Mixed-Mode Y/Z-Parameters
Mixed-mode S-parameters are used extensively by singal integrity engineers, and a lot of resources are available that document the conversion from their single-ended versions. The same is not the case for mixed-mode Y/Z parameters. These less frequently used versions come in useful in certain applications.
One example I came across is the simulation of an inductor for an LC oscillator in an electromagnetic simulator. Usually, the simulator exports single-ended Y-parameters for the two inductor terminals referenced to ground. However, what we care about is the behavior of differential inductance in the LC tank. So, the output needs to be transformed to mixed-mode to do any further analysis.
This article shows the derivation of mixed-mode Y/Z-parameters from their single-ended counterparts. It starts with a brief introduction to Y/Z-parameters followed by an analysis conducted via defining the differential and common-mode drive conditions for the two-port. At the end, a couple of example circuits demonstrate the expressions derived.
Single-Ended Y/Z-Parameters
Y/Z-parameters are used to define the behavior of linear electrical networks by specifying the relationship between the voltages and currents of its ports.
Y-parameters (also known as conductance parameters) relate the independent variables of port voltages to dependent variables of port currents. To derive the Y-parameters, we drive each port with a voltage source, while shorting all other ports and measure the currents into each port.
The Y-parameters 2-port are then defined using these measurements as the following.
\[\begin{array}{c c} Y_{11} \triangleq \left.\frac{I_1}{V_1}\right|_{V_2 = 0} & Y_{12} \triangleq \left.\frac{I_1}{V_2}\right|_{V_1 = 0} \\ Y_{21} \triangleq \left.\frac{I_2}{V_1}\right|_{V_2 = 0} & Y_{22} \triangleq \left.\frac{I_2}{V_2}\right|_{V_1 = 0} \end{array}\]These can be represented in matrix form as below.
\[\begin{align} \begin{bmatrix} I_1 \\ I_2 \end {bmatrix} &= \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} \\ \mathbf{I} &= \mathbf{Y}\cdot\mathbf{V} \end{align}\]Similarly, Z-parameters (impedance parameters) relate the independent variables of port currents to dependent variables of port voltages. This, in turn, means that we drive each port with an independent current source, while leaving open and measuring the voltage at every other port.
The Z-parameters for a 2-port can then be defined as the following.
\[\begin{array}{c c} Z_{11} \triangleq \left.\frac{V_1}{I_1}\right|_{I_2 = 0} & Z_{12} \triangleq \left.\frac{V_1}{I_2}\right|_{I_1 = 0} \\ Z_{21} \triangleq \left.\frac{V_2}{I_1}\right|_{I_2 = 0} & Z_{22} \triangleq \left.\frac{V_2}{I_2}\right|_{I_1 = 0} \end{array}\]These can be represented in matrix form as below.
\[\begin{align} \begin{bmatrix} V_1 \\ V_2 \end {bmatrix} &= \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} \\ \mathbf{V} &= \mathbf{Z} \cdot \mathbf{I} \end{align}\]Both sets of parameters completely describe the behavior of the linear circuit. They can also be converted from one to the other via matrix inversion.
\[\begin{align} \mathbf{Y} &= \mathbf{Z}^{-1} \end{align}\]Naturally, this requires the matrices to be invertible. There are some circuits for which the matrices will not be invertable, which implies that the other set of parameters tend to infinity.
Mixed-Mode Y/Z-Parameters
The label mixed-mode refers to the combination of and differential- and common-mode signaling over a pair of ports. A differential transmitter uses a balanced interface to send the singal and its inverse across a two-conductor channel, and a differential receiver responds to the voltage difference between the two conductors. In such a system, external interference appears in-phase on the two lines and the corruption of the differential-mode signal is avoided. As an additional benefit, the signal amplitude is doubled, which results in higher signal-to-noise ratio.
The differential- and common-mode voltages and currents in the case of a two-port driven with . The differential-mode is the out-of-phase (and generally the desirable) signal component, and the common-mode is the in-phase (and generally the parasitic) component.
\[\begin{array} {c c} V_{DM} = V_1 - V_2 & V_{CM} = \frac{V_1 + V_2}{2} \\ I_{DM} = \frac{I_1 - I_2}{2} & I_{CM} = I_1 + I_2 \end{array}\]The voltage and current expressions for the ports are as follows.
\[\begin{array} {c c} V_1 = \frac{V_{DM}}{2} + V_{CM} & V_2 = -\frac{V_{DM}}{2} + V_{CM} \\ I_1 = I_{DM} + \frac{I_{CM}}{2} & I_2 = -I_{DM}+\frac{I_{CM}}{2} \end{array}\]The expressions for the voltages are familiar, but the mixed-mode currents might be slightly confusing. The differential-mode current circulates into the first port and out of the second. The common-mode current is the total current going into both ports and coming out of the reference node.
