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Phasors. A phasor is a complex number that represents the magnitude and phase of a sinusoid:. Complex Exponentials. A complex exponential is the mathematical tool needed to obtain phasor of a sinusoidal function. A complex exponential is e j w t = cos w t + j sin w t

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phasors
Phasors
  • A phasor is a complex number that represents the magnitude and phase of a sinusoid:
complex exponentials
Complex Exponentials
  • A complex exponential is the mathematical tool needed to obtain phasor of a sinusoidal function.
  • A complex exponential is

ejwt= cos wt + j sin wt

  • A complex number A = z q can be represented

A = z q = z ejq = z cos q + j z sin q

  • We represent a real-valued sinusoid as the real part of a complex exponential.
  • Complex exponentials provide the link between time functions and phasors.
  • Complex exponentials make solving for AC steady state an algebraic problem
complex exponentials cont d
Complex Exponentials (cont’d)

What do you get when you multiple A withejwt

for the real part?

Aejwt = z ejqejwt = z ej(wt+q)

z ej(wt+q) = z cos (wt+q) + j z sin (wt+q)

Re[Aejwt] = z cos (wt+q)

sinusoids complex exponentials and phasors
Sinusoids, Complex Exponentials, and Phasors
  • Sinusoid:

z cos (wt+q)= Re[Aejwt]

  • Complex exponential:

Aejwt = z ej(wt+q), A= z ejq,

  • Phasor for the above sinusoid:

V = z q

phasor relationships for circuit elements
Phasor Relationships for Circuit Elements
  • Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor.
i v relationship for a capacitor

+

i(t)

v(t)

C

-

I-V Relationship for a Capacitor

Suppose that v(t) is a sinusoid:

v(t) = VM ej(wt+q)

Find i(t).

phasor relationship
Phasor Relationship
  • Represent v(t) and i(t) as phasors:

V = VMq

I = jwCV

  • The derivative in the relationship between v(t) and i(t) becomes a multiplication by jw in the relationship between V and I.
kirchhoff s laws
Kirchhoff’s Laws
  • KCL and KVL hold as well in phasor domain.
impedance
Impedance
  • AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law:

V = IZ

  • Z is called impedance.
impedance cont d
Impedance (cont’d)
  • Impedance depends on the frequency w.
  • Impedance is (often) a complex number.
  • Impedance is not a phasor (why?).
  • Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state.
  • Impedance in series/parallel can be combined as resistors
impedance example single loop circuit
Impedance Example:Single Loop Circuit

w = 377

Find VC

20kW

+

+

VC

10V  0

1mF

-

-

impedance example cont d
Impedance Example (cont’d)
  • How do we find VC?
  • First compute impedances for resistor and capacitor:

ZR = 20kW= 20kW  0

ZC = 1/j (377 1mF) = 2.65kW  -90

impedance example cont d1
Impedance Example (cont’d)

20kW  0

+

+

VC

2.65kW  -90

10V  0

-

-

impedance example cont d2

W

Ð

°

æ

ö

2

.

65

k

-

90

=

Ð

°

ç

÷

V

10V

0

C

W

Ð

°

+

W

Ð

°

2

.

65

k

-

90

20

k

0

è

ø

Impedance Example (cont’d)

Now use the voltage divider to find VC:

analysis techniques
Analysis Techniques
  • All the analysis techniques we have learned for the linear circuits are applicable to compute phasors
    • KCL&KVL
    • node analysis/loop analysis
    • superposition
    • Thevenin equivalents/Notron equivalents
    • source exchange
  • The only difference is that now complex numbers are used.
  • Phasors can then converted to corresponding sinusoidal functions to get the time-varying function.
impedance1
Impedance
  • V = IZ, Z is impedance, measured in ohms ()
  • Resistor:
    • The impedance is R
  • Inductor:
    • The impedance is jwL
  • Capacitor:
    • The impedance is 1/jwC
analysis techniques1
Analysis Techniques
  • All the analysis techniques we have learned for the linear circuits are applicable to compute phasors
    • KCL&KVL
    • node analysis/loop analysis
    • superposition
    • Thevenin equivalents/Notron equivalents
    • source exchange
  • The only difference is that now complex numbers are used.
  • Phasors can then converted to corresponding sinusoidal functions to get the time-varying function.
admittance
Admittance
  • I = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S)
  • Resistor:
    • The admittance is 1/R
  • Inductor:
    • The admittance is 1/jwL
  • Capacitor:
    • The admittance is jwC
phasor diagrams
Phasor Diagrams
  • A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes).
  • A phasor diagram helps to visualize the relationships between currents and voltages.
an example

+

+

VC

1mF

V

+

1kW

VR

An Example

2mA  40

  • I = 2mA  40
  • VR = 2V  40
  • VC = 5.31V  -50
  • V = 5.67V  -29.37
phasor diagram
Phasor Diagram

Imaginary Axis

Real Axis

V

VC

VR

homework 9
Homework #9
  • How to determine initial conditions for a transient circuit. When a sudden change occurs, only two types quantities will definitely remain the same before and after the change.
    • IL(t), inductor current
    • Vc(t), capacitor voltage
  • Find these two types of the values before the change and use them as initial conditions of the circuit after change