12 Jul 2020 - tsp

The following article should provide a short summary about the meaning and definition of Z parameters in two pole networks. I’ve written this short primer since I had a friend who hasn’t used Z (and S) parameters before and wanted to do a small electronics project together.

So what is a two port network? A two port network basically is an abstract view onto any component or group of components that has two electrical ports, every port usually having two wires or connectors (many times one of them being tied to a ground potential). It’s an easy way to modularize the view of complex circuits and is often used in RF circuit analysis. The basic idea is to split up a network into - for example - a sequence or two port networks that are connected in series or in parallel. Each two port network is analyzed independently and all equations are derived for each network on it’s own. One can then either tune network component parameters in case one thinks about components in parts (for example when one wants to combine filter network components, amplifiers, etc. - this is also often done analogue to the block diagrams in RF networks) or combine derived equations.

The two port network can be imagined as a simple blackbox with an input
port on the left and an output port on the right side. Note that for the
definition of a *port* to be fulfilled the same current that’s flowing into
a port has to flow out of a given port.

Note that any linear circuit with four terminals can be seen as a two port
network. Note that the whole formalism of two port network descriptions using
Z and ABCD parameters only works for *linear* networks due to the superposition
principle of solutions.

The basic idea of the impedance matrix - or Z parameters - is analog to the idea of an impedance in general. As one knows the impedance is the generalization of the concept of the resistivity, it models the linkage between current and voltage. For a simple ohmic resistor the resistance had been defined as

[ U=R*I ]In this case the Voltage $U$ has been proportional to the current $I$. The proportionality factor $R$ has been termed resistance. The extension to capacitors and inductor lead to the definition of the impedance that also contained information about phase shifts between voltage and current:

[ U(t)=Z*I(t) ]In this case $Z$ is a complex number that is formed by the ohmic resistance component $R$ and the reactance $X$

[ Z=R+j*X ]This works pretty well since the voltage and current of an sinusoidal wave can also be represented using Euler’s formula that states, that

[ e^{j*\omega*t} = \cos(\omega * t) + j * \sin(\omega * t) \\ \sin(\omega * t) = \frac{e^{j * \omega * t} - e^{-j* \omega * t}}{2 * j} \\ \cos(\omega * t) = \frac{e^{j * \omega * t} + e^{-j* \omega * t}}{2} ]Since Fourier’s theorem states that one can describe any waveform as the sum of an infinite number of sine or cosine waves and most of the time one’s only interested about behavior at some specific frequencies one can describe all currents and voltages using this representation using complex numbers. The usage of complex impedance thus allows to introduce the phase shift that occurs with capacitance and inductance.

The idea for a two port network goes a step further and keeps the same structure. The idea is to introduce an object, the Z parameter matrix, that links voltages and currents the same way as the impedance:

[ \left(\begin{matrix}U_{1} \\ U_{2} \end{matrix}\right) = \left(\begin{matrix}z_{11} & z_{12} \\ z_{21} & z_{22} \end{matrix}\right) * \left(\begin{matrix}I_{1} \\ I_{2} \end{matrix}\right) ]As one can see the matrix describes the coupling between both currents and both voltages:

[ U_1 = z_{11} * I_{1} + z_{12} * I_{2} \\ U_2 = z_{21} * I_{1} + z_{22} * I_{2} ]As one can see this boils down to the inductance under open and short circuit conditions.

In case of an open circuit on the second port $I_2 = 0$:

[ U_1 \mid_{I_2=0} = z_{11} * I_{1} \\ U_2 \mid_{I_2=0} = z_{21} * I_{1} ]In case of an open circuit on the first port $I_1 = 0$:

[ U_1 \mid_{I_1=0} = z_{12} * I_{2} \\ U_2 \mid_{I_1=0} = z_{22} * I_{2} ]One can see that the Z parameters of course have the same dimension as an resistor - they’re given in Ohm.

Since this description still is not totally compatible with the idea of simply chaining blocks (for example using basic operator calculus by joining the matrices/tensors by simply multiplying them) one can also use ABCD parameters. These are also sometimes called chaining parameters because of their convenient properties:

[ \left(\begin{matrix}U_{1} \\ I_{1} \end{matrix}\right) = \left(\begin{matrix}A & B \\ C & D \end{matrix}\right) * \left(\begin{matrix}U_{2} \\ -I_{2} \end{matrix}\right) ]As one can see they provide a mapping between the input voltage and current and the output voltage and current (and vice versa - which is even more useful in many cases). The definition displayed above is the most common, one might also use a positive sign for the output current and put the output port on the left hand side of the matrix which is what I’m usually doing

[ \left(\begin{matrix}U_{2} \\ I_{2} \end{matrix}\right) = \left(\begin{matrix}A & B \\ C & D \end{matrix}\right) * \left(\begin{matrix}U_{1} \\ I_{1} \end{matrix}\right) ]This allows pretty easy chaining of matrices and operators. Using this definition one can look at the meaning of the parameters $A$, $B$, $C$ and $D$:

[ U_2 = A * U_1 + B*I_1 \\ I_2 = C * U_1 + D*I_1 ]Again one can look at short- and open circuit conditions. An open circuit would lead to $I_1=0$:

