left[\begin{array}{cc}{A}_2& B_2\\ C_2& D_2\end{array}\right]\cdots \left[\begin{array}{cc}{A}_i& B_i\\ C_i& D_i\end{array}\right]\cdots \left[\begin{array}{cc}{A}_{N-1}& B_{N-1}\\ C_{N-1}& D_{N-1}\end{array}\right]\left[\begin{array}{cc}{A}_N& B_N\\ C_N& D_N\end{array}\right]$$

(4)

where

$$A_T={\displaystyle \sum a_p{\cos }^p\theta }$$

(5)

$$B_T=\frac{j}{\sin \theta }{\displaystyle \sum b_q{\cos }^q\theta }$$

(6)

$$C_T=\frac{j}{\sin \theta }{\displaystyle \sum c_q{\cos }^q\theta }$$

(7)

$$D_T={\displaystyle \sum d_p{\cos }^p\theta }$$

(8)

a_p and d_p are polynomial functions with degrees p. b_q and c_q are polynomial functions with degrees q. p and q defined in Equation (9). All polynomial functions can be determined by the even-and odd-mode characteristic impedances, respectively.

$$\left\{\begin{array}{c}p=1,3,5,\cdots ,Nq=0,2,4,\cdots ,N+1\hspace{0.17em}\mathrm{when}\hspace{0.17em}N\hspace{0.17em}\mathrm{is}\hspace{0.17em}\mathrm{odd}\hspace{0.17em}\mathrm{number}\\ p=0,2,4,\cdots ,Nq=1,3,5,\cdots ,N+1\hspace{0.17em}\mathrm{when}\hspace{0.17em}N\hspace{0.17em}\mathrm{is}\hspace{0.17em}\mathrm{even}\hspace{0.17em}\mathrm{number}\end{array}\right.$$

(9)

S-parameters of the N-section IT are

$$S_{11}=\frac{A_T{Z}_L+B-C{Z}_S{Z}_L-D{Z}_S}{A_T{Z}_L+B+C{Z}_S{Z}_L+D{Z}_S}$$

(10)

$$S_{21}=\frac{2\sqrt{Z_S{Z}_L}}{A_T{Z}_L+B+C{Z}_S{Z}_L+D{Z}_S}$$

(11)

The amplitude-squared transfer function can be expressed as

$$|S_{21}^{circuit}(\theta ){|}^2=\frac{1}{1+|F_{circuit}{|}^2}=\frac{1}{1+{\left|(A_T+B_T-k{C}_T-k{D}_T)/2\sqrt{k}\right|}^2}$$

(12)

where F_circuit is the characteristic function of the circuit:

$$F_{circuit}={\displaystyle \sum (a_p-k{d}_p){\cos }^p\theta /2\sqrt{k}}+\frac{j}{\sin \theta }{\displaystyle \sum (b_q-k{c}_q){\cos }^q\theta /2\sqrt{k}}$$

(13)

Conversely, based on the synthesis approach in [46], firstly, we define two variables: x = cosφ = α cosθ = cosθ/cosθ_c, y = cosξ = x sinθ_c/sinθ, where θ_c is the electrical length at the lower cutoff frequency fc of the passband, and α is a quantity used to determine the bandwidth of a bandpass filter. The Chebyshev transfer function can be written as

$$|S_{21}^{formula}(\theta ){|}^2=\frac{1}{1+{\left|F_{formula}\right|}^2}=\frac{1}{1+{\epsilon }^2{\cos }^2(N\varphi +\xi )}$$

(14)

where $\epsilon =\sqrt{{10}^{0.1L_A}-1}$, and L_A is the in-band ripple level factor.

The characteristic function of Chebyshev formula F_formula can be rewritten as:

$$F_{formula}=\epsilon \frac{\Sigma u_q{\cos }^q\theta }{\sin \theta }$$

(15)

From F_circuit = F_formula, the general design conditions of N-section IT are summarized as

$$\left\{\begin{array}{c}{a}_p-k{d}_p=0\\ (b_q-k{c}_q)/2\sqrt{k}=\epsilon u_q\end{array}\right.$$

(16)

Therefore, there are 2N parameters of characteristic impedances in the N-section IT, where N + 2 design conditions are used to derive the Chebyshev response in Equation (16), and N − 2 variable parameters can be designated arbitrarily.

