I. INTRODUCTION
Vector measurements of scattering parameters are indispensable in microwave engineering and require the application of a vector network analyser, which is highly complex and expensive measurement device. To avoid the use of such a sophisticated apparatus, the six-port reflectometer allowing for the vector measurement of reflection coefficient has been proposed [Reference Engen1, Reference Engen2], in which the measured value is determined utilizing four simple power measurements.
The development of such a measurement technique has also allowed for the transmission coefficient measurement which requires, however, a bit more advanced measurement setup. The most popular measuring systems utilize an external power divider and two six-ports between which a measured device is connected [Reference Hoer3–Reference Haddadi, Wang, Glay and Lasri6]. However, such an approach requires the application of at least seven power detectors among which only three are simultaneously used in a particular measurement. In [Reference Muiioz, Margineda, Martin and Rojo7] a measuring system being composed of two five-port reflectometers is proposed in which the number of required power detectors is reduced. Utilizing such an approach one cannot measure, however, the phase of transmission coefficient which is a serious limitation. Another concept of the multiport system for S-parameter measurements can be found in [Reference Brantervik and Kollberg8], where two four-ports and only two power detectors have been applied. To measure complex scattering parameters utilizing such a limited number of power readings a known variable reference has to be used, which complicates the measurement.
Further development of the multiport measurement technique has resulted in the modified measuring system containing a single multiport network, which allows for the measurement of all S-matrix coefficients [Reference Jia9–Reference Xiao-Ming11]. The main drawback of such a solution is the necessity of isolator application, which does not allow for measurements within a multioctave frequency range. An interesting isolator-free alternative has been proposed in [Reference Khouaja and Ghannouchi12], where also a single six-port has been utilized. The measurement of entire S-matrix requires the application of a highly complicated system of switches nevertheless the entire measurement procedure can be done automatically.
Recently, it has been shown that a standard 4 × 4 Butler matrix which constitutes a well-known component often used in microwave systems can also be utilized in multiport measurements of scattering parameters [Reference Staszek, Gruszczynski and Wincza13]. Moreover, the application of a Butler matrix composed of the coupled-line directional couplers allows for wideband operation. However, the system presented in [Reference Staszek, Gruszczynski and Wincza13] requires the application of seven power meters among which only four are utilized in a particular measurement. This issue has been overcome in the measuring system utilizing 8 × 8 Butler matrix, which has been proposed in [Reference Staszek, Gruszczynski and Wincza14]. The number of power meters has been decreased to five, nevertheless, the system suffers from the need of the isolator application.
Although a great variety of six-port-based measuring systems has been reported, the optimal power distribution ensured by the applied multiport network is still a matter of interest [Reference Brunetti, Fornero and Rietto15–Reference Probert and Carroll17]. It has been shown that the uniform power distribution [Reference Staszek, Gruszczynski and Wincza16] as well as the increased number of the applied power detectors [Reference Staszek, Gruszczynski and Wincza14, Reference Staszek, Gruszczynski and Wincza16, Reference Kim, Kim, Hwang, Kim and Kwon18, Reference Monzo-Cabrera, Pedreno-Molina, Lozano-Guerrero and Toledo-Moreo19] significantly improves the measurement accuracy. Further minimization of the measurement error can be achieved by a proper tuning of the mentioned power distribution, as shown in [Reference Staszek, Gruszczynski and Wincza16].
In this paper, a novel multiport system capable of reflection and transmission coefficients’ measurements is presented. It consists of two directional couplers and a standard 4 × 4 Butler matrix. The proposed system ensures a highly uniform power distribution, which additionally can be tuned in order to enhance the measurement precision in the entire operational frequency range. Moreover, no narrowband elements such as isolators are required. A fully analytical calibration procedure for transmission coefficient measurement based on the least-squares approach that can be applied to a multiport system having arbitrary number of power meters, is presented. Furthermore, a complete analysis of the simplified model for the transmission coefficient measurement is discussed and the measurement inaccuracy for reflection and transmission coefficients is estimated. The proposed measuring system has been verified experimentally in a wide frequency range 1–5 GHz, by the measurements of several microwave components. The obtained results within the magnitude range from 0 to 50 dB are very close to the reference values obtained with a commercial vector network analyser in terms of magnitude and phase, which fully confirms the usefulness of the proposed multiport system featuring tunable power distribution.
II. THEORETICAL ANALYSIS
The multiport measurement technique allows for measurement of both complex reflection and transmission coefficients using scalar power measurement [Reference Engen1–Reference Hoer4]. Such unique property results from a specific design of multiport reflectometers, which combine the reference signal from signal source with the signal dependent on the measured S-parameter [Reference Engen1, Reference Staszek, Gruszczynski and Wincza13]. For each power meter applied in the measuring system combination of reference and measurement-dependent signals is done with different magnitude and phase relations. Having at least three values of measured power, one can determine the measured reflection or transmission coefficient in terms of magnitude and phase. Higher number of power meters, as well as advantageous magnitude and phase relations provided by the applied multiport network, allow for enhancement of the measurement accuracy [Reference Staszek, Gruszczynski and Wincza14, Reference Staszek, Gruszczynski and Wincza16].
