1. Thevenin Theorem in Sinusoidal Steady Analysis Aim: To obtain a simple equivalent circuit for a 1-port circuit that consists of linear, time-invariant resistors, capacitors, inductors and independent sources. Thevenin Equivalent: + _ v i 1-port circuit + _ V I + _ Z TH V TH Z Th Thevenin impedance Equivalent impedance between terminals when sources are set to zero. V Th Open circuit voltage The voltage of the port when the port is left as open circuit. Thevenin Theorem in sinusoidal steady analysis: A 1-port circuit that consists of linear resistor, capacitor, inductors and independent sources has a Thevenin equivalent circuit in sinusoidal steady state if the port voltage phasor can be uniquely determined for a given port current phasor, in other words, if the 1-port is current- controlled.

2. Norton Theorem in Sinusoidal Steady Analysis Aim: To obtain a simple equivalent circuit for a 1-port circuit that consists of linear, time-invariant resistors, capacitors, inductors and independent sources. Norton Equivalent: + _ v i 1-port circuit Norton Theorem in sinusoidal steady analysis: A 1-port circuit that consists of linear resistor, capacitor, inductors and independent sources has a Norton equivalent circuit in sinusoidal steady state if the port current phasor can be uniquely determined for a given port voltage phasor, in other words, if the 1- port is voltage- controlled. + _ V I YNYN ININ G N Norton conductance I N Short circuit current Equivalent conductance between terminals when sources are set to zero. The current through the port when the port is short-circuited.

3. How to obtain Thevenin equivalent circuit? + _ V I 1-port circuit Connect a sinusoidal current source to the port. I*I* V*V* + _ Solve the circuit using sinusoidal steady analysis and obtain a relation between phasors I * and V *. Use I=I * and V=-V * to obtain a relation between I and V. + _ V I 1-port circuit Set the values of independent sources to zero. Calculate the equivalent impedance Z Th = V / I. Assume that I=0 (open-circuit the port) and calculate V th =V taking into account all independent sources. There exist two methods for this!

4. How to obtain Norton equivalent circuit? Connect a sinusoidal voltage source to the port. Solve the circuit using sinusoidal steady analysis and obtain a relation between phasors I * and V *. Use I=-I * and V=V * to obtain a relation between I and V. + _ V I 1-port circuit Set the values of independent sources to zero. Calculate the equivalent admitance G N = I / V. Assume that V=0 (short-circuit the port) and calculate I N =I taking into account all independent sources. There exist two methods for this! + _ V I 1-port circuit I*I* V*V* + _ +-+-

5. Thevenin Equivalent: If 1-port is not current-controlled there is no Thevenin eq.. Norton Equivalent: If 1-port is not voltage-controlled there is no Norton eq.. No Norton equivalent! No Thevenin equivalent! Interchange between Thevenin and Norton From Thevenin to Norton: From Norton to Thevenin:

6. Example 1: Find the Thevenin equivalent circuit for the following circuit!

7. Example 2: Find the Norton equivalent circuit for the following circuit!

8. Circuit(Network) Functions in Sinusoidal Steady Analysis + _ E1E1 ISIS linear, time- independent elements How does V d k affect I s ? Depending on w! Assume that there is only one source.

9. + _ E1E1 ISIS are polynomials in (jw) with real coefficients. This depends on the circuit but not on the value of I s. linear, time- independent elements Circuit(Network) Functions in Sinusoidal Steady Analysis How does V d k affect I s ? Depending on w!

10. One can define many other circuit functions: Impedance Function Input Impedance Function Voltage Transfer Function Current Transfer Function

11. Symmetries of Circuit Functions Lemma: Let be a polynomial in complex variable s with real coefficients. 1) 2) (z is called as a root of n(z).) Proof: 1) 2) From (1)

12. Circuit function: Symmetry Property: The magnitude of any circuit function is an even function of w and its phase is an odd function of w. Proof: and from Lemma Since the phase of is. Theorem: For a circuit in sinusoidal steady state, any circuit function is well-defined and is the ratio of two polynomials in (jw) with real coefficients if det(T(jw)) is nonzero.

13. + _ V s (t) 1-port circuit Result: In order to find the behaviour of the circuit for the frequency w, one should find and.