Mixed-Mode Y-Parameters
Similar to their single-ended counterparts, mixed-mode Y-parameters relate these differential- and common-mode currents (dependent variables) to voltages (independent variables).
\[\begin{array}{c c} Y_{DD} \triangleq \left.\frac{I_{DM}}{V_{DM}}\right|_{V_{CM} = 0} & Y_{DC} \triangleq \left.\frac{I_{DM}}{V_{CM}}\right|_{V_{DM} = 0} \\ Y_{CD} \triangleq \left.\frac{I_{CM}}{V_{DM}}\right|_{V_{CM} = 0} & Y_{CC} \triangleq \left.\frac{I_{CM}}{V_{CM}}\right|_{V_{DM} = 0} \end{array}\]In matrix form, mixed-mode Y-paramaters are as below.
\[\begin{bmatrix} I_{DM} \\ I_{CM} \end {bmatrix} = \begin{bmatrix} Y_{DD} & Y_{DC} \\ Y_{CD} & Y_{CC} \end{bmatrix} \begin{bmatrix} V_{DM} \\ V_{CM} \end{bmatrix}\]Converting Single-Ended to Mixed-Mode
Two conditions have to be considered to convert the single-ended Y-parameters to mixed-mode. In the first condition the ports are driven with oppsite polarity voltage sources. \( V_{CM} \) is set to zero, and the circuit is excited with \( V_{DM} \) only.
\[\begin{array}{l} V_1 = -V_2 = V \\ V_{DM} = V_1 - V_2 = 2V \\ V_{CM} = \frac{V_1 + V_2}{2} = 0 \end{array}\]We can then derive \( I_1 \) and \( I_2 \) from the single ended Y-parameters.
\[\begin{array}{l} I_1 = \left(Y_{11} - Y_{12}\right) V \\ I_2 = \left(Y_{21} - Y_{22}\right) V \end{array}\]Then, the differential and common-mode currents can be derived and \( V_{DM} \) can be substituted in for \( V \) to arrive at the final expressions.
\[\begin{array}{l} I_{DM} = \frac{I_1 - I_2}{2} = \frac{Y_{11} - Y_{12} - Y_{21} + Y_{22}}{2} V = \frac{Y_{11} - Y_{12} - Y_{21} + Y_{22}}{4} V_{DM} \\ I_{CM} = I_1 + I_2 = \left(Y_{11} - Y_{12} + Y_{21} - Y_{22}\right) V = \frac{Y_{11} - Y_{12} + Y_{21} - Y_{22}}{2} V_{DM} \end{array}\]In the second condition, the ports are driven with the same polarity voltage sources. \( V_{DM} \) is set to zero, and the circuit is excited with \( V_{CM} \) only.
\[\begin{array}{l} V_1 = V_2 = V \\ V_{DM} = V_1 - V_2 = 0 \\ V_{CM} = \frac{V_1 + V_2}{2} = V \end{array}\]Similarly, given the voltages at the ports under common-mode drive, we derive \( I_1 \) and \( I_2 \) from the single-ended Y-parameters.
\[\begin{array}{l} I_1 = \left(Y_{11} + Y_{12}\right) V \\ I_2 = \left(Y_{21} + Y_{22}\right) V \\ \end{array}\]Then, we derive the differential and common-mode currents under common-mode drive, and convert the variable V to the common-mode voltage \( V_{CM} \).
\[\begin{array}{l} I_{DM} = \frac{I_1 - I_2}{2} = \frac{Y_{11} + Y_{12} - Y_{21} - Y_{22}}{2} V = \frac{Y_{11} + Y_{12} - Y_{21} - Y_{22}}{2} V_{CM} \\ I_{CM} = I_1 + I_2 = \left(Y_{11} + Y_{12} + Y_{21} + Y_{22}\right) V = \left(Y_{11} + Y_{12} + Y_{21} + Y_{22}\right) V_{CM} \end{array}\]We can combine these expressions with their differential-mode drive counterparts in matrix form to arrive at the mixed-mode Y-parameters.
\[\begin{bmatrix} I_{DM} \\ I_{CM} \end {bmatrix} = \begin{bmatrix} \frac{Y_{11} - Y_{12} - Y_{21} + Y_{22}}{4} & \frac{Y_{11} + Y_{12} - Y_{21} - Y_{22}}{2} \\ \frac{Y_{11} - Y_{12} + Y_{21} - Y_{22}}{2} & Y_{11} + Y_{12} + Y_{21} + Y_{22} \end{bmatrix} \begin{bmatrix} V_{DM} \\ V_{CM} \end{bmatrix}\]Mixed-Mode Z-parameters
Mixed-mode Z-parameters relate the differential and common-mode voltages (dependent variable) to currents (independent variable).
