Diversity Pertaining to the Attenuation of the RF Disturbance Suppression Power Line Filters

Abstract

Filters for the suppression of the RF disturbances in power lines are passive; wideband stop filters attenuating common as well as differential disturbances should be insensitive to the variation in the impedance at the input and the output. The design and verification process of such filters differs entirely from that used for the signal RF filters. In the document CISPR 17, entitled “Methods of measurement of the suppression characteristics of passive EMC filtering devices”, two definitions of the attenuation of the common and differential modes are presented. The first is simply direct measurement, and the second is based on a calculation using the single-ended S-parameters of the multiport. These two definitions are the source of diversity. In this paper, we hypothesize that the attenuations achieved with these two methods are identical only if the filter is in perfect balance. Consequently, different attenuations can be achieved. The requirement of balance can be violated due to the tolerances of the components used in the filter. The paper begins with a simple unbalanced four-port, for which analytical formulas are derived. It is also shown that an approach with single-ended S-parameters enables the calculation of the mode conversion between the input and the output, which is called the conversion loss. This is the measure of the grade of the imbalance. Thereafter, the common-mode attenuation of an exemplary RF suppression filter is derived with the presented methods. Additionally, the conversion loss of the filter is calculated. The comparison of the conversion loss with both attenuations is satisfactory, proving the hypothesis formulated in the paper.

1. Introduction

The RF disturbance suppression power line filter differs completely from the signal filters. The first one, unlike the second, must be:

• A wideband passive stop filter, usually from 9 kHz to 30 MHz;
• Its attenuation cannot be influenced by the variation in the external impedance at the power or the load side;
• It must perform attenuation for the common as well as the differential mode.

The second and the third features are due to the fact that the RF disturbance suppression power line filters are always inserted in a place with an unknown source and load impedance. Moreover, the balance of the power line cannot be ensured. Therefore both the common and differential modes must be attenuated. Additionally, information about the conversion from the common to the differential mode in the filter is very important. This indicates the degree of its balance.

Commonly, filter attenuations are measured with the vector network analyzer; therefore, they refer to a $50 \mathsf{\Omega }$ source and a $50 \mathsf{\Omega }$ load resistance, but other formats are possible [1,2,3,4]. Typical measurement configurations in the 50 Ω/50 Ω format allow the filter characteristics to be defined. The applicability of the RF suppression filters is determined primary with their attenuation [5,6]. Filters are an essential element used in power grid facilities where there are various types of interference (affecting grid reliability) caused, among others, by the impact of overvoltages [7,8].

In [9], two methods for the verification of the reflection and attenuation of multiports were presented. One was based on direct measurement and the other on a modal S-matrix built from the single-ended S-Parameters. This paper focuses on the ambiguity due to these two definitions and provides a solution in which both attenuations are identical. Due to the fact that the parameters of the filter depend on the materials of its components, one filter was tested for all cases in the article [10,11].

In this paper, only common-mode attenuation is considered. The paper is organized as follows: In Section 2 and Section 3, a simple four-port is presented, for which the reflection as well as attenuation can be derived analytically. The direct measurement of the reflection and transmission is presented in Section 2 and the modal S-Parameters, such as the reflection, the transmission, and the conversion from the common to the differential mode, are presented in Section 3. Section 2 and Section 3 provide an introduction to Section 4, which is devoted to the metrological verification of both approaches, where the analytical formulas from Section 2 and Section 3 are verified in Section 3.1, and the real filter is verified in Section 3.2. Finally, the conclusions are presented in Section 5.

Appendix A contains the formulas for the recalculation of the S-Matrix for the arbitrary input and output impedance.

2. Direct Measurements of the Common-Mode S-Parameters of a Four-Port

In this section, the approach based on the direct measurement is presented. Its core is illustrated with a simple four-port composed of impedance $Z_L$ inserted between ports 1 and 2, along with impedance $Z_N$ inserted between ports 3 and 4, as shown in Figure 1.

For direct measurement of the common-mode reflection and transmission coefficients of the four-port presented in Figure 1, the input ports 1 and 3, as well as the output ports 2 and 4, must be short circuited, as shown in Figure 2.