2.2. The Proposed Synthesization Approach for Simplification

To simplify the conventional topology [28] and decrease the number of extra variable parameters, TLs were used to substitute CLSs in this paper.

In Figure 3, the ith CLS is substituted by a single TL for example, where Z_i and θ are the characteristic impedance and electrical length of the single TL. Z_evi = 2Z_i and Z_odi = 0 must be maintained. Then, the ABCD matrix of TL can be rewritten as

$$\left[\begin{array}{cc}{A}_i& B_i\\ C_i& D_i\end{array}\right]=\left[\begin{array}{cc}\cos \theta & j\sin \theta Z_{evi}/2\\ 2j\sin \theta /Z_{evi}& \cos \theta \end{array}\right]$$

(17)

The ABCD matrix form of the total topology will not be changed. In other words, when any single CLS (ith CLS) can be substituted by a single TL, Equation (4) can also be derived, and the characteristic impedances in all polynomial functions (a_p, b_p, c_p and d_p) will be slightly different because of Z_evi = 2Z_i and Z_odi = 0. Meanwhile, one variable parameter will be consumed, and the number of variable parameters will decrease to N − 3. Furthermore, the consumption of more variable parameters, different positions, and more CLSs can be substituted by TLs, where the maximum number of substituted TLs is no more than N − 2.

Based on the discussion above, the general procedure of the proposed synthetization approach in Figure 4 is summarized as follows: (1) according to the actual requirements, determine the section number N in the conventional IT; (2) determine the number of CLSs needing to be substituted; (3) select the substituting positions of the CLSs; (4) calculate the ABCD matrix; (5) calculate all the characteristic impedances by the Chebyshev response; (6) if the results cannot meet the design requirements, go back to (2) and re-design the circuit.

In summary, for N-section IT (N ≥ 2) in Figure 2, the basic principle of substituting TL for CLS is summarized as follows:

(I) There are 2N parameters in the N-section IT [28];

(II) N + 2 design conditions are used to derive the Chebyshev response; thus, N − 2 variable parameters can be designated arbitrarily;

(III) Consuming one variable parameter, a single TL could substitute one CLS to maintain the Chebyshev response. In other words, maximum number of TLs will reach to N − 2;

(IV) After running through all the positions of substituted TLs in every possible combination, the number of the substitutable cases are listed in

Table 1. The number of substitutable cases for N-section IT.

Number of Substituted CLSs by TLs	Number of Substitutable Combinations
1	$C_N^1$
2	$C_N^2$
…	…
N − 3	$C_N^{N-3}$
N − 2	$C_N^{N-2}$

, and the total number is:

$$C_N^1+C_N^2+\dots +C_N^{N-2}=2^N-N-2$$

(18)

3. Design Examples for the Proposed Synthetization Approach

Several substituting cases have been discussed in previous works (N = 4) [28,30]. To demonstrate the proposed synthetization approach, N = 5 was selected to design the N-section IT. By using the proposed synthetization approach for simplification, there were 25 cases in total. All the cases are listed and compared in this section. In order words, refs. [28] and [30] are two cases in the proposed synthetization approach; the former have not yet been reported.

3.1. Design Example for the Conventional 5-Section IT

The topology of 5-section IT is shown in Figure 5. From Equation (4), A_T, B_T, C_T and D_T can be calculated as follows:

$$A_T=a_5{\cos }^5\theta +a_3{\cos }^3\theta +a_1\cos \theta$$

(19)

$$B_T=\frac{j}{\sin \theta }(b_6{\cos }^6\theta +b_4{\cos }^4\theta +b_2{\cos }^2\theta +b_0)$$

(20)

$$C_T=\frac{j}{\sin \theta }(c_6{\cos }^6\theta +c_4{\cos }^4\theta +c_2{\cos }^2\theta +c_0)$$

(21)

$$D_T=d_{5}co{s}^5\theta +d_{3}co{s}^3\theta +d_{1}cos\theta$$

(22)

a_p, d_p (p = 1, 3, 5), b_q and c_q (q = 0, 2, 4, 6) are the coefficients of the polynomial functions which are determined by characteristic impedances of Z_ev1, Z_od1, Z_ev2, Z_od2, Z_ev3, Z_od3, Z_ev4, Z_od4, Z_ev5 and Z_od5 (the detailed equations are listed in

Table A1. Detailed parameter for the 5-section IT [28].

