# 3.2: Parallel Coupled Lines

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Several of the key advantages of coupled-line filters over other distributed filters is that the coupling between resonators can be strong if the pair of coupled-line resonators are close to each other, or low if they are widely separated. This enables filters to be realized with high \((30–70\%)\), medium \((10–30\%)\), or low \((2–10\%)\) bandwidths. The low-bandwidth filters require low coupling of the resonators, while the high bandwidth filters require high coupling. Coupled-line filters are also compact and can be implemented in microstrip configurations or most of the other transmission line technologies. The different transmission line forms have different \(Q\)s. Planar transmission lines have moderate \(Q\)s especially at high frequencies, due principally to

Figure \(\PageIndex{1}\): Coupled microstrip line layout: (a) schematic; and (b) layout in an EM simulator. Dimensions of the coupled lines are \(w = 500\:\mu\text{m}\), \(s = 100\:\mu\text{m}\), \(\ell = 1\text{ cm}\), \(W = 6\text{ mm}\), and \(L = 12\text{ mm}\). The metal is \(6\:\mu\text{m}\) thick gold (conductivity \(\sigma = 42.6\times 10^{6}\text{ S/m}\)) and the alumina substrate height is \(600\:\mu\text{m}\) with relative permittivity \(\varepsilon_{r} = 9.8\) and loss tangent of \(0.001\).

Figure \(\PageIndex{2}\): Insertion loss \((S_{21})\) and return loss \((S_{11})\) of the coupled line of Figure \(\PageIndex{1}\) with an \(s = 100\:\mu\text{m}\) gap.

Figure \(\PageIndex{3}\): Insertion loss and return loss over a narrow frequency range plotted on a Smith chart. The annotation at the markers is the frequency in GHz.

radiated fields. Low \(Q\) means that the insertion loss of filters will be high and the filter skirts will not be as steep as they would be if a high-\(Q\) transmission line structure was used. For example, up to the early 2000s the RF filters in the front end of cellular phones were mostly seventh-order Chebyshev filters using coupled slablines as shown in Figure \(\PageIndex{8}\) (although here only

Figure \(\PageIndex{4}\): Input impedance at Port \(1\) of the coupled microstrip lines in Figure \(\PageIndex{1}\). Port \(2\) is terminated in \(50\:\Omega\).

Figure \(\PageIndex{5}\): Capacitively coupled microstrip lines: (a) schematic; and (b) representation in a microwave CAD tool with a subcircuit representing the coupled lines. The layout of the subcircuit is shown in Figure \(\PageIndex{1}\).

Figure \(\PageIndex{6}\): Return loss and insertion loss when the coupling capacitors are \(44.8\text{ fF}\). This results in a Butterworth-like response. \(S_{21}'\) is the response without \(C_{1}\) and \(C_{2}\).

four resonators are shown for clarity). Typically the top and bottom plates of the resonators were \(2–3\text{ mm}\) apart and the area of the slabline filter was \(1\text{ cm}\times 1\text{ cm}\). This is too large for today’s thin smart phones. However, they are very good filters and cheap to produce. The electrical design procedure for parallel coupled slabline filters and for using other transmission line structures is the same as for parallel coupled microstrip filters. The difference is only in the final physical implementation.

Figure \(\PageIndex{7}\): Return loss and insertion loss when the coupling capacitors are \(37.6\text{ fF}\). This results in a Chebyshev-like response. \(S_{21}'\) is the response without \(C_{1}\) and \(C_{2}\).

Figure \(\PageIndex{8}\): Slabline: (a) cross section of a single slabline; and (b) parallel coupled-line configuration with four coupled slablines.

## 3.2.1 Coupled-Line Configurations

The coupled-line configuration considered in Example 3.1.1 is called a combline section. This is one of many that have desirable frequency-selective responses. The characteristics of many parallel coupled-line configurations having bandpass, all-pass, or all-stop characteristics are shown in Table \(\PageIndex{1}\). The most important configurations are starred \((⋆)\).