The described principle has been applied to develop the proposed measuring system, as shown in Fig. 1, which allows for the measurement of reflection and transmission coefficients. The presented system is excited by a signal source applied to the directional coupler Q 1, which allows for the measurement of the reference power P REF . The second directional coupler Q 2 is used to split the signal from Q 1 into the reference signal provided through the adjustable attenuator A to the 4th port of the 4 × 4 Butler matrix and the second signal, which is provided to the measured device DUT. The applied switches allow for measuring reflection coefficient (all switches in position 1) or transmission coefficient (all switches in position 2). In case of reflection coefficient measurement the signal reflected from DUT is transmitted through the directional coupler Q 2 and is provided to port #1 of the Butler matrix. For the transmission coefficient measurement, the signal transmitted through the measured device (DUT) is provided to port #1. Both signals applied to port #1 and port #4 of the 4 × 4 Butler matrix are summed with different phase relations and measured by four power meters connected to the output ports of the Butler matrix. It can be observed that in the proposed measuring system for each measurement all the applied power meters are utilized. It allows for enhancing the measurement precision with decreased number of power meters in comparison with the measuring systems involving two multiports, in which at least seven power meters are required.
Schematic diagram of the proposed multiport system with a single 4 × 4 Butler matrix allowing for reflection and transmission coefficients’ measurement.
Further analysis of the proposed measuring system is performed with the assumption, that all the utilized components feature ideal isolations, as well as perfect impedance match. The relations between power readings P i (i = 1–4) and the measured reflection and transmission coefficients Γ and T, respectively, are as follows:
$${p_{{\rm \Gamma }ink}} = \displaystyle{{{P_{\Gamma ink}}} \over {{P_{REF}}}} = {\left\vert {{T_1}{T_2}} \right\vert^2}{\left\vert {{C_2}{S_{in}}{\it \Gamma} + A{S_{ik}}} \right\vert^2}, $$
$${p_{Tink}} = \displaystyle{{{P_{Tink}}} \over {{P_{REF}}}} = {\left\vert {{T_1}} \right\vert^2}{\left\vert {{C_2}{S_{in}}T + A{T_2}{S_{ik}}} \right\vert^2}, $$
where i – number of Butler matrix port, at which the power is measured, n – number of Butler matrix port, to which switch SW2 is connected, k – number of Butler matrix port, to which the adjustable attenuator is connected, T x , Cx – transmission and coupling coefficients of the directional coupler Q x , A – attenuation of the adjustable attenuator A, S xx – S-parameters of the utilized 4 × 4 Butler matrix.
The usefulness of the proposed measuring system can be verified using the geometric interpretation of the multiport measurement technique, in which the measurement result is the intersection point of several circles on a complex plane [Reference Engen1, Reference Engen2, Reference Staszek, Gruszczynski and Wincza16]. The distributions of these circle centers for reflection coefficient measurements c Γ and for transmission coefficient measurements c T are as follows:
$${c_{{\rm \Gamma} ink}} = - \displaystyle{{A{S_{ik}}} \over {{C_2}{S_{in}}}}.$$
$${c_{Tink}} = - A\displaystyle{{{T_2}{S_{ik}}} \over {{C_2}{S_{in}}}}.$$
It can be observed that the applied adjustable attenuator allows for uniform scaling of the distance between the origin of a complex plane and all circle centers (called further the magnitude of circle centers’ distribution), which can be utilized in order to enhance the measurement accuracy, as it is described in detail in [Reference Staszek, Gruszczynski and Wincza16]. Moreover, it is seen that the directional coupler Q 1 does not influence the circle centers’ distributions. It is applied only to provide the reference power measurement, therefore, the requirements related to its parameters are relaxed and its nominal coupling may be very weak (e.g. 10–20 dB). Assuming that the directional coupler Q 2 is an ideal 3 dB quadrature directional coupler, (3) and (4) can be expressed as:
$${c_{\Gamma ink}} = - A\sqrt 2 \displaystyle{{{S_{ik}}} \over {{S_{in}}}},$$
$${c_{Tink}} = - jA\displaystyle{{{S_{ik}}} \over {{S_{in}}}},$$
and the magnitude of the circle centers’ distribution can be adjusted by setting the desired attenuation:
$$\left \vert {{c_\Gamma}} \right \vert = \sqrt 2 \times {10^{ - (A/20)}},$$
$$\left \vert {{c_T}} \right \vert = {10^{ - (A/20)}},$$
where attenuation A is expressed in dB.
The above discussion is related to the applied directional couplers and adjustable attenuator. As it has been shown the magnitude of circle centers’ distribution can be scaled by the applied attenuator. However, the mutual angular distance between circle centers arg[c] results only from the 4 × 4 Butler matrix's parameters and can be chosen by a proper selection of the Butler matrix's input ports. Tables 1 and 2 present all possible configurations of the proposed system for the directional coupler Q 2 having 3 dB coupling level and attenuation of the adjustable attenuator being equal to 0 dB. It can be seen that some of the possible configurations cannot be utilized in measurements, since the circle centers overlap, leading to ambiguous measurements. This fact can be explained with the use of schematic diagram of the 4 × 4 Butler matrix seen in Fig. 1. To ensure four different circle centers, the signal related to the measured value and the reference signal have to excite different internal directional couplers of the Butler matrix, which ensures the maximum variety of the phase shifts for these two signals seen at the output ports of the 4 × 4 Butler matrix.
Circle centers’ distributions for reflection coefficient measurement, assuming A = 0 dB, C 2 = 3 dB, n, port of Butler matrix, to which switch SW2 is connected, and k, port of Butler matrix, to which attenuator is connected.
Circle centers’ distributions for transmission coefficient measurement, assuming A = 0 dB, C 2 = 3 dB, n, port of Butler matrix, to which switch SW2 is connected, and k, port of Butler matrix, to which attenuator is connected.