$a_5=\frac{1}{T_1{T}_2{T}_3{T}_4{T}_5}[S_1(S_1+S_2)(S_2+S_3)(S_3+S_4)(S_4+S_5)]$ $a_3=-\frac{1}{T_1{T}_2{T}_3{T}_4{T}_5}[(S_2+S_3)(S_3+S_4)(S_4+S_5)T_1^2+S_1(S_3+S_4)(S_4+S_5)T_2^2+S_1(S_1+S_2)(S_4+S_5)T_3^2+S_1(S_1+S_2)(S_2+S_3)T_4^2]$ $a_1=\frac{1}{T_1{T}_2{T}_3{T}_4{T}_5}[S_1{T}_2^2{T}_4^2+(S_2+S_3)T_1^2{T}_4^2+(S_4+S_5)T_1^2{T}_3^2]$

$b_6=-\frac{1}{2T_1{T}_2{T}_3{T}_4{T}_5}[S_1{S}_5(S_1+S_2)(S_2+S_3)(S_3+S_4)(S_4+S_5)]$ $\begin{array}{l}{b}_4=\frac{1}{2T_1{T}_2{T}_3{T}_4{T}_5}[S_5(S_2+S_3)(S_3+S_4)(S_4+S_5)T_1^2+S_1{S}_5(S_3+S_4)(S_4+S_5)T_2^2+S_1{S}_5(S_1+S_2)(S_4+S_5)T_3^2+S_1{S}_5(S_1+S_2)(S_2+S_3)T_4^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+S_1(S_1+S_2)(S_2+S_3)(S_3+S_4)T_5^2]\hfill \end{array}$ $b_2=-\frac{1}{2T_1{T}_2{T}_3{T}_4{T}_5}[(S_2+S_3)(S_3+S_4)T_1^2{T}_5^2+S_5(S_4+S_5)T_1^2{T}_3^2+S_5(S_2+S_3)T_1^2{T}_4^2+S_1(S_1+S_2)T_3^2{T}_5^2+S_1(S_3+S_4)T_2^2{T}_5^2+S_1{S}_5{T}_2^2{T}_4^2]$ $b_0=\frac{T_1{T}_3{T}_5}{2T_2{T}_4}$

$c_6=-\frac{2}{T_1{T}_2{T}_3{T}_4{T}_5}(S_1+S_2)(S_2+S_3)(S_3+S_4)(S_4+S_5)$ $c_4=\frac{2}{T_1{T}_2{T}_3{T}_4{T}_5}\{(S_1+S_2)(S_2+S_3)(S_3+S_4)(S_4+S_5)+(S_3+S_4)(S_4+S_5)T_2^2+(S_1+S_2)(S_4+S_5)T_3^2+(S_1+S_2)(S_2+S_3)T_4^2\}$ $c_2=-\frac{2}{T_1{T}_2{T}_3{T}_4{T}_5}[(S_3+S_4)(S_4+S_5)T_2^2+(S_1+S_2)(S_4+S_5)T_3^2+(S_1+S_2)(S_2+S_3)T_4^2+T_2^2{T}_4^2]$ $c_0=\frac{2T_2{T}_4}{T_1{T}_3{T}_5}$

$d_5=\frac{1}{T_1{T}_2{T}_3{T}_4{T}_5}{S}_5(S_1+S_2)(S_2+S_3)(S_3+S_4)(S_4+S_5)$ $d_3=-\frac{1}{T_1{T}_2{T}_3{T}_4{T}_5}[S_5(S_3+S_4)(S_4+S_5)T_2^2+S_5(S_1+S_2)(S_4+S_5)T_3^2+S_5(S_1+S_2)(S_2+S_3)T_4^2+(S_1+S_2)(S_2+S_3)(S_3+S_4)T_5^2]$ $d_1=\frac{1}{T_1{T}_2{T}_3{T}_4{T}_5}[S_5{T}_2^2{T}_4^2+(S_1+S_2)T_3^2{T}_5^2+(S_3+S_4)T_2^2{T}_5^2]$

in Appendix A).