The PCL section of Filter (a) in Table \(\PageIndex{1}\) is called an interdigital section and is a bandpass filter. The dual filter is Filter (b), with what is called a parallel coupled section, and both can be used to realize narrow \((2–3\%)\) to wide (up to \(30\%\)) bandwidth filters with close to symmetrical responses around the center frequency \(f_{0}\). For a wide bandwidth filter the lines are close and for a narrow bandwidth filter the parallel lines are widely separated so that there is low-level coupling. Filter (c) is also a bandpass filter, however, the frequency selectivity is not as good as with Filters (a) and (b). Filters (d), (e), and (f) are all-pass filters. These filters are used to adjust the phase of a transmitted signal, usually to accommodate phase dispersion (e.g., phase variation with respect to frequency) that was introduced somewhere else. While this analog function was once important, it is less so now as DSP techniques can usually equalize the phase sufficiently. Filter (g) is an allstop filter, meaning that negligible signal is transmitted. This filter has no practical use. Filters (h) and (i) are inherently all-stop filters. An example of the response of the combline section of Filter (h) was seen in Figure \(\PageIndex{2}\) of Example \(\PageIndex{1}\), where the all-stop response is centered at \(4\text{ GHz}\). The response is too broad to be useful as a bandstop filter (in any practical application).

Attributes |
Circuit |
Response |
---|---|---|

\(⋆\) (a) Interdigital section. Bandpass. Narrow to wide bandwidth. \(\ell = \lambda /4\) at \(f_{0}\). Symmetrical passband. | ||

\(⋆\) (b) Parallel coupled section. Bandpass. Narrow to wide bandwidth. \(\ell = \lambda /4\) at \(f_{0}\). Symmetrical passband. | ||

(c) Bandpass. Moderate to wide bandwidth. \(\ell = \lambda /4\) at \(f_{0}\). | ||

(d) All-pass. \(0 <\ell ≤ \lambda /2\) at \(f_{0}\). | ||

(e) All-pass. \(0 <\ell ≤ \lambda /2\) at \(f_{0}\). | ||

(f) All-pass. \(0 <\ell ≤ \lambda /2\) at \(f_{0}\). | ||

(g) All-stop (no practical use). | ||

\(⋆\) (h) Combline section. All-stop without matching. Bandpass with matching. Moderate to wide bandwidth. \(\ell = \lambda /4\) at \(f_{0}\). Compact. Asymmetrical response. | ||

(i) All-stop without matching. Bandpass with matching. Moderate to wide bandwidth. \(\ell = \lambda /4\). Compact. Asymmetrical response. |

Table \(\PageIndex{1}\): Responses of the nine coupled-line configurations with narrow \((2–3\%)\), moderate \((3– 10\%)\), and wide \((10\%–30\%)\) bandwidth. The most important configurations are starred \((⋆)\).

However, with minimal matching, such as a series capacitor at each of ports (see Figure \(\PageIndex{5}\)), the combline section becomes a very good bandpass filter (see Figures \(\PageIndex{6}\) and \(\PageIndex{7}\)). The combline bandpass filter, Filter (h), is more compact than Filters (a) and (b). However, it has an asymmetrical bandpass response (which can be seen in Figure \(\PageIndex{7}\)(a)), whereas Filters (a) and (b) have a symmetrical frequency response. Which type of response is preferred depends on the application. Filter (i) is the dual of Filter (h), but in practice

Attributes |
Circuit |
Response |
---|---|---|

\(⋆\) (j) Lowpass. Narrow to moderate bandwidth. \(\ell = \lambda /2\) at \(f_{0}\). | ||

\(⋆\) (k) Lowpass. Narrow to moderate bandwidth. \(\ell = \lambda /4\) at \(f_{0}\). | ||

\(⋆\) (l) Lowpass. Narrow bandwidth. \(\ell = \lambda /4\) at \(f_{0}\). | ||

\(⋆\) (m) Bandstop. Narrow bandwidth. \(\ell = \lambda /4\) at \(f_{0}\). |

Table \(\PageIndex{2}\): Responses of coupled-line configurations having lowpass and bandstop responses.

the combline configuration of Filter (h) is preferred.

