Propagation constant measurements of reflection-asymmetric and nonreciprocal microwave networks from S-parameters without using a reflective standard

Highlights

• Our method evaluates propagation constants of non-reciprocal microwave networks.

• It implements this evaluation without resorting to any calibration technique.

• It uses scattering parameters of the line/network connection and the thru connection.

• It eliminates errors arising from fabrication tolerance and thickness of the second line.

• It is applicable for reflection-asymmetric and nonreciprocal microwave networks.

Abstract

We propose a simple method for determination of forward and backward propagation constants of nonreciprocal uniform networks or lines with asymmetric reflections. The wave cascading matrix presentation is used to diagonalize and decompose the line in terms of impedance and amplitude/phase transitions. The proposed method uses scattering parameters of the line/network connection and the thru connection, thus eliminating errors arising from fabrication tolerance and thickness of the second line. Besides, compared with conventional methods which necessitate calibration techniques such as the thru-reflect-line, our method has the advantage that it determines propagation constants without resorting the measurement of any reciprocal reflective standard. Propagation constant measurements of a reciprocal network (different-length X-band waveguide straights) and a nonreciprocal network (microwave phase shifter) were first carried out and then compared with those measured by different methods to validate and monitor the accuracy of our method.

Introduction

Electromagnetic characterization of microwave networks such as waveguides or transmission lines is important for understanding their reflection and transmission responses for an applied electromagnetic stimulus. Such a characterization can be carried out using their scattering (S-) matrix, ABCD matrix, wave cascading matrix (WCM), or impedance or admittance matrices [1], [2]. Of these matrices, the WCM presentation along with S-matrix presentation is especially useful for characterization of cascaded microwave networks at high frequencies. Vector network analyzers (VNAs) measure the S-matrix of a network/line in terms of pseudo-waves traveling in opposite directions defined for a reference impedance $Z_{\mathrm{ref}}$, which can be different than the characteristic impedance $Z_0$ of that network/line [3]. This is the methodology used in many calibration techniques involving line standard measurements such as the thru-reflect-line (TRL) technique, the multiline TRL technique, the line-reflect-line (LRL) technique, and line-reflect-match (LRM) technique [2], [4], [5], [6], [7]. These techniques share the common point that the WCM presentation of a two-port network will have non-zero diagonal elements (and zero off-diagonal elements) for a phase-amplitude change transition region in the transformation at one port from one reference-impedance to another reference-impedance. This common point is exploited by the aforementioned calibration techniques for determination of propagation constant ($\gamma$) of reciprocal nonreflecting [2], [5], [6], [8], [9] and symmetrically-reflecting [10] uniform lines. In our recent study, we extended these studies to $\gamma$ determination of uniform reciprocal lines with asymmetric reflections [11].

On the other hand, methods not requiring the measurement of reciprocal reflective calibration (called line-line (LL) methods from now on) also implement electromagnetic characterization of sample-loaded lines or networks through propagation constant measurements [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. Compared with conventional methods which use calibrated S-parameters for material characterization via reciprocal reflective calibration standard [32], [33], [34], [35], [36], [37], these methods not only can eliminate measurement errors arising from inaccurate reflective calibration standards but also decrease the workload bypassing the traditional calibration procedure such as the TRL [2]. LL methods measure uncalibrated S-parameter measurements of two identical but different-length network/line sections. There are two main issues that should be discussed related to these methods. First, the available LL methods, nonetheless, assume that these network/line sections are reciprocal with symmetrical ($S_{11}=S_{22}$) [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] or asymmetric reflections ($S_{11}\ne S_{22}$) [11]. To our best knowledge, all LL methods in the literature in their current forms cannot extract forward and backward $\gamma$ of nonreciprocal network/lines. Such a study could be important for investigation of nonreciprocal networks such as microwave phase shifter [1] or of chiral (meta) materials [38]. Second, LL methods necessitate using identical networks/lines with different lengths for propagation constant determination. However, in real measurements, it is difficult to realize such a requirement especially for lines whose properties change with frequency. Besides, any fabrication error present in the second line could additionally decrease the accuracy of measurements. Therefore, there is a need for a method for propagation constant determination of nonreciprocal networks/lines using uncalibrated S-parameter measurements of one network or line only. In this study, we propose a simple and useful method for propagation constant determination of nonreciprocal networks/lines with asymmetric reflections using uncalibrated S-parameter measurements of the network/line connection together with the thru connection.

The remainder of the manuscript is organized as follows. First, the WCM presentation of a nonreciprocal uniform network/line with asymmetric reflections in terms of S-parameters from its ABCD matrix parameters in Section 2.1. Next, in Section 2.2 it will be shown that this WCM matrix can be diagonalized and decomposed into impedance transition and amplitude/phase transition matrices using its eigenvalues, eigenvectors, and eigenbases. Then, the proposed method based on uncalibrated S-parameters of the nonreciprocal network/line in addition to uncalibrated S-parameters of the thru connection is presented in Section in 3. After, the proposed method is validated and its accuracy is compared by propagation constant measurements of one nonreciprocal and one reciprocal microwave networks in Section 4.Finally, important points of our method are recapitalized in Section 5.