It must be emphasized that the circle centers’ distribution of the proposed multiport system can be described using simple formulas (5) and (6). As seen the location of a given circle center depends on the coupling and transmission coefficients of the directional coupler Q 2 and on only two S-parameters of the 4 × 4 Butler matrix. It is to be noted that in case of the multiport systems involving Butler matrices published recently in [Reference Staszek, Gruszczynski and Wincza13, Reference Staszek, Gruszczynski and Wincza14], the location of a given circle center depends on four S-parameters of the utilized Butler matrix, therefore, the deviation of the circle centers’ distribution can be four times larger than the amplitude and phase imbalance of the utilized Butler matrix. In the proposed multiport system such a simple relation between the S-parameters of the utilized components and the circle centers’ distribution decreases the influence of an imperfect realization of the utilized directional coupler Q 2 and Butler matrix on the circle centers’ distribution, which reflects in higher measurement accuracy in comparison with the previously published multiport systems involving Butler matrices. Moreover, in the proposed measuring system only five power meters (including reference power measurement) are applied, which ensure four uniformly distributed circle centers for both reflection and transmission coefficient measurements. Thus the obtained accuracy is higher than the one for the system described in [Reference Staszek, Gruszczynski and Wincza13], where for a particular measurement only three circle centers exist.
III. SYSTEM CALIBRATION
Precise measurements with the use of an arbitrary multiport system need to be preceded by a suitable calibration procedure, which is the substantial problem related to multiport measurement technique. Since the proposed system allows for measuring both reflection and transmission coefficients it requires separate calibration procedures for each type of measurement.
To measure the reflection coefficient all switches must be set in position 1. The relation between power measured at the output ports of Butler matrix and reflection coefficient Γ given by (1) has to be augmented by imperfect impedance match of the measuring port as well as non-ideal isolations. Therefore, it can be expressed with the well-known formula [Reference Engen1]:
$${p_i} = {q_i}{\left \vert {\displaystyle{{1 + {A_i}\Gamma} \over {1 + {A_0}\Gamma}}} \right \vert ^2},$$
where q i , Ai , and A 0 are the system constants, which have to be found during the calibration procedure. It is seen that (9) has the same form as in case of classic multiport reflectometers, hence the proposed system can be calibrated with the use of the calibration procedure for multiport reflectometers having increased number of ports, e.g. [Reference Staszek, Kaminski, Rydosz, Gruszczynski and Wincza20].
The transmission coefficient measurement requires a separate calibration procedure. As it has been mentioned, equation (2) shows the relation between measured power and transmission coefficient T with assumed ideal impedance match and perfect isolations of all utilized components. However, this condition cannot be fulfilled in practice. Hence, to achieve a satisfying measurement accuracy the developed model has to take into account the imperfect parameters. Figure 2(a) presents the signal flow, in which all non-ideal parameters of the proposed measuring system are seen. As it can be observed, the connection of directional couplers (together with switches in position 2) is represented by the three-port network the S-parameters of which are indicated as S Cxx . The utilized 4 × 4 Butler matrix has been simplified to the three-port network (involving ports i, k and n described in Section II) with S-parameters indicated as S Bxx . It has to be emphasized, that such a simplification does not provide any inaccuracy and this model can be used separately for each output port of the Butler matrix, at which the power is measured (#5–#8). Both described three-ports are connected by the measured two-port device (DUT) and the adjustable attenuator A.
Signal flow in the proposed multiport system for transmission coefficient measurement: graph of the entire measuring system with all imperfect parameters (a), graph of the measuring system with neglected imperfect isolations (b), graph of the measuring system with imperfect isolations and with assumed perfect impedance match of all components (c).
It can be observed in Fig. 2(a), that the accurate model of a real measuring system is complex and would require a highly sophisticated calibration procedure involving large number of parameters. Therefore, the imperfect isolations of both three-ports (S C23, S C32, S Bnk , and S Bkn marked gray in Fig. 2(a) have been neglected, which significantly simplifies the signal flow to the form as shown in Fig. 2(b). It has to be underlined that such a simplification introduces some inaccuracy however, as it is shown in Section IV, the relative power reading error caused by the described simplification is significantly lower in comparison to the measurement inaccuracy of the commercially available power meters.
Based on the simplified signal flow as shown in Fig. 2(b) one can derive the relation between the measured transmission coefficient and the power measured at the output ports of the utilized Butler matrix:
$${p_i} = {q_i} \vert 1 + {A_i}T{ \vert ^2},$$
where q i , and A i are the calibration constants to be found during the calibration procedure and coefficient T is given as follows:
$$T = \displaystyle{{{S_{21}}} \over {(1 - {\Gamma _1}{S_{11}})(1 - {\Gamma _2}{S_{22}}) - {S_{12}}{S_{21}}{\Gamma _1}{\Gamma _2}}},$$
where Γ1 and Γ2 are the reflection coefficients seen at the measuring ports between which the measured device is inserted. Moreover, assuming that the impedance match of both measuring ports is about 20 dB equation (11) may be simplified:
$${T^{\ast}} = \displaystyle{{{S_{21}}} \over {(1 - {\Gamma _1}{S_{11}})(1 - {\Gamma _2}{S_{22}})}}.$$
It is seen, therefore, that to obtain the entire S-matrix of a given two-port one has to perform four measurements. First, the reflection coefficient S 11 has to be found (switches SW1 and SW2 in Fig. 1 in position 1). Subsequently, all switches must be switched into position 2 allowing for the measurement of transmission coefficient T* given by (12). Further, DUT has to be reversed and the same procedure has to be repeated, resulting in determination of reflection coefficient S 22 and reverse transmission coefficient T* rev . After these four measurements the exact transmission coefficients S 21 and S 12 can be calculated. It must be emphasized, that for reversed DUT equation (12a) takes the modified form:
$$T_{rev}^{\ast} = \displaystyle{{{S_{12}}} \over {(1 - {\Gamma _1}{S_{22}})(1 - {\Gamma _2}{S_{11}})}}.$$
If the measured device is reciprocal, i.e. S 21 = S 12, only one measurement of transmission coefficient is necessary, therefore, the number of measurements can be reduced to three.