Table \(\PageIndex{2}\) presents coupled-line sections having lowpass and bandstop responses. All of these filters have desirable characteristics and so all are starred \((⋆)\). Filter (l) is not a PCL filter but it is used in trade-off among the three lowpass filters (i.e., Filters (j), (l) and (m)). The usual application of a bandstop filter is to notch out an undesired signal (e.g. to prevent an LO from appearing where it is not desired). As such, the desired frequency response is narrowband and Filter (m) is a good choice for a bandstop filter.

Filter design using PCL sections begins with the choice of a PCL configuration having the essential desired frequency response. Each of the PCL sections in Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) have two resonators and can implement bandpass, bandstop, and lowpass filters having second-order responses.\(^{1}\) To meet specifications it is generally necessary to replicate the basic section. So treating the two resonator configurations as unit cells, a multi-cell filter can be realized. The multi-cell forms of the main bandpass configurations are shown in Table \(\PageIndex{3}\). The multi-cell implementation of the interdigital, parallel

Name |
Unit Cell |
Multi-cell Form |
---|---|---|

(a) Interdigital bandpass filter. \(\ell = \lambda /4\). | ||

(b) Parallel edge-coupled bandpass filter. \(\ell = \lambda /4\). | ||

(c) Parallel edge-coupled hairpin bandpass filter. \(\ell > \lambda /4\). | ||

(d) Combline bandpass filter. \(\ell = \lambda /4\). Matching required. |
||

(e) Combline bandpass filter with extended stopband. \(\ell = \lambda /4\). \(\ell_{2}\approx \lambda /8\) (typically). Matching required. |

Table \(\PageIndex{3}\): Multi-cell forms of bandpass parallel coupled-line configurations. The wavelength, \(\lambda\), is at the center frequency of the filter.

edge-coupled bandpass filters, (a) and (b) in Tables \(\PageIndex{3}\), is straightforward. A variation is shown in Filter (c), which folds the parallel edge-coupled sections to realize a compact multi-cell form called a hairpin filter [1, 2, 3]. One of the consequences of using transmission line segments to realize a filter is that there are spurious passbands. This results because a \(\lambda /4\) section of line looks the same electrically as a \(3\lambda /4\) section. Consequently there are normally spurious passbands at the odd-harmonic frequencies. If the basic resonator is \(\lambda /2\) long, then the spurious passbands would be at all of the harmonic frequencies. A solution is to incorporate capacitors in the

Figure \(\PageIndex{9}\): Network model of a pair of coupled lines.

resonators as shown in Filter (e) in Table \(\PageIndex{3}\) [2, 4, 5]. Thus each of the original transmission line resonators now becomes a combination of a capacitor and a shorter transmission line segment. If the new transmission line segment is \(\lambda /8\) long, the first spurious passband will now be at \(5f_{0}\) rather than \(3f_{0}\). The spurious passbands can be pushed further up in frequency by using even shorter transmission line lengths, but the performance of the filter at the passband frequency, \(f_{0}\), will be compromised.

There are many variations on the PCL filter and several different design techniques have been developed [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Refer to these citations to explore alternative PCL configurations and alternative methods of design to that presented here. This chapter presents one of the common approaches to synthesizing PCL filters and the scheme accommodates the use of capacitive loading used to extend the stopband of a passband filter.

Before launching into the synthesis procedure for a PCL filter, a comment on synthesis versus optimization is warranted. With a simple topology such as the second-order filter topologies shown in Table \(\PageIndex{1}\), an optimization procedure could be used to design the widths and lengths of the lines and the two or four variables describing the required external matching networks that are not shown. A global optimization is certainly feasible. However, it is not feasible to use global optimization of, for example, a seventh-order filter required to meet typical cell phone specifications. An exception is if a design for a very similar specification is available and the changes required are small. Competitive filter design requires synthesis. This will lead to optimum performance, and the insight gained can be used in topology modifications.