We note that the device being tested between the input I and output O, presented in Figure 2, is actually a two-port; therefore, the modal analysis of the reflection and transmission does not make sense. Simply speaking, there exists only one mode, named the common mode.

For the circuitry presented in Figure 2, the following formula for the common-mode reflection is valid

$$S_{II}^{CC}=S_{OO}^{CC}=\frac{Z_L{Z}_N}{Z_L{Z}_N+2Z_0{Z}_L+2Z_0{Z}_N},$$

(1)

along with the formula for the common-mode transmission

$$S_{IO}^{CC}=S_{OI}^{CC}=\frac{2Z_0(Z_L+Z_N)}{Z_L{Z}_N+2Z_0{Z}_L+2Z_0{Z}_N}.$$

(2)

Notice that all the modal S-Parameters discussed so far refer to the impedance $Z_0$.

3. Derivation of the Modal S-Parameters from the Single-Ended Ones by a Four-Port

3.1. Relation between the Single-Ended and Modal S-Parameters

The single-ended S-matrix of a multiport is an S-matrix for an arbitrarily chosen pair of ports. The single-ended S-Parameters of the four-port presented in Figure 1 are as follows:

$$\begin{array}{ccc}\hfill S_{11}=S_{22}& =& \frac{Z_L}{2Z_0+Z_L}\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill S_{33}=S_{44}& =& \frac{Z_N}{2Z_0+Z_N}\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill S_{12}=S_{21}& =& \frac{2Z_0}{2Z_0+Z_L}\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill S_{34}=S_{43}& =& \frac{2Z_0}{2Z_0+Z_N}\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill S_{13}=S_{31}& =& 0\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill S_{24}=S_{42}& =& 0\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill S_{14}=S_{41}& =& 0\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill S_{23}=S_{32}& =& 0.\hfill \end{array}$$

(10)

Due to [9], the relations between the single-ended S-Parameters presented above and the common-mode input and common-mode output modal S-Parameters $S^{CC}$ yield

$$\begin{array}{c}\hfill \begin{array}{c}{S}^{CC}=\left[\begin{array}{cc}{S}_{II}^{CC}& S_{IO}^{CC}\\ \\ S_{OI}^{CC}& S_{OO}^{CC}\end{array}\right]\hfill \\ \\ =\frac{1}{2}\left[\begin{array}{cc}{S}_{11}+S_{31}+S_{13}+S_{33}& S_{12}+S_{32}+S_{14}+S_{34}\\ \\ S_{21}+S_{41}+S_{23}+S_{43}& S_{22}+S_{42}+S_{24}+S_{44}\end{array}\right].\hfill \end{array}\end{array}$$

(11)

For the specific case presented in Figure 1, the general formula presented by Equation (11), with the introduction of the formulas from Equations (3)–(10), can be simplified as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{S}^{CC}\left(\frac{Z_0}{2}/\frac{Z_0}{2}\right)=\left[\begin{array}{cc}{S}_{II}^{CC}& S_{IO}^{CC}\\ \\ S_{OI}^{CC}& S_{OO}^{CC}\end{array}\right]\hfill \\ \\ =\frac{1}{2}\left[\begin{array}{cc}\frac{Z_L}{2Z_0+Z_L}+\frac{Z_N}{2Z_0+Z_N}& \frac{2Z_0}{2Z_0+Z_L}+\frac{2Z_0}{2Z_0+Z_N}\\ \\ \frac{2Z_0}{2Z_0+Z_L}+\frac{2Z_0}{2Z_0+Z_N}& \frac{Z_L}{2Z_0+Z_L}+\frac{Z_N}{2Z_0+Z_N}\end{array}\right].\hfill \end{array}\end{array}$$

(12)

Note that the parameters in Equation (12) refer to the impedances $Z_0/2$ at the input and the output.