The proposed system for transmission coefficient measurement can be calibrated using several different sections of matched transmission lines having electrical length φ k (length of the first transmission line can be assumed zero) and two matched loads. It is worth mentioning that the requirements related to the impedance match of the needed calibration standards are not excessive. Since the total isolation between two measuring ports to which DUT is connected is a sum (if expressed in dB) of Butler matrix's isolation, external coupler's isolation, and attenuation of the utilized attenuator, the waves reflected from the imperfect calibration standards are almost undetectable by the applied power meters (more comprehensive analysis is presented in Section IV).
It can be observed, that having both measuring ports terminated with a matched load (T = 0), the measured power p i, L is equal to the corresponding coefficients q i :
$${q_i} = {p_{i,L}},$$
where i indicates the number of port at which the power is measured (i = 1, 2, … , N). In order to find the remaining coefficients A i , (10) can be rewritten as:
$${p_{i,k}} = {q_i}{\left \vert {1 + {A_i}{W_k}} \right \vert ^2},$$
where W k indicates the known transmission coefficient of kth calibration standard (k = 1, 2, … , K) and p i,k is the power measured at ith port, when kth calibration standard is applied. The coefficient W k , given by:
$${W_k} = \displaystyle{{\exp ( - j{\varphi _k})} \over {1 - {\Gamma _1}{\Gamma _2}\exp ( - 2j{\varphi _k})}},$$
corresponds to the kth section of matched transmission line having electrical length φ k . Having measured K values of power p i,k for each output port of the Butler matrix, one can obtain the coefficients A i as follows:
$${A_i} = \displaystyle{{\beta {\delta _i} - \gamma {\varepsilon _i} + j(\gamma {\delta _i} - \alpha {\varepsilon _i})} \over {\alpha \beta - {\gamma ^2}}},$$
where
$$\alpha = \sum\limits_{k = 1}^K {\sum\limits_{n = k + 1}^K {{{({\mathop{\rm Re}\nolimits} [{W_k} - {W_n}])}^2}}}, $$
$$\beta = \sum\limits_{k = 1}^K {\sum\limits_{n = k + 1}^K {{{({\mathop{\rm Im}\nolimits} [{W_k} - {W_n}])}^2}}}, $$
$$\gamma = \sum\limits_{k = 1}^K {\sum\limits_{n = k + 1}^K {({\mathop{\rm Re}\nolimits} [{W_k} - {W_n}])({\mathop{\rm Im}\nolimits} [{W_k} - {W_n}])}}, $$
$${\delta _i} = \sum\limits_{k = 1}^K {\sum\limits_{n = k + 1}^K {\left( {{\mathop{\rm Re}\nolimits} [{W_k} - {W_n}]\displaystyle{{{p_{i,k}} - {p_{i,n}}} \over {2{q_i}}}} \right)}}, $$
$${\varepsilon _i} = \sum\limits_{k = 1}^K {\sum\limits_{n = k + 1}^K {\left( {{\mathop{\rm Im}\nolimits} [{W_k} - {W_n}]\displaystyle{{{p_{i,k}} - {p_{i,n}}} \over {2{q_i}}}} \right)}}, $$
and k and n are the indices of the calibration standards (k = 1, 2, … , K, n = 2, 3, … , K). The genuine circle centers for transmission coefficient measurement can be derived using the calculated coefficients as follows:
$${c_i} = - \displaystyle{1 \over {{A_i}}}.$$
It can be observed that the presented calibration procedure can be applied to a measuring system having arbitrary number of ports at which the power is measured (N). Moreover, the number of sections of transmission lines utilized in calibration (K) can be also arbitrarily chosen; however, at least two are required. It is worth mentioning that a higher number of calibration standards decreases the influence of the measurement inaccuracy of the applied power meters on the calibration results.
Having the measuring system calibrated one can proceed to the measurements of transmission coefficient. The measured transmission coefficient T given by either (11) or (12) can be calculated as follows:
$$T = \displaystyle{{\mu \rho - \kappa \nu + j(\eta \rho - \mu \nu )} \over {\eta \kappa - {\mu ^2}}},$$
where
$$\eta = 2\sum\limits_{i = 1}^N {\sum\limits_{j = i + 1}^N {{{\left( {\displaystyle{{{\mathop{\rm Re}\nolimits} [{A_i}]} \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{{{\mathop{\rm Re}\nolimits} [{A_j}]} \over { \vert {A_j}{ \vert ^2}}}} \right)}^2}}}, $$
$$\kappa = 2\sum\limits_{i = 1}^N {\sum\limits_{j = i + 1}^N {{{\left( {\displaystyle{{{\mathop{\rm Im}\nolimits} [{A_i}]} \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{{{\mathop{\rm Im}\nolimits} [{A_j}]} \over { \vert {A_j}{ \vert ^2}}}} \right)}^2}}}, $$
$$\mu = 2\sum\limits_{i = 1}^N {\sum\limits_{j = i + 1}^N {\left[ {\left( {\displaystyle{{{\mathop{\rm Re}\nolimits} [{A_i}]} \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{{{\mathop{\rm Re}\nolimits} [{A_j}]} \over { \vert {A_j}{ \vert ^2}}}} \right)} \right. \cdot}} \left. {\left( {\displaystyle{{{\mathop{\rm Im}\nolimits} [{A_i}]} \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{{{\mathop{\rm Im}\nolimits} [{A_j}]} \over { \vert {A_j}{ \vert ^2}}}} \right)} \right],$$
$$\eqalign{& \nu = \sum\limits_{i = 1}^N {\sum\limits_{j = i + 1}^N {\left[ {\left( {\displaystyle{{{\mathop{\rm Re}\nolimits} [{A_i}]} \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{{{\mathop{\rm Re}\nolimits} [{A_j}]} \over { \vert {A_j}{ \vert ^2}}}} \right) \cdot} \right.}} \cr & \quad \left. {\left( {\displaystyle{1 \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{1 \over { \vert {A_j}{ \vert ^2}}} - \displaystyle{{{p_i}} \over {{q_i} \vert {A_i}{ \vert ^2}}} + \displaystyle{{{p_j}} \over {{q_j} \vert {A_j}{ \vert ^2}}}} \right)} \right],} $$
