S-parameters allow a two-port description not just by permuting (I_{1}), (U_{1}), (I_{2}), (U_{2}), but using a superposition, leading to a power view of the port (We only look at two-ports here, because multi-ports are not (yet?) implemented.).

You may start with the effective power, being negative or positive

[\begin{array}{ll} {P = u \cdot i} & \ \end{array}]

The value of (P) may be the difference of two real numbers, with (K) being another real number.

[\begin{array}{ll} {ui = P = a^{2} - b^{2} = \left( {a + b} \right)\left( {a - b} \right) = \left( {a + b} \right)\left( {KK^{- 1}} \right)\left( {a - b} \right) = \left{ {K\left( {a + b} \right)} \right}\left{ {K^{- 1}\left( {a - b} \right)} \right}} & \ \end{array}]

Thus you get

[\begin{array}{ll} {K^{- 1}u = a + b} & \ \end{array}]

[\begin{array}{ll} {Ki = a - b} & \ \end{array}]

and finally

[\begin{array}{ll} {a = \frac{u + K^{2}i}{2K}} & \ \end{array}]

[\begin{array}{ll} {b = \frac{u - K^{2}i}{2K}} & \ \end{array}]

By introducing the reference resistance (Z_{0}: = K^{2} > 0) we get finally the Heaviside transformation

[\begin{array}{ll} {a = \frac{u + Z_{0}i}{2\sqrt{Z_{0}}}, b = \frac{u - Z_{0}i}{2\sqrt{Z_{0}}}} & \ \end{array}]

In case of our two-port we subject our variables to a Heaviside transformation

[\begin{array}{ll} {a_{1} = \frac{U_{1} + Z_{0}I_{1}}{2\sqrt{Z_{0}}} b_{1} = \frac{U_{1} - Z_{0}I_{1}}{2\sqrt{Z_{0}}}} & \ \end{array}]

[\begin{array}{ll} {a_{2} = \frac{U_{2} + Z_{0}I_{2}}{2\sqrt{Z_{0}}} b_{2} = \frac{U_{2} - Z_{0}I_{2}}{2\sqrt{Z_{0}}}} & \ \end{array}]

The s-matrix for a two-port then is

[\begin{array}{ll} {\left( \begin{array}{l} b_{1} \ b_{2} \ \end{array} \right) = \left( \begin{array}{ll} s_{11} & s_{12} \ s_{21} & s_{22} \ \end{array} \right)\left( \begin{array}{l} a_{1} \ a_{2} \ \end{array} \right)} & \ \end{array}]

Two obtain (s_{11}) we have to set (a_{2} = 0). This is accomplished by loading the output port exactly with the reference resistance (Z_{0},) which sinks a current (I_{2} = - U_{2}/Z_{0}) from the port.

[\begin{array}{ll} {s_{11} = \left( \frac{b_{1}}{a_{1}} \right){a{2} = 0}} & \ \end{array}]

[\begin{array}{ll} {s_{11} = \frac{U_{1} - Z_{0}I_{1}}{U_{1} + Z_{0}I_{1}}} & \ \end{array}]

Loading the input port from an ac source (U_{0}) via a resistor with resistance value (Z_{0}), we obtain the relation

[\begin{array}{ll} {U_{0} = Z_{0}I_{1} + U_{1}} & \ \end{array}]

Entering this into 77, we get

[\begin{array}{ll} {s_{11} = \frac{2U_{1} - U_{0}}{U_{0}}} & \ \end{array}]

For (s_{21}) we obtain similarly

[\begin{array}{ll} {s_{21} = \left( \frac{b_{2}}{a_{1}} \right){a{2} = 0}} & \ \end{array}]

[\begin{array}{ll} {s_{21} = \frac{U_{2} - Z_{0}I_{2}}{U_{1} + Z_{0}I_{1}} = \frac{2U_{2}}{U_{0}}} & \ \end{array}]

Equations 79 and 81 now tell us how to measure (s_{11}) and (s_{21}): Measure (U_{1}) at the input port, multiply by 2 using an E source, subtracting (U_{0}), which for simplicity is set to 1, and divide by (U_{0}). At the same time measure (U_{2}) at the output port, multiply by 2 and divide by (U_{0}). Biasing and measuring is done by subcircuit S_PARAM. To obtain (s_{22}) and (s_{12}), you have to exchange the input and output ports of your two-port and do the same measurement again. This is achieved by switching resistors from low ((1m\Omega)) to high ((1T\Omega)) and thus switching the input and output ports.