An analogous derivation can be performed for the common-mode input and the differential-mode output modal S-Parameters $S^{CD}$, referring to [9],

$$\begin{array}{c}\hfill \begin{array}{c}{S}^{CD}=\left[\begin{array}{cc}{S}_{II}^{CD}& S_{IO}^{CD}\\ \\ S_{OI}^{CD}& S_{OO}^{CD}\end{array}\right]\hfill \\ \\ =\frac{1}{2}\left[\begin{array}{cc}{S}_{11}+S_{31}-S_{13}-S_{33}& S_{12}+S_{32}-S_{14}-S_{34}\\ \\ S_{21}+S_{41}-S_{23}-S_{43}& S_{22}+S_{42}-S_{24}-S_{44}\end{array}\right].\hfill \end{array}\end{array}$$

(13)

For the specific case presented in Figure 1, the general formula presented by Equation (13), with the introduction of the formulas from Equations (3)–(10), can be simplified as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{S}^{CD}\left(\frac{Z_0}{2}/2Z_0\right)=\left[\begin{array}{cc}{S}_{II}^{CD}& S_{IO}^{CD}\\ \\ S_{OI}^{CD}& S_{OO}^{CD}\end{array}\right]\hfill \\ \\ =\left[\begin{array}{cc}\frac{Z_0(Z_L-Z_N)}{(2Z_0+Z_L)(2Z_0+Z_N)}& \frac{Z_0(Z_N-Z_L)}{(2Z_0+Z_L)(2Z_0+Z_N)}\\ \\ \frac{Z_0(Z_N-Z_L)}{(2Z_0+Z_L)(2Z_0+Z_N)}& \frac{Z_0(Z_L-Z_N)}{(2Z_0+Z_L)(2Z_0+Z_N)}\end{array}\right].\hfill \end{array}\end{array}$$

(14)

Note that the parameters in Equation (14) refer to the impedance $Z_0/2$ at the input and $2Z_0$ at the output.

3.2. Power Waves

The incident power wave a can be introduced either with the incident voltage $U^{+}$ or incident current $I^{+}$ as follows:

$$a=\frac{U^{+}}{\sqrt{Z_0}}=I^{+}\sqrt{Z_0}.$$

(15)

Analogously, the reflected power wave b can be introduced either with the reflected voltage $U^{-}$ or reflected current $I^{-}$ as follows:

$$b=\frac{U^{-}}{\sqrt{Z_0}}=I^{-}\sqrt{Z_0}.$$

(16)

3.3. Modal Common Mode

With the modal common mode, the approach with the incident and reflected currents is more convenient, as illustrated in Figure 3.

The current in the return path in Figure 3a is actually the double common-mode incident current at the input

$$I_I^{C+}=\frac{EMF}{2Z_0}.$$

(17)

According to Figure 3b, the common-mode current $I^C$ yields

$$I^C=\frac{EMF}{2}\left(\frac{1}{2Z_0+Z_L}+\frac{1}{2Z_0+Z_N}\right).$$

(18)

The common mode current reflected at the input $I_I^{C-}$ complements the incident current $I_I^{C+}$, such that $I^C=I_I^{C+}-I_I^{C-}$, yielding

$$I_I^{C-}=\frac{EMF}{4Z_0}\left(\frac{Z_L}{2Z_0+Z_L}+\frac{Z_N}{2Z_0+Z_N}\right).$$

(19)

The common-mode incident power wave at the input port yields

$$a_I^C=I^{C+}\sqrt{Z_0}=\frac{EMF}{2Z_0}\sqrt{Z_0}.$$

(20)

The common-mode reflected power wave at the input port yields

$$b_I^C=I^{C-}\sqrt{Z_0}=\frac{EMF}{4Z_0}\left(\frac{Z_L}{2Z_0+Z_L}+\frac{Z_N}{2Z_0+Z_N}\right)\sqrt{Z_0}.$$

(21)

Finally, the common-mode reflection coefficient at the input $S_{II}^{CC}=b_I^C/a_I^C$ is identical, as proven in Equation (12).