$$\eqalign{& \rho = \sum\limits_{i = 1}^N {\sum\limits_{j = i + 1}^N {\left[ {\left( {\displaystyle{{{\mathop{\rm Im}\nolimits} [{A_i}]} \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{{{\mathop{\rm Im}\nolimits} [{A_j}]} \over { \vert {A_j}{ \vert ^2}}}} \right) \cdot} \right.}} \cr & {\kern 1pt} \quad \left. {\left( {\displaystyle{1 \over { \vert {A_i}{ \vert ^2}}} - \displaystyle{1 \over { \vert {A_j}{ \vert ^2}}} - \displaystyle{{{p_i}} \over {{q_i} \vert {A_i}{ \vert ^2}}} + \displaystyle{{{p_j}} \over {{q_j} \vert {A_j}{ \vert ^2}}}} \right)} \right],} $$
and i and j are the indices of the Butler matrix output ports with power meter connected (i = 1, 2, …, N, j = 2, 3, …, N).
IV. INFLUENCE OF THE MODEL SIMPLIFICATION ON TRANSMISSION COEFFICIENT MEASUREMENT
As it is said in Section III, the applied simplification of the measuring system model for transmission coefficient measurement introduces some inaccuracy which has to be investigated in order to confirm the usefulness of such an approach. The utilized model takes into account the imperfect impedance match of all used components and only imperfect isolations have been neglected. Therefore, to ensure clarity of the following analysis, the impact of the introduced simplification of the proposed model has been verified with the assumed ideal impedance match of all components, as shown in Fig. 2(c). It has to be emphasized that influence of the imperfect isolations on the measurement results is dependent on almost each S-parameter distinguished in Fig. 2(a) including the reflection coefficients seen at the measuring ports and measured scattering parameters of DUT. However, the consideration of such a complex dependence requires too many parameters and the presentation of obtained results would not be legible. Therefore, the presented analysis of the imperfect isolation's impact on transmission coefficient measurements is rather an estimate than the exact value of maximum error, nevertheless, assuming the reasonable impedance match of the utilized components (say 20 dB) it can be sufficiently accurate.
Utilizing the signal flow as shown in Fig. 2(c) one can derive the expression for the normalized measured power as:
$${p_i} = {\left \vert {\displaystyle{{{S_{C31}}{S_{A21}}{S_{Bik}}} \over {1 - {S_{A21}}{S_{Bnk}}{S_{12}}{S_{C32}}}} + \displaystyle{{{S_{C21}}{S_{21}}{S_{Bin}}} \over {1 - {S_{21}}{S_{Bkn}}{S_{A12}}{S_{C23}}}}} \right \vert ^2}.$$
Assuming the reciprocity of utilized components and measured device one can introduce:
$$T = {S_{21}} = {S_{12}},$$
$$I = {S_{Bkn}}{S_{A12}}{S_{C23}} = {S_{Bnk}}{S_{A21}}{S_{C32}},$$
where I is the isolation between measuring ports derived utilizing signal flow as shown in Fig 2(c). Further (29) can be expressed as follows:
$${p_i} = {\left \vert {\displaystyle{{{S_{C31}}{S_{A21}}{S_{Bik}}} \over {1 - TI}}} \right \vert ^2}{\left \vert {1 + \displaystyle{{{S_{C21}}{S_{Bin}}} \over {{S_{C31}}{S_{A21}}{S_{Bik}}}}T} \right \vert ^2},$$
which can be simplified to the form:
$${p_i} = \displaystyle{{q_i^{\ast}} \over {{{\left \vert {1 - TI} \right \vert} ^2}}}{\left \vert {1 + A_i^{\ast} T} \right \vert ^2}.$$
It can be observed that (33) has the form being close to (10). For analyzing (33) one can see that the imperfect isolation can be considered as a factor modifying coefficients q i as shown in (10) without affecting remaining coefficients A i . Since the coefficients q i are linearly related to the measured power p i , one can introduce the relative power reading error caused by the imperfect isolations:
$${e_P} = \left \vert {\displaystyle{1 \over {{{\left \vert {1 - TI} \right \vert} ^2}}} - 1} \right \vert. $$
It is seen that e P depends on both isolation I and the measured transmission coefficient T. Further investigation reveals that the maximum relative error for a given magnitude of the measured reflection coefficient can be obtain if:
$$\arg [TI] = 0.$$
Therefore, the maximum relative power reading error can be expressed as a function of magnitudes:
$${e_{P\max}} = \vert T \vert \vert I \vert \displaystyle{{2 - \vert T \vert \vert I \vert} \over {{{(1 - \vert T \vert \vert I \vert )}^2}}}.$$
To verify the impact of the model simplification on the measurement accuracy the obtained values of e Pmax given by (36) have to be compared with the measurement inaccuracy of the power meters, as shown in Fig. 3. A typical measurement accuracy of the high-class microwave power meters is about 0.1 dB, which corresponds to the relative power reading error being equal to 0.0233 indicated by the gray line. It is seen that if the isolation between measuring ports is not worse than 40 dB the power reading error caused by simplification of the measuring system model is lower than the error resulting from the power meters uncertainty and decreases with smaller magnitudes of the measured transmission coefficient.