After the complicated arithmetical operation from Equations (10) to (13), F_circuit can be written as

$$\begin{array}{c}{F}_{circuit}=\frac{1}{T_1{T}_2{T}_3{T}_4{T}_5}(X_5{\cos }^5\theta +X_3{\cos }^3\theta +X_1\cos \theta )\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{j}{T_1{T}_2{T}_3{T}_4{T}_5\sin \theta }(Y_6{\cos }^6\theta +Y_4{\cos }^4\theta +Y_2{\cos }^2\theta +Y_0)\end{array}$$

(23)

where

$$X_p=(a_p-k\cdot d_p)/2\sqrt{k}$$

(24)

$$Y_q=(b_q-k\cdot c_q)/2\sqrt{k}$$

(25)

Similarly, from Equations (10) to (13), F_formula can be written as

$$F_{formula}=\epsilon \cos (5\phi +\xi )=\frac{\epsilon }{\sin \theta }[u_6{\cos }^6\theta +u_4{\cos }^4\theta +u_2{\cos }^2\theta +u_0]$$

(26)

where

$$u_6=\frac{16}{{\cos }^6{\theta }_c}(1+\sqrt{1-{\cos }^2{\theta }_c})$$

(27)

$$u_4=-\frac{4}{{\cos }^4{\theta }_c}(5+7\sqrt{1-{\cos }^2{\theta }_c})$$

(28)

$$u_2=\frac{1}{{\cos }^2{\theta }_c}(5+13\sqrt{1-{\cos }^2{\theta }_c})$$

(29)

$$u_0=-1$$

(30)

From F_circuit = F_formula, the seven design conditions can be summarized as follows to maintain the Chebyshev response.

$$\left\{\begin{array}{l}{a}_5-k\cdot d_5=0\\ a_3-k\cdot d_3=0\\ a_1-k\cdot d_1=0\end{array}\right.\hspace{1em}\mathrm{and}\hspace{1em}\left\{\begin{array}{l}(b_6-k\cdot c_6)/2\sqrt{k}=\epsilon u_6\\ (b_4-k\cdot c_4)/2\sqrt{k}=\epsilon u_4\\ (b_2-k\cdot c_2)/2\sqrt{k}=\epsilon u_2\\ (b_0-k\cdot c_0)/2\sqrt{k}=\epsilon u_0\end{array}\right.$$

(31)

For the section number N = 5, there are 10 parameters (Z_ev1, Z_od1, Z_ev2, Z_od2, Z_ev3, Z_od3, Z_ev4, Z_od4, Z_ev5 and Z_od5) corresponding to seven design conditions for the Chebyshev response; thus, three variable parameters can be utilized to simplify the circuit topology.

3.2. Design Example for Proposed Synthesization Approach

Since there were 3 variable parameters, different 25 cases could be divided into three categories, Categories I, II and III, which will be discussed in this section.

3.2.1. Category I: One CLS Is Substituted by TL

When one CLS is substituted by TL, there are two variable parameters, and the number of the substitutable cases is $C_5^1=5$. Any of the CLSs (first, second, third, fourth, fifth) can be substituted by TL. All the possible cases are listed in

Table 2. The substitutable cases in category I.

The ith CLS Substituted By TL	1st	2nd	3rd	4th	5th
Index of cases	Case 1	Case 2	Case 3	Case 4	Case 5

. This is the simplest category; therefore, the detailed discussion will be omitted.

3.2.2. Category II: Two CLSs Are Substituted by TL

When two CLSs are substituted by TLs, there is only one variable parameter left. The number of substitutable cases is $C_5^2=10$, and all the cases are listed in

Table 3. The substitutable cases in category II.

	The ith CLS Is Substituted by TL
The jth CLS is substituted by TL		1st	2nd	3rd	4th	5th
	1st		Case 6	Case 7	Case 8	Case 9
	2nd			Case 10	Case 11	Case 12
	3rd				Case 13	Case 14
	4th					Case 15
	5th

. Case 11 is selected as an example and discussed in detail. The topology of case 11 is shown in Figure 6. For fabrication convenience, the odd-mode characteristic impedance of the third CLS is fixed to 0.6 Ω (Z_od3 = 0.6 Ω). Since the second and fourth CLSs are substituted by TLs, Z_ev2 = 2Z₂, Z_od2 = 0 and Z_ev4 = 2Z₄, Z_od4 = 0 can be derived.