## 3.2.2 Coupled-Line Circuit Models

An approximate coupled-line model of a pair of coupled lines was presented in Section 5.9.5 of [29]. This model is repeated in Figure \(\PageIndex{9}\). The parameters of the network model are related to the model impedances as follows:

\[\label{eq:1}n=\frac{1}{K}=\frac{Z_{0e}+Z_{0o}}{Z_{0e}-Z_{0o}} \]

\[\label{eq:2}Z_{0S}=\sqrt{(Z_{02}Z_{0o})} \]

\[\label{eq:3}Z_{01}=\frac{Z_{0S}}{\sqrt{1-K^{2}}} \]

\[\label{eq:4}Z_{02}=Z_{0S}\frac{\sqrt{1-K^{2}}}{K^{2}} \]

Various terminating arrangements of the coupled lines result in several useful filter elements. One arrangement is shown in Figure \(\PageIndex{10}\). Also shown in this figure is the development of the network model based on the model in Figure \(\PageIndex{9}\). The final network model is a transmission line of characteristic impedance \(Z_{01}\) in cascade with an open-circuited stub. Consider what happens at the resonant frequency, \(f_{r}\) (the frequency at which the lines are one-quarter wavelength long). At lower frequencies, \(f ≪ f_{r}\), the \(Z_{02}\) line

Figure \(\PageIndex{10}\): Lowpass distributed network section derived from a pair of coupled lines with Port \(1\) open-circuited. The open circuit is indicated by a node (open circle) with a line through it. The final network model is a transmission line of characteristic impedance \(Z_{01}\) and an open-circuited stub of characteristic impedance \(Z_{02}\). The lines and stubs are one-quarter wavelength long at the corner frequency. (Thus with the stub \(f_{r} = f_{0}\), and the characteristic impedance of the stub is as shown.)

Figure \(\PageIndex{11}\): Parallel coupled-line section with Ports \(1\) and \(3\) open-circuited and network models. (For the stub, the characteristic impedance of the stub is shown and \(f_{r} = f_{0}\).)

is an open circuit and signals travel along \(Z_{01}\). At resonance the \(Z_{02}\) stub becomes a short circuit and signals do not pass. This is a crude verification that this is a lowpass structure. The process is visual and is expected to be self-explanatory. Other examples are shown in Figures \(\PageIndex{11}\) to \(\PageIndex{13}\).

The model of a combline section is shown in Figure \(\PageIndex{13}\). The final network reduction is repeated in Figure \(\PageIndex{14}\), and it will be shown that the model in Figure \(\PageIndex{14}\)(b) is equivalent to the model in Figure \(\PageIndex{14}\)(a). In the synthesis of a combline filter, the network of Figure \(\PageIndex{14}\)(b) is obtained and this can be

Figure \(\PageIndex{12}\): Interdigital section and network models. (For the stub, the characteristic impedance of the stub is shown and \(f_{r} = f_{0}\).)

Figure \(\PageIndex{13}\): Combline section and network models. (For the stubs, the characteristic impedances of the stubs are shown and \(f_{r} = f_{0}\).)

Figure \(\PageIndex{14}\): Equivalent models of a section of combline. (For the stubs, the characteristic impedances of the stubs are shown and \(f_{r} = f_{0}\).)

related back to the dimensions of the coupled line. The equivalence is done using \(ABCD\) parameters and, as will be seen, the equivalence will not be at just one frequency but will be broadband. The \(ABCD\) parameters of the network in Figure \(\PageIndex{14}\)(a) are obtained by cascading the \(ABCD\) parameters of three two-ports (the \(ABCD\) parameters of which are given in Table 2.4.1 of [30]). The \(ABCD\) parameters of the network in Figure \(\PageIndex{14}\)(a) are (from the multiplication of three \(ABCD\) parameter matrices)

\[\begin{align} \label{eq:5}T_{A}&=T_{\text{TRANSFORMER}}T_{\text{SERIES STUB}}T_{\text{SHUNT STUB}} \\ \label{eq:6} &=\left[\begin{array}{cc}{1/n}&{0}\\{0}&{n}\end{array}\right]\left[\begin{array}{cc}{1}&{\jmath Z_{02}\tan\theta}\\{0}&{1}\end{array}\right]\left[\begin{array}{cc}{1}&{0}\\{-\jmath /(Z_{01}\tan\theta)}&{1}\end{array}\right] \\ \label{eq:7}&=\left[\begin{array}{cc}{1/n}&{\jmath Z_{02}\tan\theta /n} \\ {0}&{n}\end{array}\right]\left[\begin{array}{cc}{1}&{0}\\{-\jmath /(Z_{01}\tan\theta)}&{1}\end{array}\right] \\ \label{eq:8}&=\left[\begin{array}{cc}{\frac{1}{n}\left(1+\frac{Z_{02}}{Z_{01}}\right)}&{\jmath Z_{02}\tan\theta /n}\\{-\jmath n/(Z_{01}\tan\theta)}&{n}\end{array}\right]\end{align} \]

