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Transcript
Additivity and Multiplicativity Theorem: (Additivity) Group 1 Consider a circuit with linear resistors and Group 2 independent sources. Solve the circuit when only sources in Group 1 are active whereas sources in Group 2 are set to zero i ,v 1 1 2 2 Solve the circuit when only sources in Group 2 are active whereas sources in Group 1 are set to zero i ,v Then, the solution when all the sources are active is Proof: wT w1 w2 , Circuit equations when all sources are active: A 0 M 0 I N 0 iT 0 AT vT 0 0 eT wT ~ A x bT ~ xT A 1bT iT A v 0 T eT M 0 I N iT i1 i2 , vT v1 v2 0 AT 0 1 ~ 1 0 0 xT A w1 w2 0 0 wT If only group 1 is active: A 0 M 0 I N 0 i 0 AT v 0 0 e w1 ~ A x b1 ~ x1 A 1b1 i1 A v 0 1 e1 M 0 I N 1 0 0 w1 1 0 0 w2 0 AT 0 ~ 1 0 x1 A w1 If only group 2 is active: A 0 M 0 I N 0 i 0 AT v 0 0 e w2 i1 A v 0 1 e1 M 0 I N ~ 1 0 x2 A w2 ~ 1 0 ~ 1 0 ~ 1 0 0 x1 x2 A A A xT w1 w2 w1 w2 ~ A x b2 ~ 1 x1 A b2 0 AT 0 Theorem: (Multiplicativity) Consider a circuit with linear resistors and independent sources vs . Assume that solutions are i, v . If independent sources are set to k vs , solutions are k i, k v . Proof: ........... 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: RTH i 1-port circuit + v _ + _ i + VTH v _ i RTH + _ + v VTH _ v(t ) RTH i(t ) vTH (t ) RTH Thevenin resistance Equivalent resistor between terminals when sources are set to zero. VTH 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: i 1-port circuit + v _ i + iN GN v _ i + iN GN v _ i(t ) GN v(t ) iN (t ) GN Norton conductance Equivalent conductance between terminals when sources are set to zero. iN Short circuit current 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. v (t ) v (t ) v (t ) vTH iN vTH i (t ) iN i (t ) i (t ) How to obtain Thevenin equivalent circuit? i + 1-port circuit v v* _ + i + 1-port circuit v _ _ i* • Connect a current 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. • Set the values of independent sources to zero. • Calculate the equivalent resistance Rth = v / i. • Assume that i=0 and calculate Vth taking into account all independent sources . How to obtain Norton equivalent circuit? i 1-port circuit + + v v* _ _ i + 1-port circuit v _ + - i* • 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. • Set the values of independent sources to zero. • Calculate the equivalent conductance GN = i/v. • Assume that v=0 and calculate IN taking into account all independent sources . Interchange between Thevenin and Norton V = RTH I +VTH • Thevenin Equivalent: If 1-port is not current-controlled there is no Thevenin eq.. • Norton Equivalent: I = GNV + I N If 1-port is not voltage-controlled there is no Norton eq.. From Thevenin to Norton: • RTH ¹ 0 1 VTH I= VRTH RTH GN RTH = 0, No Norton equivalent! IN From Norton to Thevenin: • GN ¹ 0 1 IN V= IGN GN RTH VTH GN = 0, No Thevenin equivalent!