lecture 26 review steady state sinusoidal response phasor representation of sinusoids phasor...
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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