Then, the relationship among the characteristic impedances of CLSs (Z_evi and Z_odi) and TLs (Z_i), impedance ratio k (where 1.5 < k < 5.0), and electrical length θ_c (where 30° < θ_c < 60°) are shown in Figure 7. As k increase and θ_c decrease, the coupling strengths C_i of all CLSs will become stronger, where C_i = −20log(Z_evi − Z_odi)/(Z_evi + Z_odi), i = 1, 3, and 5. Therefore, the sum of all coupling strengths, C_∑, will become smaller, where C_∑ = C₁ + C₃ + C₅, and the availability section is about 1.5 < k < 3.5 and 30° < θ_c < 60°

3.2.3. Category III: Three CLSs Are Substituted by TL

When three CLSs are substituted by TLs, there is no variable parameter. The number of substitutable cases is $C_5^3=10$, and all the cases are listed in

Table 4. The substitutable cases of category III.

	The ith CLS Is Not Substituted by TL
The jth CLS is Not substituted by TL		1st	2nd	3rd	4th	5th
	1st		Case 16	Case 17	Case 18	Case 19
	2nd			Case 20	Case 21	Case 22
	3rd				Case 23	Case 24
	4th					Case 25
	5th

. Case 21 is selected as an example and discussed in detail. The topology of case 21 is shown in Figure 8. Since the first, third, and fifth CLSs are substituted by TLs, Z_ev1 = 2Z₁, Z_od1 = 0; Z_ev3 = 2Z₃, Z_od3 = 0; and Z_ev5 = 2Z₅, Z_od5 = 0 can be derived.

The relationships among characteristic impedances of CLSs (Z_evi and Z_odi) and TLs (Z_i), impedance ratio k (where 1.5 < k < 5.0), and electrical length θ_c (where 30° < θ_c < 60°) are shown in Figure 9. The same results can be summarized accordingly: as k increases and θ_c decreases, the coupling strengths C₂ and C₄ will become stronger, and C_∑ will become smaller, where C_∑ = C₂ + C₄. The first, third, and fifth CLSs are substituted by TLs; therefore, the availability section is narrower than case 11, which is about 1.5 < k < 2.0 and 30° < θ_c < 35°.

3.3. Comparison and Summary

To compare these three categories, seven selected examples are listed in

Table 5. Typical simplification examples with detailed parameters, under the condition of Z_S = 1.5 Ω, Z_L = 1.0 Ω, k = 1.5, θ = 90 at f₀ = 2 GHz, θ_c = 45°, RL = 20 dB.

Number of Variable Parameters	Substitutable Cases	Characteristic Impedances of theith CLSs or TLs (Ω)					Symmetrical/ Asymmetrical Topology
		1st	2nd	3rd	4th	5th
3	Conventional topology [28] Former work	CLS Zev1 = 4.1030 Zod1 = 0.4371	CLS Zev2 = 4.7546 Zod2 = 0.6000	CLS Zev3 = 4.7474 Zod3 = 0.6000	CLS Zev4 = 4.2000 Zod4 = 0.6000	CLS Zev5 = 2.8441 Zod5 = 0.1826	Symmetrical
2	Case 2 This work	CLS Zev1 = 3.8756 Zod1 = 0.6643	TL Z2 = 1.5268	CLS Zev3 = 3.8798 Zod3 = 0.6643	CLS Zev4 = 3.9323 Zod4 = 0.6000	CLS Zev5 = 2.8214 Zod5 = 0.2053	Asymmetrical
1	Case 11 in Figure 5 This work	CLS Zev1 = 3.8447 Zod1 = 0.6953	TL Z2 = 1.4813	CLS Zev3 = 3.0487 Zod3 = 0.6000	TL Z4 = 1.1271	CLS Zev5 = 2.6067 Zod5 = 0.4200	Symmetrical
	Case 10 This work	CLS Zev1 = 3.7642 Zod1 = 0.7758	TL Z2 = 1.1076	TL Z3 = 0.9148	CLS Zev4 = 3.0366 Zod4 = 0.6000	CLS Zev5 = 2.7415 Zod5 = 0.2852	Asymmetrical
0	Case 18 This work	CLS Zev1 = 3.8158 Zod1 = 0.7242	TL Z2 = 1.2486	TL Z3 = 1.4696	CLS Zev4 = 5.5788 Zod4 = 1.5118	TL Z5 = 1.5133	Asymmetrical
	Case 19 [30] Former work	CLS Zev1 = 3.7113 Zod1 = 0.8287	TL Z2 = 0.7888	TL Z3 = 0.4592	TL Z4 = 0.6071	CLS Zev5 = 2.4931 Zod5 = 0.5336	Symmetrical
	Case 21 in Figure 7 This work	TL Z1 = 2.2700	CLS Zev2 = 8.5051 Zod2 = 2.1307	TL Z3 = 2.8948	CLS Zev4 = 5.8497 Zod4 = 1.2408	TL Z5 = 1.5133	Symmetrical

