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# Phasors PowerPoint PPT Presentation

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

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).

### 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.

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

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

20kW  0

+

+

VC

2.65kW  -90

10V  0

-

-

W

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65

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V

10V

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W

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+

W

Ð

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2

.

65

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20

k

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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.

• I = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S)

• Resistor:

• Inductor:

• Capacitor:

### 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

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