\[\begin{array}{c c} Z_{DD} \triangleq \left.\frac{V_{DM}}{I_{DM}}\right|_{V_{CM} = 0} & Z_{DC} \triangleq \left.\frac{V_{DM}}{I_{CM}}\right|_{V_{DM} = 0} \\ Z_{CD} \triangleq \left.\frac{V_{CM}}{I_{DM}}\right|_{V_{CM} = 0} & Z_{CC} \triangleq \left.\frac{V_{CM}}{I_{CM}}\right|_{V_{DM} = 0} \end{array}\]In matrix form, mixed-mode Z-paramaters are as below.
\[\begin{bmatrix} V_{DM} \\ V_{CM} \end {bmatrix} = \begin{bmatrix} Z_{DD} & Z_{DC} \\ Z_{CD} & Z_{CC} \end{bmatrix} \begin{bmatrix} I_{DM} \\ I_{CM} \end{bmatrix}\]Converting Single-Ended to Mixed-Mode
Similar to the analysis for the Y-parameters, there are two conditions of consideration. In the first condition, the two-port terminals are driven with current sources with opposite polarity. This excites the circuit with \(I_{DM}\) while setting \(I_{CM}\) to zero.
In this case, the port currents are as below.
\[I_1 = -I_2 = I_{DM}\]The port voltages, \(V_1\) and \(V_2\), then can be calculated from the single-ended parameters.
\[\begin{align} V_1 &= (Z_{11} - Z_{12})I_{DM} \\ V_2 &= (Z_{21} - Z_{22})I_{DM} \end{align}\]We can then derive the differential- and common-mode voltages for the ports.
\[\begin{align} V_{DM} &= V_1 - V_2 = (Z_{11} - Z_{12} - Z_{21} + Z_{22}) I_{DM} \\ V_{CM} &= \frac{V_1 + V_2}{2} = \frac{Z_{11} - Z_{12} - Z_{21} + Z_{22}}{2} I_{DM} \end{align}\]For the scond condition, the two-port terminals are driven with current sources with the same polarity. This excites the circuit with \(I_{CM}\) while setting \(I_{DM}\) to zero.
The port voltages, \(V_1\) and \(V_2\), then can be calculated from the single-ended parameters.
\[\begin{align} V_1 &= (Z_{11} + Z_{12})\frac{I_{CM}}{2} \\ V_2 &= (Z_{21} + Z_{22})\frac{I_{CM}}{2} \end{align}\]We can then finally derive the remaining differential- and common-mode voltages for the ports.
\[\begin{align} V_{DM} &= V_1 - V_2 = \frac{Z_{11} + Z_{12} - Z_{21} - Z_{22}}{2} I_{CM} \\ V_{CM} &= \frac{V_1 + V_2}{2} = \frac{Z_{11} + Z_{12} + Z_{21} + Z_{22}}{4} I_{CM} \end{align}\]We can combine these expressions with their differential-mode drive counterparts in matrix form to arrive at the mixed-mode Z-parameters.
\[\begin{bmatrix} V_{DM} \\ V_{CM} \end {bmatrix} = \begin{bmatrix} Z_{11} - Z_{12} - Z_{21} + Z_{22} & \frac{Z_{11} + Z_{12} - Z_{21} - Z_{22}}{2} \\ \frac{Z_{11} + Z_{12} - Z_{21} - Z_{22}}{2} & \frac{Z_{11} + Z_{12} + Z_{21} + Z_{22}}{4} \end{bmatrix} \begin{bmatrix} I_{DM} \\ I_{CM} \end{bmatrix}\]Examples
Single-Ended Termination
The Z-parameters for the differential termination is as follows.
\[\begin{bmatrix} V_1 \\ V_2 \end {bmatrix} = \begin{bmatrix} R & 0 \\ 0 & R \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}\]Using the expressions above, these can be converted to mixed-mode Z-parameters.
\[\begin{bmatrix} V_1 \\ V_2 \end {bmatrix} = \begin{bmatrix} 2R & 0 \\ 0 & \frac{R}{2} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}\]The differential impedance seen into the differential port is doubled and the common-mode impedance is halved. In single-ended systems that use 50Ω characteristic impedance, the differential impedance is (as expected) 100Ω, and the common-mode impedance is 25Ω.
Differential Termination
We could just as well terminate the system differentially, as follows:
The Y-parameters for the differential termination is below. Notice how similar this looks to the MNA stamp of a resistor.
\[\begin{bmatrix} I_1 \\ I_2 \end {bmatrix} = \begin{bmatrix} G & -G \\ -G & G \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}\]The mixed-mode Y-parameters for the differential resistor is:
\[\begin{bmatrix} I_{DM} \\ I_{CM} \end {bmatrix} = \begin{bmatrix} G & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} V_{DM} \\ V_{CM} \end{bmatrix}\]The common-mode input conductance of the two-port is zero, implying an infinite common-mode input impedance. For the same reason, the Y-parameters cannot be converted to Z-parameters with the above expression as the determinant of the Y-parameter matrix is zero.