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: V  V m e j 0   V m  0 
• The phasor current is:
I 
Vm
200
• So that
2
e
 j 45 

Vm
200
2
  45 
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:
•
V  Vm  0 , I 
Vm
200
2
  45 
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:
45
Imaginary
Input
Vm
Vm
V
Vm
200 2
Real
45
Time
Vm
200 2
I
Response
Example 1 – Circuit analysis using phasors
• Use phasors to determine the steady state current i(t) in the
circuit below if Vs(t) = 12cos(120t). 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: i ( t )  I e j 120  t
Example 1: Phasor diagram
• Input voltage phasor:
V
s
 12  0 V
• Output current phasor:
I  0 . 116  15  A
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:
• Conversion to phasor:
vR ( t )  V Re
iR ( t )  I Re
j t
j t
• Voltage-current relation:
• Voltage-current relation:
V
Re
V
j t
R
 R  I Re
 RIR
j t
Resistor phasor voltage-current relations
• Phasor voltage-current
relation for resistors:
V
R
 RIR
• Phasor diagram:
• Note: voltage and current
have same phase for resistor
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:
• Conversion to phasor:
v L ( t )  V Le
iL ( t )  I Le
j t
j t
• Voltage-current relation:
• Voltage-current relation:
V Le
V
j t
L
 L ( j ) I L e
 j L  I L
j t
Inductor phasor voltage-current relations
• Phasor voltage-current
relation for inductors:
V
L
 j L  I L
• Phasor diagram:
• Note: current lags voltage by
90 for inductors
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:
• Conversion to phasor:
vC ( t )  V C e
iC ( t )  I C e
j t
j t
• Voltage-current relation:
• Voltage-current relation:
ICe
V
C
j t

 C ( j  )V
1
j C
IC  
Ce
j
C
j t
IC
Capacitor phasor voltage-current relations
• Phasor voltage-current
relation for capacitors:
V
C

1
j C
IC  
j
C
IC
• Phasor diagram:
• Note: voltage lags current by
90 for capacitors
Capacitor voltage-current waveforms
• Notes: Current and voltage are 90 out of phase; derivative
associated with energy storage causes voltage to lag current