Lecture 27 Review Phasor voltage-current relations for circuit elements Impedance and admittance Steady-state sinusoidal analysis Examples Related educational modules: –Section 2.7.3, 2.7.4

Phasor voltage-current relations

Impedance Define the impedance,, of a circuit as: Notes: Impedance defines the relationship between the voltage and current phasors The above equations are identical in form to Ohm’s Law Units of impedance are ohms (  )

Impedance – continued Impedance is a complex number Where R is called the resistance X is called the reactance Impedance is not a phasor There is no sinusoidal waveform it is describing

Circuit element impedances Our phasor circuit element voltage-current relations can all be written in terms of impedances

Admittance Admittance is the inverse of impedance Admittance is a complex number Where G is called the conductance B is called the susceptance

Why are impedance and admittance useful? The analysis techniques we used for time domain analysis of resistive networks are applicable to phasor circuits E.g. KVL, KCL, circuit reduction, nodal analysis, mesh analysis, Thevenin’s and Norton’s Theorems… To apply these methods: Impedances are substituted for resistance Phasor voltages, currents are used in place of time domain voltages and currents

Steady state sinusoidal (AC) analysis KVL, KCL apply directly to phasor circuits Sum of voltage phasors around closed loop is zero Sum of current phasors entering a node is zero Circuit reduction methods apply directly to phasor circuits Impedances in series, parallel combine exactly like resistors in series, parallel Voltage, current divider formulas apply to phasor voltages, currents

AC analysis – continued Nodal, mesh analyses apply to phasor circuits Node voltages and mesh currents are phasors Impedances replace resistances Superposition applies in frequency domain If multiple signals exist at different frequencies, superposition is the only valid frequency domain approach Summation of individual contributions must be done in the time domain (unless all contributions have same frequency)

AC analysis – continued Thévenin’s and Norton’s Theorems apply to phasor circuits v oc and i sc become phasors ( and ) The Thévenin resistance, R TH, becomes an impedance, Maximum power transfer: To provide maximum AC power to a load, the load impedance must be the complex conjugate of the Thévenin impedance

Example 1 Determine i(t) and v(t), if v s (t) = 100cos(2500t)V

Example 2 In the circuit below, v s (t) = 5cos(3t). Determine: (a) The equivalent impedance seen by the source (b) The current delivered by the source (c) The current i(t) through the capacitor

Example 2 – part (a) (a) Determine the impedance seen by the source

Example 2 – part (b) (b) Determine current delivered by the source

Example 2 – part (c) (c) Determine current i(t) through the capacitor

Example 3 Use nodal analysis to determine the current phasors and if

; On previous slide: – Set up reference node, independent node – Write KCL at independent node – Solve for node voltage

Example 3 – continued

Example 3 – continued again What are i c (t) and i R (t)? What are i c (t) and i R (t) if the frequency of the input current is 5000 rad/sec?

Example 3 – revisited Can example 3 be done more easily?

Example 4 Use mesh analysis to determine.

Example 4 – continued