The common-mode current transmitted from the input to the output is identical to the actual common-mode current

$$I^C=\frac{EMF}{2}\left(\frac{1}{2Z_0+Z_L}+\frac{1}{2Z_0+Z_N}\right).$$

(22)

The reflected common-mode power wave at the output $b_O^C$ yields

$$b_O^C=I^C\sqrt{Z_0}=\frac{EMF}{2}\left(\frac{1}{2Z_0+Z_L}+\frac{1}{2Z_0+Z_N}\right)\sqrt{Z_0}.$$

(23)

Finally, the common-mode transmission coefficient between the output and the input $S_{IO}^{CC}=b_O^C/a_I^C$ is identical, as proven in Equation (12).

3.4. Modal Common Mode to Differential Mode Conversion

According to Figure 3b the differential-mode current $I^D$ yields

$$I^D=\frac{EMF}{2}\frac{Z_N-Z_L}{(2Z_0+Z_L)(2Z_0+Z_N)}.$$

(24)

Evidently, $b_O^D=I^D$ and $b_I^D=-I^D$ because there is no source at the output. Therefore,

$$\begin{array}{ccc}\hfill S_{IO}^{CD}=S_{OI}^{CD}& =& \frac{b_O^D}{a_I^C}\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill S_{II}^{CD}=S_{OO}^{CD}& =& \frac{b_I^D}{a_I^C}.\hfill \end{array}$$

(26)

Finally, the conversion matrix of the common mode at the input to the differenial mode at the output is identical, as proven in Equation (14).

4. Verification and Application of Both Methods

The transmission coefficients of the matrix for conversion of the common mode at the input/output referred to $25 \mathsf{\Omega }$, and to the common mode at the output/input, they referred to $25 \mathsf{\Omega }$ resistance; they were calculated according to Equation (11), which yielded

$$\begin{array}{c}\hfill \begin{array}{ccc}{S}_{IO}^{CC}(25\mathsf{\Omega }/25\mathsf{\Omega })& =& \frac{1}{2}\left(S_{12}+S_{32}+S_{14}+S_{34}\right)\\ \\ S_{OI}^{CC}(25\mathsf{\Omega }/25\mathsf{\Omega })& =& \frac{1}{2}\left(S_{21}+S_{41}+S_{23}+S_{43}\right)\end{array}.\end{array}$$

(27)

They were identical due to reciprocity.

In order to refer the S-Parameters formulated in Equation (27) to $50 \mathsf{\Omega }$ resistance at the input and to $50 \mathsf{\Omega }$ resistance at the output, they must be transposed to a Z-Matrix, according to Equation (A1); thereafter, the Z-Matrix must be transposed to an S-Matrix referring to $Z_{01}=Z_{02}=50 \mathsf{\Omega }$ reference impedance with Equation (A3), obtaining $S_{IO}^{CC}(50 \mathsf{\Omega }/50 \mathsf{\Omega })$ and $S_{IO}^{CC}(50 \mathsf{\Omega }/50 \mathsf{\Omega })$.

The transmission coefficients of the matrix for conversion of the common mode at the input/output referred to $25 \mathsf{\Omega }$, and to the differential mode at the output/input, they referred to $100 \mathsf{\Omega }$ resistance; they were calculated according to Equation (13), which yielded

$$\begin{array}{c}\hfill \begin{array}{ccc}{S}_{IO}^{CD}(25\mathsf{\Omega }/100\mathsf{\Omega })& =& \frac{1}{2}\left(S_{12}+S_{32}-S_{14}-S_{34}\right)\\ \\ S_{OI}^{CD}(100\mathsf{\Omega }/25\mathsf{\Omega })& =& \frac{1}{2}\left(S_{21}+S_{41}-S_{23}-S_{43}\right)\end{array}.\end{array}$$

(28)

They were also identical due to reciprocity.