Estimated maximum relative power reading error versus magnitude of the measured transmission coefficient (black lines) for four different values of isolations I compared with the 0.1 dB uncertainty of the power meter (gray line).
It is worth mentioning that the obtained maximum relative power reading error is an estimate derived on the basis of the simplified signal flow as shown in Fig 2(c). In practice, however, the imperfect impedance match of measuring port and measured device can increase this error. On the other hand, in the presented analysis of the simplified signal flow as shown in Fig. 2(c) the worst case given by (35) has been assumed which almost never occurs in practice, what justifies the presented analysis.
Another analysis has to be performed for the simplification of the formula for transmission coefficient (11) into the form given by (12). Since the ratio of (11) and (12) is equal to:
$$\displaystyle{{{T^{\ast}}} \over T} = 1 - \displaystyle{{{S_{12}}{S_{21}}{\Gamma _1}{\Gamma _2}} \over {(1 - {\Gamma _1}{S_{11}})(1 - {\Gamma _2}{S_{22}})}},$$
the maximum magnitude and phase error resulting from such a simplification can be easily estimated. It can be observed that (12) takes into account the imperfect impedance match of the measured device, therefore, for better clarity of this investigation it has been assumed that S 11 = S 22 = 0. Moreover, it can be concluded from (37) that the maximum error will occur for high magnitude of transmission coefficients of DUT. Hence, zero-insertion loss has been assumed (S 21 = S 12 = 1). The reflection coefficients seen at the measuring ports Γ1 and Γ2 are assumed to be equal (called further Γ MS ). The mentioned assumptions allow one to rewrite (37) as follows:
$$\displaystyle{{{T^{\ast}}} \over T} = 1 - \Gamma _{MS}^2. $$
Since (38) represents the ratio of the exact and approximated values of transmission coefficient, it allows for estimation of the maximum measurement error caused by the introduced simplification of (11) into (12) in terms of both magnitude and phase. The maximum magnitude error expressed in dB is equal to:
$${\Delta _{mag}} = \pm 20{\log _{10}}\left( {1 - {{\left \vert {{\Gamma _{MS}}} \right \vert} ^2}} \right),$$
whereas the maximum phase error can be estimated as follows:
$${\Delta _{phase}} \cong \pm \arctan ( \vert {\Gamma _{MS}}{ \vert ^2}).$$
The estimated errors versus magnitude of the impedance match of the measured ports have been plotted in Fig. 4. It is seen, that for the impedance match not worse than 20 dB the simplification given by (12) introduces the magnitude error not exceeding 0.1 dB and the phase error being lower than 1°. Moreover, it has to be emphasized that for any greater insertion loss of DUT these errors are smaller.
Estimated maximum error of transmission coefficient measurement introduced by the utilization of simplified formula (12) for an ideally matched two-port having zero-insertion loss (S 21 = S 12 = 1).
V. ACCURACY INVESTIGATION OF THE ADJUSTABLE MEASUREMENT SYSTEM
The proposed measuring system features adjustable circle centers’ distribution for both reflection and transmission coefficient measurements. The distance between circle centers and the origin of a complex plane can be tuned according to (7) and (8), which allows for a significant enhancement of the measurement accuracy. The concept of scalable circle centers’ distribution as well as the procedure allowing to estimate the maximum magnitude and phase errors for the multiport reflectometers featuring the given circle centers’ distribution have been widely described in [Reference Staszek, Gruszczynski and Wincza16].
Utilizing this approach the accuracy of the proposed measuring system has been estimated for different attenuation values of the applied attenuator A, which provide the magnitudes of circle centers in range from 0.1 to 1. It has to be underlined that the accuracy of the transmission coefficient measurement can be estimated exactly with the same manner as the accuracy of the reflection coefficient measurement (the same number of circles with the same mutual arrangement). Moreover, as it has been shown in Section IV, the transmission coefficient measurement is significantly more affected by the applied power meters inaccuracy rather than by imperfect isolations, which are not taken into account in the developed model. Therefore, the measurement error for both types of the scattering parameters, i.e. reflection and transmission coefficients, can be estimated with the use of the same analysis. The estimated maximum magnitude and phase measurement errors obtained for different magnitudes of circle centers are presented in Figs 5 and 6. It can be seen that for the measurement of highly reflective elements (|S 11| ≈ 1) or low-insertion loss elements (|S 21| ≈ 1) the maximum magnitude of circle centers’ distribution ensures the highest accuracy of the measured phase, whereas, when well-matched devices or high-insertion loss two-ports are to be measured, the measurement precision can be increased by decreasing the circle centers’ distribution magnitude.
The maximum magnitude measurement error versus magnitude of measured S-parameter for the proposed multiport system with the adjustable circle centers’ distribution. The assumed power detectors uncertainty Δ PD = 0.1 dB.
The maximum phase measurement error versus magnitude of measured S-parameter for the proposed multiport system with the adjustable circle centers’ distribution. The assumed power detectors uncertainty Δ PD = 0.1 dB.