under the same design conditions (θ = 90° at f₀ = 2 GHz, k = 1.5, θ_c = 45° and RL = 20 dB). Their circuit simulation results are shown in Figure 10. The following conclusions are clearly summarized as:

(1) Although all the topologies (25 cases) are quite different, by using the proposed synthetization approach, the same Chebyshev frequency responses (S₁₁, S₂₁ and group delay) can be derived.

(2) Symmetrical or asymmetrical topology will not affect the Chebyshev frequency responses, which are only determined by all the characteristic impedances of CLSs and TLs.

(3) When the category is determined, considering coupling strengths and impedance ratio, CLSs are preferred to connect the source and load terminals in symmetrical topology.

(4) More variable parameters will provide more flexible design procedures and wider realization ranges. When the number of variable parameters is zero, the most simplification topology appears with the unique solution of characteristic impedances. Although the realization range is quite limited, the circuit is highly convenient for fabrication.

4. Experimental Results

To validate the proposed synthetization approach, a 75 Ω–50 Ω IT (case 11) was designed, fabricated, and measured, where θ_c = 45°, RL = 20 dB, and the center frequency f₀ = 2 GHz. The characteristic impedances and the physical parameters of the fabricated IT are listed in

Table 6. Parameters of the fabricated IT.

Characteristic Impedance (Ω)	Coupling Strength (dB)	Physical Parameters (mm)
Zev1 = 192.23 Zod1 = 34.77	3.18	W1 = 4.0 L1 = 27.5 S1 = 10.8
Z2 = 69.06		W2 = 1.4 L2 = 27.0
Zev3 = 152.43 Zod3 = 30.00	3.46	W3 = 4.9 L3 = 26.7 S3 = 8.4
Z4 = 56.35		W4 = 2.1 L4 = 26.0
Zev5 = 130.33 Zod5 = 20.10	2.82	W5 = 7.6 L5 = 26.4 S5 = 10.3

and Table A1.

The slot-coupled directional (SCD) structure was used in the circuit fabrication. The 3D view of the proposed IT is shown in Figure 11. The structure was constructed on the Roger’s RT/duroid 5880 substrate with a dielectric constant (ε_r) of 2.2, a loss tangent (tanδ) of 0.0009, and a thickness of 0.787 mm.

Figure 12a,b shows the front- and back-side photographs of the fabricated IT. Its circuit simulation, EM simulation, and measured results are shown in Figure 13. The S₁₁ of EM simulation and measured results were lower than −17.1 dB and −14.5 dB in the passband, respectively. The errors between the EM simulation and measured results were mainly caused by fabrication error and parasitic effect around the terminal 75 Ω resistor. Four reflection zeros within the passband can clearly be observed in the measurement results.

5. Conclusions

In this paper, a novel synthetization approach for the N-section IT has been proposed. Based on the principles of this proposed approach, TLs could substitute CLSs for flexible circuit design, simplified circuit topology, and easy fabrication; moreover, the same Chebyshev response will be maintained. Furthermore, through analyzing the relationship between the number of variable parameters and the number of design conditions, at most, N − 2 CLSs can be substituted by TLs, and the substituting positions can be selected flexibly. Therefore, both symmetric and asymmetric substituting structures can be realized. To validate the proposed theory, a 75 Ω–50 Ω filtering IT was fabricated and measured. The measured results met the circuit simulation and the EM simulation effectively. This work has proposed a novel synthetization approach for IT design, and a detailed mathematical approach has been presented, which can be significant in IT and the design of other components. However, parts of the circuits were not easy to fabricate because of the strong coupled strength. In our future works, we will focus on changing the structure to decrease the coupled strength and provide extra transmission zeros to improve the selectivity of the IT.