Lecture 26•Review

• Steady state sinusoidal response• Phasor representation of sinusoids

•Phasor diagrams•Phasor representation of circuit elements•Related educational modules:

– Section 2.7.2, 2.7.3

Steady state sinusoidal response – overview

• Sinusoidal input; we want the steady state response• Apply a conceptual input consisting of a complex

exponential input with the same frequency, amplitude and phase• The actual input is the real part of the conceptual input

• Determine the response to the conceptual input• The governing equations will become algebraic

• The actual response is the real part of this response

Review lecture 25 example

• Determine i(t), t, if Vs(t) = Vmcos(100t).

• Let Vs(t) be:

• Phasor:• The phasor current is:

• So that

00m

jm VeVV

4522002200

45 mjm Ve

VI

Phasor Diagrams• Relationships between

phasors are sometimes presented graphically• Called phasor diagrams• The phasors are

represented by vectors in the complex plane

• A “snapshot” of the relative phasor positions

• For our example:

• , 0mVV 452200

mVI

Phasor Diagrams – notes

• Phasor lengths on diagram generally not to scale• They may not even share the same units• Phasor lengths are generally labeled on the diagram

• The phase difference between the phasors is labeled on the diagram

Phasors and time domain signals• The time-domain (sinusoidal) signals are completely

described by the phasors• Our example from Lecture 25:

Real

Imaginary

Vm

2200mV

V

I

45Time

45

Input

Response

Vm

2200mV

Example 1 – Circuit analysis using phasors

• Use phasors to determine the steady state current i(t) in the circuit below if Vs(t) = 12cos(120t). Sketch a phasor diagram showing the source voltage and resulting current.

Example 1: governing equation

Example 1: Apply phasor signals to equation

• Governing equation:

• Input:

• Output: tjeI)t(i 120

Example 1: Phasor diagram

• Input voltage phasor:

• Output current phasor:

VV s 012

A.I 151160

Circuit element voltage-current relations

• We have used phasor representations of signals in the circuit’s governing differential equation to obtain algebraic equations in the frequency domain

• This process can be simplified:• Write phasor-domain voltage-current relations for circuit

elements• Convert the overall circuit to the frequency domain• Write the governing algebraic equations directly in the

frequency domain

Resistor i-v relations• Time domain:

• Voltage-current relation:

• Conversion to phasor:

• Voltage-current relation:tj

Rtj

R eIReV

RR IRV

tjRR eV)t(v

tjRR eI)t(i

Resistor phasor voltage-current relations• Phasor voltage-current

relation for resistors:• Phasor diagram:

• Note: voltage and current have same phase for resistor

RR IRV

Resistor voltage-current waveforms

• Notes: Resistor current and voltage are in phase; lack of energy storage implies no phase shift

Inductor i-v relations• Time domain:

• Voltage-current relation:

• Conversion to phasor:

• Voltage-current relation:tj

Ltj

L eI)j(LeV

LL ILjV

tjLL eV)t(v

tjLL eI)t(i

Inductor phasor voltage-current relations• Phasor voltage-current

relation for inductors:• Phasor diagram:

• Note: current lags voltage by 90 for inductors

LL ILjV

Inductor voltage-current waveforms

• Notes: Current and voltage are 90 out of phase; derivative associated with energy storage causes current to lag voltage

Capacitor i-v relations• Time domain:

• Voltage-current relation:

• Conversion to phasor:

• Voltage-current relation:

tjCC eV)t(v

tjCC eI)t(i

tjC

tjC eV)j(CeI

CCC ICj

ICj

V

1

Capacitor phasor voltage-current relations

• Phasor voltage-current relation for capacitors:

• Phasor diagram:

• Note: voltage lags current by 90 for capacitors

CCC ICj

ICj

V

1

Capacitor voltage-current waveforms

• Notes: Current and voltage are 90 out of phase; derivative associated with energy storage causes voltage to lag current