Phasors
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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

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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

    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)

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

  • 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

  • Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor.


+

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).


Computing the Current


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.


I-V Relationship for an Inductor

V = jwLI

+

i(t)

v(t)

L

-


Kirchhoff’s Laws

  • KCL and KVL hold as well in phasor domain.


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 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

w = 377

Find VC

20kW

+

+

VC

10V  0

1mF

-

-


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’d)

20kW  0

+

+

VC

2.65kW  -90

10V  0

-

-


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

  • 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.


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 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

  • 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

  • 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.


+

+

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

Imaginary Axis

Real Axis

V

VC

VR


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


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