Sinusoidal Steady-State Analysis Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Contents Introduction Nodal Analysis Mesh Analysis Superposition Theorem Source Transformation Thevenin and Norton Equivalent Circuits OP-amp AC Circuits Applications

Introduction Steps to Analyze ac Circuits: –Transform the circuit to the phasor (frequency) domain. –Solve the problem using circuit techniques (nodal/mesh analysis, superposition, etc.). –Transform the resulting phasor to the time domain.

Nodal Analysis Variables = Node Voltages Applying KCL to each node gives each independent equation If supernodes included, –Applying KCL to each supernode gives 1 equation. –Applying KVL at each supernode gives 1 more equation. Supernode

Example 1 Find i x.

Example 1 (Cont’d)

Example 2 Applying KCL for the supernode gives 1 equations. Applying KVL at the supernode gives 1 equations. 2 variables solved by 2 equations.

Example 2 (Cont’d)

Mesh Analysis Supermesh Excluded Variables = Mesh Currents Applying KVL to each mesh gives each independent equation If supermeshes included, –Applying KVL to each supermesh gives 1 equation. –Applying KCL at each supernode gives 1 more equation.

Example 1 Find I o.

Example 2 Find V o. Applying KVL for mesh 1 & 2 gives 2 equations. Applying KVL for the supermesh gives 1 equations. Applying KCL at node A gives 1 equations. 4 variables solved by 4 equations

Example 2 (Cont’d)

Superposition Theorem Since ac circuits are linear, the superposition theorem applies to ac circuits as it applies to dc circuits. The theorem becomes important if the circuit has sources operating at different frequencies. –Different frequency-domain circuit for each frequency –Total response = summation of individual responses in the time domain –Total response  summation of individual responses in the phasor domain

Example 1 Find I o. = +

Example 1 (Cont’d)

Example 2

Example 2 (Cont’d)

Source Transformation

Example 1 Find V x.

Thevenin & Norton Equivalent Circuits

Example 1

Example 2

Example 2 (Cont’d)

Example 3

Example 3 (Cont’d)

OP AMP AC Circuits: Example 1 Ideal op amps assumed –Zero input current & zero differential input voltage

Example 2

Applications: Capacitance Multiplier

Applications: Oscillators Barkhausen criteria must be meet for oscillators –(1) Overall gain  1 –(2) Overall phase shift = 0