**Introduction**

Scattering parameters (S-parameters) represent the way voltage waves propagate through an electrical network. High frequency linear networks can be fully characterised by these parameters which are measured at the ports of the network. S-parameters are a particularly useful measurement tool as there is no need to identify the components in the network to characterise it.

**Two-Port Network Theory**

Two-port networks are often used to describe how S-parameters are calculated, although S-parameters can be used to characterise electrical networks with *any* number of ports. The theory remains the same for different numbers of ports, only the number of parameters change. There are always N^{2} S-parameters for an N-port network, each one representing a potential input-output path. S-parameters are complex numbers. In a two port network there are four S-parameters.

In linear networks, when a voltage wave is incident on a port, the energy splits up:

- some energy goes into the port and exists
- some energy goes into the port and scatters to the other ports
- some energy goes into the port and is reflected back out

S-parameters represent the voltage ratios in the network and are are usually displayed in a matrix format. The first and second subscript numbers in an S-parameter for a two-port network is the output port number and the input port number respectively. The relationship between the incident voltage waves and the reflected waves is given by equation:

(1)

Where the variables are described as:

- a
_{1 - }voltage wave incident on port 1 - a
_{2 -}voltage wave incident on port 2 - b
_{1 - }voltage wave reflected from port 1 - b
_{2 - }voltage wave reflected from port 2 - S
_{11}- input reflection coefficient - S
_{12}- reverse voltage gain - S
_{21}- forward voltage gain - S
_{22}- output reflection coefficient

**Measuring S-parameters**

S-parameters are typically measured using a voltage source and a matched load Z_{0} (load with the same value as system impedance). S-parameters are measured by successively opening and shorting different input/output ports of the network. The matrix in the equation above can be expanded to give the two equations:

(2)

These equations can then be rearranged for each S-parameter and using the maximum power transfer theorem (inserting a matched load at a port sets that ports voltage to zero), the equations for the parameters can be calculated using the following equations:

(3)

So for example, to calculate the input reflection coefficient of port 1 (S_{11}), the user would apply a voltage across the port 1, place a matched load at (or ground) a_{2}, then measure the voltage across b_{1 }and a_{1}. Dividing b_{1} by a_{1} gives the input reflection coefficient at port 1.

**Example**

The following example demonstrates how to extract the S-parameters from a two-port SAW in OnScale.

This SAW model has two ports each made up of two finger pairs.

To calculate the model's S-parameters, two simulations must be run to acquire all of the information to solve the set of equations (3):

- one that drives port 1 and grounds port 2 (saw_p1.flxinp)
- one that drives port 2 and grounds port 1 (saw_p2.flxinp)

- Open the post processing tool
- Load in both time histories (saw_p1.flxhst and saw_p2.flxhst)
- Select pize idt1_act:Voltage in the results manager
- Select the S-parameters button in the Home tab of the ribbon
- Assign time histories to the appropriate records as follows:
- Select Calculate
- The results will appear in the results manager under the Frequency History section
- S-parameters can be plot by double clicking on the frequency history:

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