VI. MEASUREMENTS
The proposed measuring system has been build for an experimental verification. As the directional coupler Q 1, the coupled-line multisection directional coupler as shown in [Reference Gruszczynski and Wincza21] has been utilized. It features the nominal coupling being equal to 8.34 dB and operates in the frequency range 1–5 GHz. Since that coupler is used for the reference power measurement, a low coupling level is advantageous. The 3 dB coupled-line directional coupler reported previously in [Reference Gruszczynski and Wincza22] which operates in the same frequency range has been used as the coupler Q 2. For the signals summation the broadband 4 × 4 Butler matrix, presented in [Reference Gruszczynski and Wincza23] having the same operational bandwidth has been applied. In order to provide scalable circle centers’ distribution, SMA attenuators have been utilized. For power measurements five USB Mini-Circuits Power Sensors PWR-8GHS, featuring power measurement uncertainty equal to 0.1 dB, have been applied. Photographs of the developed system in configuration for reflection and transmission coefficients measurements are shown in Fig. 7.
Photograph of the developed measuring system in configuration for reflection coefficient measurement (a) and for transmission coefficient measurement (b).
To ensure proper measurements the developed multiport system has been calibrated using two separate calibration procedures for the measurement of reflection and transmission coefficients. The former has been performed following the procedure presented in [Reference Staszek, Kaminski, Rydosz, Gruszczynski and Wincza20], utilizing four different calibration standards, i.e. open, short, offset-open and offset-short, and additionally matched load, whereas for transmission reflection measurement, the procedure described in Section III has been applied. Both calibrations have been performed for different attenuations of the adjustable attenuator A, in order to provide circle centers’ distributions having appropriately scaled magnitudes. The values of attenuations as well as theoretical magnitudes of circle centers’ distributions are listed in Table 3.
Attenuations of the attenuator and corresponding theoretical magnitude of circle centers’ distributions for reflection and transmission coefficient measurements.
The circle centers’ distributions obtained during the calibration procedures performed in frequency range 1–5 GHz are shown in Figs 8 and 9. As it can be observed for both calibrations, regardless of the chosen attenuation of the adjustable attenuator, a highly uniform arrangement of the circle centers has been achieved.
Circle centers’ distribution of the proposed measuring system for reflection coefficient measurements. The results are obtained during the calibration process in the frequency range 1–5 GHz for different theoretical magnitudes of circle centers’ distributions: 1.0 (a), 0.8 (b), 0.7 (c), and 0.5 (d).
The circle centers’ distribution of the proposed measuring system for transmission coefficient measurements. The results are obtained during the calibration process in the frequency range 1–5 GHz for different theoretical magnitudes of circle centers’ distributions: 1.0 (a), 0.8 (b), 0.7 (c), and 0.5 (d).
The obtained circle centers’ distributions allow for proper measurements of scattering parameters. However, due to the simplification of the measuring system for transmission coefficient measurement, the isolation between the measuring ports, to which the measured device is connected, as well as their impedance match have to be considered. Figure 10 presents the isolation between the measuring ports together with their impedance match. One can observe that the mentioned isolation is not worse than 55 dB in the entire considered frequency range. As it is shown in Fig. 3, such a level of isolation introduces power measurement error being at least ten times lower than the one resulting from the power meters inaccuracy. Therefore, the proposed simplification can be applied in the developed measuring system. Moreover, a good impedance match of both measuring ports allows for using the simplified formula (12) for the measured transmission coefficient.
Measured reflection coefficients seen at the measuring ports and the measured isolation between the measuring ports.
Performance of the developed measuring system has been verified by measurements of reflection and transmission coefficients of several attenuators. For the reflection coefficient measurements eight shorted attenuators having attenuation from 1 to 20 dB have been used. For the transmission coefficient measurement 12 attenuators have been measured. Their attenuations vary from 1 to 50 dB. To examine the measurement accuracy the reference measurements with the use of commercial vector network analysed N5224A by Agilent have been performed. VNA has been calibrated with the use of a standard SOLT technique, with 85052D Calibration Kit by Agilent.
The measurement accuracy of the proposed system has been estimated using the procedure described in [Reference Staszek, Gruszczynski and Wincza16] for the circle centers’ distribution having magnitude 0.5 which corresponds to the highest accuracy among all circle centers’ distributions listed in Table 3. However, in the utilized procedure the circle centers’ distribution is assumed to be ideal (without any fluctuations in terms of frequency), therefore, it is rather an estimate than the exact maximum error. The measurement results are presented in Figs 11 and 12, whereas, the maximum measurement errors are shown in Figs 13 and 14. The measured values and the results obtained with VNA are in a very good agreement within the entire operational frequency range and the reference values in each case are located within the estimated accuracy intervals of the proposed system. A negligible discrepancy seen for smaller magnitudes of measured values is caused by an imperfect linearity of the applied power meters and the noise present during the measurement.
Measured reflection coefficients of shorted attenuators: magnitude (a) and phase (b). Solid lines indicate the measurement results obtained with the use of the proposed system, dashed lines present the reference values obtained with Agilent N5224A VNA. Measurement accuracy of the proposed system is marked black, whereas the accuracy of VNA is marked gray.
Measured transmission coefficients of the attenuators: magnitude (a) and phase (b). Solid lines indicate the measurement results obtained with the use of the proposed system, dashed lines present the reference values obtained with Agilent N5224A VNA. Measurement accuracy of the proposed system is marked black, whereas the accuracy of VNA is marked gray (not seen due to small values).
Maximum magnitude (a) and phase (b) measurement error versus the magnitude of the measured reflection coefficient for different magnitudes of circle centers’ distribution. As the reference values the measurements with N5224A VNA have been used.
Maximum magnitude (a) and phase (b) measurement error versus the magnitude of the measured transmission coefficient for different magnitudes of circle centers’ distribution. As the reference values the measurements with N5224A VNA have been used.
