# Quality factor of a transmission line coupled coplanar waveguide resonator

- Ilya Besedin
^{1, 2}Email authorView ORCID ID profile and - Alexey P Menushenkov
^{2}

**Received: **3 October 2017

**Accepted: **12 January 2018

**Published: **22 January 2018

## Abstract

We investigate analytically the coupling of a coplanar waveguide resonator to a coplanar waveguide feedline. Using a conformal mapping technique we obtain an expression for the characteristic mode impedances and coupling coefficients of an asymmetric multi-conductor transmission line. Leading order terms for the external quality factor and frequency shift are calculated. The obtained analytical results are relevant for designing circuit-QED quantum systems and frequency division multiplexing of superconducting bolometers, detectors and similar microwave-range multi-pixel devices.

## Keywords

## 1 Introduction

Low loss rates provided by superconducting coplanar waveguides (CPW) and CPW resonators are relevant for microwave applications which require quantum-scale noise levels and high sensitivity, such as mutual kinetic inductance detectors [1], parametric amplifiers [2], and qubit devices based on Josephson junctions [3], electron spins in quantum dots [4], and NV-centers [5]. Transmission line (TL) coupling allows for implementing relatively weak resonator-feedline coupling strengths without significant off-resonant perturbations to the propagating modes in the feedline CPW. Owing to this property and benefiting from their simplicity, notch-port couplers are extensively used in frequency multiplexing schemes [6], where a large number of CPW resonators of different frequencies are coupled to a single feedline. The geometric design of such resonators determines the resonant frequencies of their modes, loss rates and coupling coefficients of these modes to other elements of the circuit. 3D electromagnetic simulation software provides excellent means for complete characterization of such structures by finite element analysis. However, in the case of simple structures, analytical formulas can be devised that are invaluable for engineering of large multi-pixel resonator arrays.

The external quality factor depends not only on the resonator characteristics, but also the coupling elements and port impedances and can be calculated in case of a capacitive [7], inductive [8], and both capacitive and inductive [9] couplings. More complex setups with bandpass filters for qubit applications have been developed for capacitive coupling [10]. In the present article we aim to analytically describe the TL coupled coplanar waveguide (CPW) resonator in an arbitrary-impedance environment.

Conformal mapping is an established tool in engineering that uses the analytic property of complex-variable functions to transform the boundary shape in boundary value problems of Laplace’s equation in two dimensions. Classical results for mode impedances in microstrip and coplanar TL structures can be systematically derived using the Schwarz-Christoffel mapping [11, 12], yielding explicit formulas in terms of elliptic integrals. The same approach has been successfully applied to finite-width ground plane CPWs and coupled microstrip lines [13] and coupled CPWs [14]. These results rely on the presence of at least one of the structures’ symmetry planes, which reduce these problems to two conductors, also yielding explicit formulas with elliptic integrals. If the conductor plane symmetry is broken, as in the case of finite-thickness and conductor-backed dielectrics [14] and finite-thickness conductors [13], approximate formulas can account for the change in the effective dielectric constant and effective conductor surface area.

As a rule of thumb, an explicit formula in terms of elliptic integrals for a structure’s per-unit-length capacitance and inductance can be obtained, if the boundary value problem for Laplace’s equation can be divided into separate decoupled homogeneous domains, and the number of distinct conductors in each domain is no more than two. In the present paper we use a general approach based on implicit expressions with hyperelliptic integrals [15, 16], recently revisited in [17]. In contrast to the more established techniques, it is not constrained by the requirement that each domain contains no more than two conductors, but still requires them to be homogeneous. This allows to obtain expressions for the mutual capacitance and inductance matrices of coupled transmission lines even if they have completely arbitrary conductor and gap widths. The effect of finite thickness conductors has been recently investigated [18].

### 1.1 Resonant scattering

From the decay rate of an excitation in the resonant mode one cannot distinguish between loss channels, however external loss can be measured directly by probing the microwave amplitude at the TL ports, or, more practically, connecting the TL to a vector network analyzer and measuring a scattering parameter \(S_{21}\).

*a*,

*α*,

*τ*are introduced to characterize transmission through the cables and other components of the measurement system that are not directly connected to the resonator and \(Q_{l}\), \(Q_{e}\) and \(f_{r}\) depend on the coupling element and the resonator.