[ U_2 \mid_{I_1=0} = A * U_1 \to A = \frac{U_2}{U_1} \mid_{I_1=0} \\ I_2 \mid_{I_1=0} = C * U_1 \to C = \frac{I_2}{U_1} \mid_{I_1=0} ]In case of a short circuit on the input port the potential has to vanish, i.e. $U_1=0$:

[ U_2 \mid_{U_1=0} = B*I_1 \to B = \frac{U_2}{I_1} \mid_{U_1=0} \\ I_2 \mid_{U_1=0} = D*I_1 \to D = \frac{I_1}{U_1} \mid_{U_1=0} ]As mentioned earlier this description allows easy chaining of multiple matrices - just remember matrix multiplication works from right to left. In case an input signal $U_1$, $I_1$ first enters an network $\hat{\alpha}$ and then it’s output another network $\hat{\beta}$ one can simply describe the total output $U_3$, $I_3$ as

[ \left(\begin{matrix}U_{3} \\ I_{3} \end{matrix}\right) = \hat{\beta} * \hat{\alpha} * \left(\begin{matrix}U_{1} \\ I_{1} \end{matrix}\right) ]Even more interesting in RF engineering (for example while designing filters
and amplifiers) are scattering parameters - i.e. parameters that describe the
amount of energy that gets reflected and transmitted when entering a two port
in a given direction. These are usually measured using either a spectrum analyzer
and a tracking generator as well as directional couplers or more conveniently
but *way* more expensive using a vector network analyzer.

Using a spectrum analyzer one would first inject a signal using a tracking generator and measure it’s transmission on the output port, then attach a dummy load to the output port and measure reflected energy on the input port by decoupling the reflected wave using a directional coupler. Then one would swap input and output ports and perform the same measurement again. Of course the measurements don’t result in single numbers but in a frequency dependent component of course.

When using a VNA the network analyzer transmits a single pulse into the first port of the device under test, performs a Fourier analysis of the transmitted and reflected part simultaneously and then does the same from the other side.

To measure S parameters one has to calibrate the measurement setup in any case - this means also calibrating the setup to compensate for the effect of used cabling and couplers using test kits of perfectly matched shorts and open terminations (these are rather expensive too). One even has to account for correct connection or SMA cables like the applied force and humidity in the environment when going into the regime of multi GHz areas. And then of course there is an effect of the way one couples the signals onto the PCBs or into the waveguides, etc.

[ \left(\begin{matrix}b_1 \\ b_2 \end{matrix}\right) = \left(\begin{matrix}S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix}\right) * \left(\begin{matrix}a_{1} \\ a_{1} \end{matrix}\right) ]In this case $a_1$ is the incident and $b_1$ the reflected wave at port $1$, $a_1$ the incident and $b_2$ the reflected wave at port $b$.

As one can see this representation also doesn’t allow easy chaining of the parameter matrices since the reflected waves of both ports are represented on one side of the matrix, the incident waves on the other side. One can do the same trick as with $ABCD$ parameters and separate the ports on both sides of the matrix:

[ \left(\begin{matrix}b_1 \\ a_1 \end{matrix}\right) = \left(\begin{matrix}T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix}\right) * \left(\begin{matrix}a_{2} \\ b_{2} \end{matrix}\right) ]These $T$ parameters are then called transfer parameters. Keep in mind that the vectors on the input and output side have their incident and reflected components exchanged to allow for easy chaining - the wave leaving the output port of an upstream component is the wave entering the input port of the next downstream component. As one can see the measurement of $T$ parameters is not as “trivial” as the measurement of scattering parameters in any way - but they can be derived from known $S$ parameters pretty easily (calculation is a little bit longer but it’s a simple system of linear equations):

[ T_{11} = -\frac{S_{22}}{S_{21}} \\ T_{12} = \frac{1}{S_{21}} \\ T_{21} = \frac{S_{12} * S_{21} - S_{11} * S_{22}}{S_21} \\ T_{22} = \frac{S_{11}}{S_{21}} ]Of course the transformation is also easy to invert:

[ S_{11} = \frac{1}{T_{22}} \\ S_{12} = - \frac{T_{21}}{T_{22}} \\ S_{21} = \frac{T_{12}}{T_{22}} \\ S_{22} = \frac{T_{11} * T_{22} - T_{12} * T_{21}}{T_{22}} ]Just a short word about the traveling direction of waves: One can model waves traveling into different directions using different signs for their time evolution or their frequency components. I.e. in case the incident wave is

[ A*e^{j * \omega * t + \vec{k} * \vec{x}} ]the corresponding reflected wave in case of complete reflection without phase change could be modeled as

[ B*e^{-j * \omega * t + \vec{k} * \vec{x}} ]The whole state at a specific point at the transmission line would then of course be the sum of incident and reflected wave:

[ A*e^{j * \omega * t + \vec{k} * \vec{x}} B*e^{-j * \omega * t + \vec{k} * \vec{x}} ]As should be well known this is one of the forms of generic solutions often seen when calculating the response of a potential barrier or the penetration into a potential - the coefficients $A$ and $B$ are directly related to the transmission parameters. This should give an intuitive view why the scattering- and transmission parameters are of particular interest when performing theoretical calculations about semiconductors (who are essentially formed by different potential barriers) or devices that are based on tunneling effect.

This article is tagged: Basics, Electronics, Tutorial

Dipl.-Ing. Thomas Spielauer, Wien (webcomplains389t48957@tspi.at)

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