Furthermore, it is seen in Figs 13 and 14, that the scalable circle centers’ distribution significantly improves the measurement accuracy for both reflection and transmission coefficient measurements. In general the presented maximum magnitude and phase measurement errors for both reflection and transmission coefficient measurements clearly correspond to the theoretical predictions are shown in Figs 5 and 6, however, some discrepancy is seen. It must be emphasized that the theoretical predictions are shown in Figs 5 and 6 have been estimated assuming precisely known circle centers’ distribution. However, in practical measuring systems the circle centers’ distribution is calculated in calibration procedure which always introduces inaccuracy due to power measurement error. Furthermore, as seen in Fig. 11 the vector network analyser used for measuring the reference values features the inaccuracy being comparable to the proposed system. Hence, the measurement error as shown in Fig. 13 is not the difference between measured and genuine values of the reflection coefficients, but in fact is the difference between measured values and the reference values which are somehow deteriorated. It can be proved by the analysis of results of the transmission coefficient measurement. In Fig. 12 it is seen, that the utilized VNA features significantly lower measurement inaccuracy comparing to the proposed measuring system. Since the reference values are very close to the genuine ones, the calculated measurement error as shown in Fig. 14 is in agreement with the theoretical assumption presented in Figs 5 and 6. Finally, as described in Section V, the measurement accuracy strictly depends on the circle centers’ distribution. The measurements have been performed within the frequency range covering more than 2 octaves, therefore, some deterioration of circle centers’ location occurs. Since each circle center fluctuates independently, the mutual arrangement of these four points somehow changes in terms of frequency which affects the measurement accuracy. Therefore, the maximum measurement errors insignificantly differ from the estimated ones.
For further verification of the proposed measuring system a high-pass filter has been measured. The obtained results are presented in Fig. 15. As it is seen the measurement results are very close to the reference values obtained with the use of VNA. The negligible discrepancy between measured S-parameters and the reference values as well as the distortion of the transmission coefficient for magnitudes smaller than 35 dB are caused by two major factors: (i) utilization of the non-ideal SMA adaptor, which was necessary to connect the measured filter to the measuring system, and (ii) imperfect matched load featuring VSWR ≈ 1.1 which has been used for filter termination in reflection coefficient measurement. Nevertheless, the proposed measuring system allows for a precise measurement of S-parameters having magnitude not smaller than −35 dB, which is more than sufficient in many practical applications.
Measured S-parameters of the high-pass filter. Magnitude of reflection coefficient S 11 and transmission coefficient S 21 (a), phase of reflection coefficient S 11 (b) and phase of transmission coefficient S 21 (c). Solid lines indicate the measurement results obtained with the use of the proposed system, dashed lines present the reference values obtained with Agilent N5224A VNA.
VII. CONCLUSION
In this paper, a novel multiport system for S-parameter measurements has been proposed. It is composed of two directional couplers and a standard 4 × 4 Butler matrix. The proposed system does require neither additional power dividers nor isolators, therefore, the multioctave operation can be achieved. It has been shown that the proposed measuring system provides scalable circle centers’ distribution for both reflection and transmission coefficient measurements ensuring the enhanced measurement accuracy. Moreover, the impact of the imperfect isolations on the measurement accuracy for transmission coefficient measurement has been discussed and the estimated maximum power reading errors are given. Furthermore, the analytical calibration procedure for transmission coefficient measurement involving the least-square approach has been shown. The presented calibration can be applied in the multiport system having an arbitrary number of ports at which the power is measured. The system performance has been verified in a wide frequency range 1–5 GHz by the measurements of several microwave components. The measurement results show a very good agreement with the ones obtained with the use of a commercial VNA. Moreover, by the utilization of the scalable circle centers’ distributions a significant enhancement of the measurement accuracy has been achieved for reflection and transmission coefficient measurements over the entire frequency range, which proves the usability of the proposed measuring system.
ACKNOWLEDGEMENT
This work was supported by the National Science Centre under grant no. DEC-2013/09/N/ST7/01219.
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Kamil Staszek received his M.Sc. and Ph.D. degrees in Electronics Engineering from the AGH University of Science and Technology, Cracow, Poland, in 2011 and 2015, respectively. His main scientific interests are multiport measurement techniques and design of the broadband passive components. He has coauthored 18 journal and 20 conference scientific papers. Currently he is with Department of Electronics at AGH University of Science and Technology.
Slawomir Gruszczynski received his M.Sc. and Ph.D. degrees in Electronics and Electrical Engineering from the Wroclaw University of Technology, Poland, in 2001 and 2006, respectively. From 2001 to 2006 he was with Telecommunications Research Institute, Wroclaw Division, and from 2005 to 2009, he worked at the Institute of Telecommunications, Teleinformatics and Acoustics, Wroclaw University of Technology. In 2009, he joined the Faculty of Informatics, Electronics and Telecommunications at AGH University of Science and Technology where he became a Head of the Department of Electronics in 2012. He has coauthored more than 40 journals and more than 50 conference scientific papers. He is a member of the IEEE, and a member of Young Scientists’ Academy at Polish Academy of Sciences (PAN) and Committee of Electronics and Telecommunications at Polish Academy of Sciences (PAN).
Krzysztof Wincza received his M.Sc. and Ph.D. degrees in Electronics and Electrical Engineering from the Wroclaw University of Technology, Poland, in 2003 and 2007, respectively. In 2007, he joined the Institute of Telecommunications, Teleinformatics and Acoustics, Wroclaw University of Technology. In 2009, he joined the Faculty of Electronics at AGH University of Science and Technology and became an Assistant Professor. Dr., Wincza was the recipient of The Youth Award presented at the 10th National Symposium of Radio Sciences (URSI) and the Young Scientist Grant awarded by the Foundation for Polish Science in 2001 and 2008, respectively. He has coauthored more than 40 journals and more than 50 conference scientific papers.