The \(ABCD\) parameters of the network in Figure \(\PageIndex{14}\)(b) are

\[\begin{align}\label{eq:9}T_{B}&=T_{\text{SHUNT STUB}}T_{\text{SERIES STUB}}T_{\text{SHUNT STUB}} \\ \label{eq:10}&=\left[\begin{array}{cc}{1}&{0}\\{-\jmath /(Z_{011}\tan\theta )}&{1}\end{array}\right]\left[\begin{array}{cc}{1}&{\jmath Z_{012}\tan\theta}\\{0}&{1}\end{array}\right]\left[\begin{array}{cc}{1}&{0}\\{-\jmath /(Z_{022}\tan\theta)}&{1}\end{array}\right] \\ \label{eq:11}&=\left[\begin{array}{cc}{1}&{\jmath Z_{012}\tan\theta}\\{-\jmath /(Z_{011}\tan\theta)}&{1+Z_{012}/Z_{011}}\end{array}\right]\left[\begin{array}{cc}{1}&{0}\\{-\jmath (Z_{022}\tan\theta)}&{1}\end{array}\right] \\ \label{eq:12} &= \left[\begin{array}{cc}{1+\frac{Z_{012}}{Z_{022}}}&{\jmath Z_{012}\tan\theta}\\{\frac{-\jmath}{\tan\theta}\left(\frac{1}{Z_{011}}+\frac{1}{Z_{022}}+\frac{Z_{012}}{Z_{011}Z_{022}}\right)}&{1+\frac{Z_{012}}{Z_{011}}}\end{array}\right] \end{align} \]

Equating Equations \(\eqref{eq:8}\) and \(\eqref{eq:12}\) yields

\[\begin{align}\label{eq:13}1+\frac{Z_{012}}{Z_{022}}&=\frac{1}{n}\left(1+\frac{Z_{02}}{Z_{01}}\right) \\ \label{eq:14} Z_{012}&=Z_{02}/n \\ \label{eq:15}\left(\frac{Z_{011}+Z_{022}+Z_{012}}{Z_{011}Z_{022}}\right)&=\frac{n}{Z_{01}} \\ \label{eq:16}1+\frac{Z_{012}}{Z_{011}}&=n\end{align} \]

which have the solution

\[\begin{align}\label{eq:17}Z_{011}&=\frac{Z_{012}}{n-1}=\frac{Z_{02}}{n(n-1)} \\ \label{eq:18}Z_{012}&=\frac{Z_{02}}{n} \\ \label{eq:19}Z_{022}&=\frac{Z_{01}Z_{02}}{Z_{02}-(n-1)Z_{01}}\end{align} \]

Rearranging these, expressions for \(Z_{01},\: Z_{02},\) and \(n\) can be obtained:

\[\label{eq:20}n=1+\frac{Z_{012}}{Z_{011}},\quad Z_{01}=\left(\frac{nZ_{011}Z_{022}}{Z_{011}+Z_{022}+Z_{012}}\right),\quad\text{and}\quad Z_{02}=nZ_{02}Z_{012} \]

Using these, Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\), and the coupled-line analysis of Section 5.6 of [29], the geometric parameters of the combline coupled-line section can be obtained corresponding to the stub circuit of Figure \(\PageIndex{14}\)(b).

Thus the equivalent circuit of the combline section, the top left figure in Figure \(\PageIndex{13}\), has the equivalent circuit shown in Figure \(\PageIndex{14}\). Filter synthesis can be directed at developing circuit structures like that in Figure \(\PageIndex{14}\)(b) and from this electrical design, the physical design consisting of combline sections can be developed.

## Footnotes

[1] Recall that the order designation comes from the lowpass prototype so that here a second-order bandpass filter actually has two resonators, each having an \(LC\)-like response.