Lecture 27 Review Phasor voltage-current relations for circuit elements Impedance and admittance...

Lecture 27•Review

• Phasor voltage-current relations for circuit elements

•Impedance and admittance•Steady-state sinusoidal analysis

• Examples•Related educational materials:

– Chapter 10.4, 10.5

Phasor voltage-current relations

LL ILjV RR IRV CC ICj

V1

+

-RV

RI

R

Real

ImaginaryIRV

I

+

-LV

LI

Lj

Real

Imaginary

ILjV

I

+

-CV

CI

Cj1

Real

Imaginary

ICj

V1

I

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 ()

Z

IV

Z ZIV

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

jXRZ

Circuit element impedances• Our phasor circuit element voltage-current relations

can all be written in terms of impedances

RZR LjZL Cj

ZC 1

Admittance

• Admittance is the inverse of impedance

• Admittance is a complex number

• Where• G is called the conductance• B is called the susceptance

ZY

1

jBGY

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• voc and isc become phasors ( and )

• The Thévenin resistance, RTH, 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

OCV SCI

THZ

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

Example 2• In the circuit below, vs(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

ifCI RI

2010SI

;• 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 ic(t) and iR(t)? • What are ic(t) and iR(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 . V

Example 4 – continued