### 1.2 Multiconductor transmission lines

*n*conductors can be described by the multiconductor TL equations [20]:

*z*is the coordinate along the TL,

**V**and

**I**the time and coordinate dependent voltages and currents on each of the conductors, and

**L**,

**C**, are the \(n \times n\) per-unit-length inductance and capacitance matrices. For simplicity we neglect finite dielectric conductance and conductor resistance as their effect is small for high Q-factor resonators and to first order they only affect the internal quality factor and can be accounted for separately.

*λ*is the inverse phase velocity of the propagating mode and the corresponding eigenvector

**b**is the voltage and current amplitude of the mode.

**1**is the \(n\times n\) identity matrix. If this property holds,

**A**can be diagonalized with

**Z**as

**Z**is symmetric and can be diagonalized with an orthogonal transform, yielding mode impedances.

## 2 Results and discussion

### 2.1 Conformal mapping of the coupled CPW

Here we follow a general analytical method [16] to calculate the per-unit-length capacitance and inductance matrices of two coupled CPWs with arbitrary lateral dimensions and an infinite ground plane.

*ϵ*and the magnetic permeability is

*μ*. On the conductors between the vacuum and the substrate the tangential component of the electric field and normal component of the magnetic field are zero. Due to the problem symmetry, the normal component of the electric field and tangential component of the magnetic field at the vacuum-substrate interface are also zero. The latter property renders the equations for the upper half-plane and the lower half-plane independent from each other. At either of the half-planes, the electric and magnetic fields can be expressed in terms of the electrostatic potential and the component of the magnetostatic vector potential along the TL, which are up to a constant equal to each other. In either of the half-planes, both the potentials satisfy Laplace’s equation:

*x*-coordinates of the right and left boundaries of the conductor cross-sections such that \(a_{0}< b_{0}< a_{1}< b_{1}<\cdots<a_{n}<b_{n}\), \(j=0,\ldots,n\). If the

*i*-th conductor has a non-zero electric potential \(\phi_{i}\) and all the others are grounded, the boundary conditions at the interface are given by

*i*-th conductor and a zero magnetic flux through any ray originating from any other conductor. For conformal mapping we introduce the complex variable \(z=x+iy\). Following the general approach, we define the points \({c_{j}}, j \in\{ 1,\ldots,n\}\setminus\{i, i+1\}\) such that \(a_{j}< c_{j}< b_{j}\) where the electric field vector component along the real axis changes its sign. These points must exist since \(a_{j}\) and \(b_{j}\) lie on the

*j*-th and \(j+1\)-th conductors respectively, both of which are grounded, so the integral of the electric field along any contour connecting them must be equal to zero. The conformal mapping

*w*-plane and the vacuum-surface interface into lines parallel to the imaginary axis of the

*w*-plane. Due to its analytic property, \(\operatorname{Im}{ w(z)}\) satisfies (16). The values of \(c_{j}\) are implicitly defined by the boundary conditions at the grounded conductors:

*w*-plane. After accounting for both the substrate and vacuum half-planes and multiplying by the vacuum permittivity (permeability), we obtain for the capacitance matrix and the inverse inductance matrix

Finite dimensions of the structure, in particular, finite conductor thickness, have been considered by an additional conformal mapping in a compatible fashion elsewhere [18]. We do not consider this case here, as it introduces two additional key complications. The first complication is related to the conformal mapping itself, as in this case the boundary value problem cannot be exactly decoupled for the upper and lower half-planes. The second complication is that the relation (11) no longer holds, and different modes of propagation no longer have the same phase velocity.

### 2.2 Boundary condition equation matrix of the TL model and perturbative analysis

**M**is a square matrix of order

*r*,

**a**is the nodal voltage, branch current and internal degree of freedom amplitude vector. Poles \(f_{p}\) of the S-matrix as a function of external perturbation frequency

*f*appear whenever the determinant of

**M**is zero:

**M**can be permuted such that it becomes a block diagonal matrix with each block corresponding to its own subsystem. In this case the determinant of the entire matrix is the product of the block determinants and can be computed efficiently provided the blocks are small enough.

Coupling introduces non-zero terms in the off-diagonal blocks. Provided that the coupling between the subsystems is weak, the determinant can be expanded in some parameters *κ* characterizing the coupling strength. The *m*-th order of the expansion can be expressed as the sum of \(C(m,r)\) determinants of **M** where in all but *m* of *r* rows *κ* is zero. Such determinants are more difficult to compute as *m* blocks merge together, however if *m* and the individual blocks are small enough computation is still efficient.