In order for the S-Parameters formulated in Equation (28) to refer to $50 \mathsf{\Omega }$ resistance at the input and to $50 \mathsf{\Omega }$ resistance at the output, they must be transposed to the Z-Matrix, according to Equation (A1); thereafter, the Z-Matrix must be transposed to an S-Matrix referring to $Z_{01}=Z_{02}=50 \mathsf{\Omega }$ reference impedance with Equation (A3) obtaining $S_{IO}^{CD}(50 \mathsf{\Omega }/50 \mathsf{\Omega })$ and $S_{IO}^{DC}(50 \mathsf{\Omega }/50 \mathsf{\Omega })$.

The verification of both approaches was conducted with the two-port Vector Network Analyser ZVL (9 kHz–3 GHz), which was calibrated with the calibration kit ZV- Z270, both manufactured by R&S.

4.1. Verification of the Approach with the Simple Four-Port

The features of the simple four-port shown in Figure 1, built for verification of the approach presented here, are shown in Figure 4 and Figure 5 and are described below.

The four-port was built with two single paths. Each path possessed two adapters, each composed of an N-connector to a laboratory cable 4 mm socket, mounted in the L-shape aluminum profile, which were placed face to face at the height of h = 3 cm above the reference plate and connected with a copper cylindrical rod with a diameter of $\varphi$ = 2 mm and a length of L = 14 cm.

In each adapter, the resistance between the N-connector and the laboratory socket was mounted as shown in Figure 4b. This corresponded to the $Z_L=0 \mathsf{\Omega }$ and $Z_N=200 \mathsf{\Omega }$ of the four-port shown in Figure 1.

Examples of the measurement setups are illustrated in Figure 5.

The results for the modal common mode are presented in Figure 6.

The theoretical reflection and transmission coefficients corresponding to the direct measurements, as presented in Figure 2, were equal to $S_{II}^{CC}=S_{OO}^{CC}=0.0$ and $S_{IO}^{CC}=S_{OI}^{CC}=1.0$, respectively; see the red lines in Figure 6a,b along with Equations (1) and (2).

The theoretical reflection and transmission coefficients calculated with Equation (12) were distinctively different to those calculated with Equations (1) and (2); compare the green and red lines in Figure 6a,b. This was due to the imbalance of the investigated circuit.

The reflection and transmission coefficients derived from the single paths’ measurements using Equation (11) converged to the theoretical reflection and transmission coefficients; compare the green lines and the blue curves in Figure 6a,b.

The theoretical reflection coefficient of the matrix for conversion from the common/differential mode at the input/output to the differential/common mode at the output/input, calculated with Equation (14), was $S_{II}^{CD}=S_{OO}^{DC}=-0.33$; see the green line in Figure 7a.

The theoretical transmission coefficient of the matrix for conversion from the common/differential mode at the input/output to the differential/common mode at the output/input, calculated with Equation (14), was $S_{IO}^{CD}=S_{OI}^{DC}=0.33$; see the green line in Figure 7b.

The reflection and transmission coefficients derived from the single paths’ measurements using Equation (13) converged to the theoretical reflection and transmission coefficients; compare the green lines and the blue curves in Figure 7a,b.

Concerning the reference impedance $Z_0$, we notice the following:

• With the direct calculation by hand, using the formulas of Equations (1) and (2), applied to the circuit presented in Figure 2, $Z_0=50 \mathsf{\Omega }$ was set and the resulting modal S-Parameters also referred to $Z_0=50 \mathsf{\Omega }$;
• With the direct calculation by hand, using the modal common-mode S-Parameters and Equation (12), $Z_0=100 \mathsf{\Omega }$ must be set in order for the resulting modal S-Parameters to refer to $Z_0=50 \mathsf{\Omega }$;
• The reflection and transmission coefficients of the matrix for conversion from the common/differential mode at the input/output to the differential/common mode at the output/input calculated by hand using Equation (14) and calculated numerically using Equation (13) referred to the resistance $Z_0=25 \mathsf{\Omega }$ at the input and $Z_0=100 \mathsf{\Omega }$ at the output.

The recalculation of the reference impedance by the matrices Equations (14) and (13) were omitted.

