SSS Thévenin/Norton Networks
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In this problem, we'll explore the Thévenin and Norton equivalence of networks operating in sinusoidal steady state. All networks considered here are comprised of linear resistors, capacitors, and inductors; and voltage and current sources all operating at the same frequency \omega. Therefore, all branch currents and voltages operate at the frequency \omega.
1) Equivalence
Consider the following two networks:
| Network 1 | Network 2 |
|---|---|
|
|
Determine expressions for \tilde{Z}_{\rm T} and \tilde{V}_{\rm T} in terms of \tilde{Z}_{\rm N} and \tilde{I}_{\rm N} which must hold for the current/voltage relationships at the terminals of these two networks to be identical when operating at sinusoidal steady state.
Enter your answers in terms of Z_N and/or I_N.
2) Another Circuit
Consider the following circuit:
Determine expressions for \tilde{Z}_{\rm T} and \tilde{V}_{\rm T}, and for \tilde{Z}_{\rm N} and \tilde{I}_{\rm N}, that would make this circuit equivalent to networks 1 and 2, respectively, when all are operating at sinusoidal steady state.
Enter your answers in terms of R, C, L, V (\tilde{V}), j (the
imaginary unit), omega, and/or any numbers you need. You may specify things
completely or you may use par(x, y, z, ...) with any number of arguments to
specify parallel combinations of resistances/impedances.