*m*of the scattering amplitude pole correction in the coupled system can be expressed as

The analytical formula for scattering coefficients and nodal voltages and branch currents induced by externally applied signals involves the cofactor matrix to **M** and can be computed in a similar fashion.

### 2.3 Boundary condition equation matrix of a TL coupled quarter wavelength resonator

*κ*is a dimensionless coupling coefficient, \(\gamma= (1-\kappa^{2})^{-1/2}\). If \(\kappa=0\), \(Z_{1}\) and \(Z_{2}\) become the characteristic impedances of the resonator and the feedline, respectively. The length of the coupler is \(l_{c}\).

### 2.4 Quality factor and frequency shift of a transmission-line-coupled resonator

We consider the general cases of a CPW resonator coupled to a TL (Figure 2(d)) with termination impedances (a) \(Z_{t1}=0\), \(Z_{t2}=\infty\), (b) \(Z_{t1}=0\), \(Z_{t2}=0\) and (c) \(Z_{t1}=\infty\), \(Z_{t2}=\infty\). Both the notch-port and capacitive and inductive butt-port coupled CPW resonators are described with this schematic.

Following the approach introduced in the previous section, we expand the determinant Δ to first order in \(Z_{2}-Z_{r}\) and to second order in *κ*.

*f*,

*κ*and \(Z_{2}\) evaluated at the

*p*-th resonance frequency are given by

*θ*and

*ψ*are phase variables that can be calculated from the zeroth-order resonance frequency (5) with

In the case of matched input and output port impedances the quality factor has no leading-order dependence on the position of the coupler section. This result arises due to the equal contribution of inductive and capacitive coupling in (11) and equal current and voltage amplitude in the feedline. For unmatched ports, this symmetry is broken and standing waves in the coupler section of the feedline can both increase and decrease the quality factor of the resonator.

### 2.5 Comparison with numerical simulation

Compared to the 3D simulation, the TL model yields systematically higher estimates of the quality factor. This can be attributed to the presence of spurious coupling between the resonator and feedline, primarily of the resonator conductor arcs attached to the coupler, which leads to a larger effective coupler length \(l_{c}\). The error of the quality factor calculation is within 70% over the entire range of simulated quality factors, which spans over three orders of magnitude from 10^{3} to 10^{6}. The deviation can be reduced by increasing the length of the coupler section while reducing the conductor width \(w_{3}\) and reducing the arc radius. The small frequency shift dependence on \(w_{3}\) predicted by our analytical model cannot be reliably reproduced with our 3D simulation due to precision and meshing issues even for large simulation sizes.

Apart from the errors that can be identified and quantified by EM simulation, practical devices are also plagued by the infamous problem of standing waves. Non-ideal connectors lead to frequency-dependent port impedances, which in turn enter the formulas (36), (39), (41) and lead to unpredictable variations of the quality factor. This issue, if not properly handled by input and output RF circulators or isolators, or use of high-precision connectors, is arguably the main source of quality factor deviations in devices.

## 3 Conclusions

In this article we have presented an analytical model for transmission line coupled high quality factor coplanar waveguide resonators. We consider two main configurations, the butt-port and notch-port coupled CPW resonators. Using a perturbative expansion of the equation for the resonance frequencies, we obtain the external quality factors and frequency shifts for \(\lambda/2\) and \(\lambda/4\) resonators due to coupling to an arbitrary-impedance environment. 3D simulation of a sample design shows that the method’s accuracy is limited by spurious couplings in the actual layout.

The obtained analytical results can be applied to accelerate the design of large circuit-QED quantum systems and frequency division multiplexing of superconducting bolometers, detectors and similar microwave-range multi-pixel devices, eliminating the need for full 3D electrodynamic simulations.

## Declarations

### Acknowledgements

The authors would like to thank A. Ustinov for sharing his comments on an early version of the manuscript, and V. Chichkov, G. Fedorov, I. Khrapach for their efforts on the design, fabrication and characterization of CPW resonators, which have motivated this work.

### Availability of data and materials

Example calculations and 3D EM simulation results are available at https://github.com/ooovector/cpw_coupling. Source code for reproducing the main analytical results is included in the additional file 1.

### Funding

This work was supported by the Russian Science Foundation (grant No. 16-12-00095).

### Authors’ contributions

The authors’ contributions are equal. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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