1. Additivity and Multiplicativity Theorem: (Additivity) Consider a circuit with linear resistors and independent sources. Group 1 Group 2 Solve the circuit when only sources in Group 1 are active whereas sources in Group 2 are set to zero Solve the circuit when only sources in Group 2 are active whereas sources in Group 1 are set to zero Then, the solution when all the sources are active is Proof: Circuit equations when all sources are active:

2. If only group 1 is active: If only group 2 is active:

3. Theorem: (Multiplicativity) Consider a circuit with linear resistors and independent sources. Assume that solutions are. If independent sources are set to, solutions are. Proof:...........

4. Thevenin (1883) and Norton (1926) Theorems Aim: To obtain a simple equivalent circuit for a 1-port circuit that consists of linear, time-invariant n-port resistors and independent sources. What does «equivalent» mean?..................................................................... Thevenin Equivalent: + _ v i 1-port circuit + _ v i + _ R TH V TH

5. + _ v i + _ R TH V TH R TH Thevenin resistance Equivalent resistor 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: A 1-port circuit has a Thevenin equivalent circuit if the port voltage can be uniquely determined for a given port current, in other words, if the 1-port is current-controlled. Norton Equivalent: + _ v i 1-port circuit + _ v i GNGN iNiN

6. + _ v i GNGN 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. Norton Theorem: A 1-port circuit has a Norton equivalent circuit if the port current can be uniquely determined for a given port voltage, in other words, if the 1-port is voltage-controlled.

7. How to obtain Thevenin equivalent circuit? + _ v i 1-port circuit Connect a current source to the port. i*i* v*v* + _ Solve the circuit and obtain a relation between 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 resistance R th = v / i. Assume that i=0 and calculate V th taking into account all independent sources.

8. How to obtain Norton equivalent circuit? Connect a voltage source to the port. Solve the circuit and obtain a relation between 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 conductance G N = i/v. Assume that v=0 and calculate I N taking into account all independent sources. + _ v i 1-port circuit i*i* v*v* + _ +-+-

9. 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: