Sinusoid Phasor

Lemma 1: (Uniqueness) Proof:

Lemma 2: (Linearity) Proof:

Lemma 3: (Differentiation) Proof:

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

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

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

KVL for the loop : 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

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

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

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

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.

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

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

v