4.2. Application to the Exemplary RF Suppression Filter

The modal common-mode attenuations in the $dB$ scale were related to the corresponding modal common-mode transmission coefficients as follows:

$$\begin{array}{c}\hfill \begin{array}{ccc}{A}_{IO}^{CC}(50\mathsf{\Omega }/50\mathsf{\Omega })& =& -20\log \left[|S_{IO}^{CC}(50\mathsf{\Omega }/50\mathsf{\Omega })|\right]\\ \\ A_{OI}^{CC}(50\mathsf{\Omega }/50\mathsf{\Omega })& =& -20\log \left[|S_{OI}^{CC}(50\mathsf{\Omega }/50\mathsf{\Omega })|\right]\end{array}.\end{array}$$

(29)

The modal insertion loss in the $dB$ scale was related to the corresponding modal conversion coefficients as follows:

$$\begin{array}{c}\hfill \begin{array}{ccc}{L}_{IO}^{CD}(50\mathsf{\Omega }/50\mathsf{\Omega })& =& -20\log \left[|S_{IO}^{CD}(50\mathsf{\Omega }/50\mathsf{\Omega })|\right]\\ \\ L_{OI}^{DC}(50\mathsf{\Omega }/50\mathsf{\Omega })& =& -20\log \left[|S_{OI}^{DC}(50\mathsf{\Omega }/50\mathsf{\Omega })|\right]\end{array}.\end{array}$$

(30)

The modal insertion loss is the measure of the balance of the investigated circuit.

The exemplary setups for the measurements of the single paths’ S-Parameters are presented in Figure 8. The setup for the direct measurement of the modal common-mode attenuation is presented in Figure 9.

The comparison of the direct measurement and the derivation of the modal common-mode attenuation is presented in Figure 10 by the red and blue curves, respectively.

From 10 kHz up to ca. 80 kHz, there was a slight difference between the curves. Between ca. 80 kHz and 1 MHz, the attenuation of the filter was larger than the dynamic of the measurement equipment. From 1 MHz upwards, the curves drifted apart.

The explanation of the behavior of the curves in Figure 10 can be easily interpreted with the modal insertion loss presented in Figure 11.

In the direct measurement of the common-mode attenuation, the filter conversion from the common to the differential mode is switched off. In other words, the modal insertion loss by this measurement is infinite.

In the calculation of the attenuation with the single-path measurements, the modal insertion loss is infinite, i.e., better than the dynamic of the measurement apparatus for a perfectly balanced circuit.

From 10 kHz up to ca. 80 kHz, the insertion loss of the filter, presented in Figure 11, was modest and smaller than the dynamic of the measurement apparatus. Between ca. 80 kHz and 1 MHz, it was larger than the dynamic of the measurement equipment. From 1 MHz upwards, the insertion loss decayed.

The modal insertion loss is the measure of the grade of the imbalance of the circuit. The common-mode attenuation measured directly was identical to that calculated with the single-path measurements only when the circuit was in perfect balance.

The discrepancy of the curves in Figure 10 are due to the imbalance of the filter.

5. Conclusions

The research results presented in this article clearly showed that using two definitions of the common mode and two definitions of the differential multiport mode according to [9], we obtained two different values of the attenuation.

The above conclusion was supported by the results for the simple four-port shown in Figure 4, gathered in Figure 6 and Figure 7. The theoretical reflections using the direct approach (red lines) differed distinctively from the ones using the approach with single paths (green lines). The measurement results obtained using the approach with the single paths (blue curves) were close to the green lines.

In the first common mode, measured directly as shown in Figure 2, the transverse to longitudinal conversion [12] was switched off. This was not the case in the derivation of the common-mode S-Parameters from the single-path S-Parameters.

The comparison of the conversion losses with both attenuations in Figure 11 confirms the thesis presented in this article. Figure 11 justifies the different filter attenuation values in the frequency range above 1 MHz, which can be seen in Figure 10.

The comparison of Equations (1) and (2) with Equation (12) implies that the modal S-Parameters measured directly and derived from the single-path S-Parameters were identical only when $Z_L=Z_N$.

The attenuations measured directly and derived from the single-path S-Parameters were identical only when the filter was in balance.

This conclusion can be generalized for balanced multiwire lines.