1. Sinusoid Phasor

2. Lemma 1: (Uniqueness) Proof:

3. Lemma 2: (Linearity) Proof:

4. Lemma 3: (Differentiation) Proof:

5. Solving State Equations Using Phasors Differential Equation Phasor Algebraic EquationSolution of the Algebraic Equation Solutions as Phasors Sinusoid Solutions as Sinusoid

6. Example: Find the particular solution of the differential equation! Answer:

7. Circuit Equations in Sinusoidal Steady-State Analysis Consider a circuit that consists of linear time-invariant elements and driven by sinusoidal sources. KCL for Node 1: From linearity and uniqueness If generalized to all nodes A matrix with real coefficients A vector of complex numbers L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

8. KVL for the loop 1-2-3-1: If generalized L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York From linearity and uniqueness

9. Linear and Time-invariant Circuit Elements L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York Resistor Inductor Capacitor Voltage controlled voltage source Voltage controlled current source Current controlled voltage source Current controlled current source Gyrator Ideal Transformator

10. Impedance and Admittance Goal: Modelling a linear and time-invariant 1-port circuit characteristic in phasor form. N 1-port circuit + _ v isis Input impedance for N: resistance reactance

11. N 1-port circuit + + _ v i Input admittance for N: conductancesusceptance

12. and the circuit is in sinusoidal steady state. a)Draw phasor diagrams for circuit elements. b)Find the phasor for and calculate the sinusoidal signal.

13. Paralel and Serial Connections of 1-ports L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

14. Circuit Equations (Tableau Equations) KCL: KVL: EE:

15. v 1 2 3 4