1. Power in Sinusoidal Steady State All currents and voltages are sinusoids with frequency “w”. Some relations: * ** *** Instantaneous power and average power (as time functions): Complex power (as phasors): Active power [Watt] Reactive power [VAR]

2. Instantaneous Power and Average Power For a resistor: The power delivered to the resistor (by the source): from * Instantaneous power oscillates twice in one period between and. Average power: Why always..................?

3. Instantaneous Power and Average Power For a capacitor: The power delivered to the capacitor (by the source): from *** Instantaneous power oscillates twice in one period between and. Average power:....................................

4. Instantaneous Power and Average Power For an inductor: By duality, the power delivered to the capacitor (by the source): Instantaneous power oscillates twice in one period between and. Average power:

5. Instantaneous Power and Average Power For a general 1-port: i + _ v 1-port circuit in SSS G from *** The power delivered to the 1-port (by the source): Average power: The average power does not only depend on the magnitude of sinusoids but also on their phase differences. : Power factor affects the average power.

6. Remember that the phase of an impedance element V=ZI is given by If effective values are used,....

7. Complex Power i + _ v 1-port G Complex power delivered to the port by the source is defined as Active power [Watt] Reactive power [VAR] [VAR]-VoltAmperReactive

8. L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

9. Conservation of Complex Power KCL+KVL Tellegen’s theorem Power is conserved. But what about the complex power? Proof: L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York Phasors that satisfy KVL Phasors that satisfy KCL KCL From Tellegen’s Theorem Theorem Consider a linear time-invariant circuit driven by a number of independent sources, all sinusoidal at the same frequency w. We assume that the circuit is in the sinusoidal steady state. Then the sum of the complex power delivered by each independent source to the circuit is equal to the sum of the complex power absorbed by each element of the circuit.

10. Maximum Power Transfer Aim: For a circuit in SSS, find a Z L such that the active power delivered from the source to a Z L is maximum. Assume that Conservation of complex powerConservation of active power Active power delivered from the source Active power consumed by Z G P L is a function of Ø I L and I Lm. (R G and E G are constants.)

11. For a maxiumum P L I L ‘s maximum value:

12. Since and only %50 of source energy is transmitted to the load. Nevertheless this is the best case as we do not have any controll on. Maximum power transfer theorem: Assume that a 1-port circuit is operating in the sinusoidal steady state at frequency w and that this circuit is driving a load impedance Z L. This one-port is specified by its Thevenin equivalent (E G, R G + jX G ). where R G > 0. The load impedance Z, will receive from the one-port a maximum average power if and only if Z L = Z G. In this case, the average power delivered to the load is given by. Example: Calculate the impedance value Z for the maximum power transfer to Z. Determine the power of Z in this case!