E E 2415

Lecture 9Phasor Circuit Analysis,

Effective Value and Complex Power: Watts, VAR’s and Volt-

Amperes

Effective Value of a Sinusoid (1/3)

Average Power:

i(t) 10

Effective Value of a Sinusoid (2/3)

In our example:

Also:

The effective value is also called the Root Mean Square value or rms value.

Effective Value of a Sinusoid (3/3)

R-C Circuit Example (1/6)

Capacitive Reactance

vs(t)

40

88.42 F

+ vR - +vc-

i

Using rms phasor for voltage source.

R-C Circuit Example (2/6)

120

40

-j30 + VR -

~+VC-

~I~

Calculate Real Power:

And Reactive Power:

Apparent power is the product of voltage and current of the source.

Also:

R-C Circuit Example (3/6)

Power Factor is the ratio of real power to apparent power:

Power Factor is also the Cosine of the anglebetween the load voltage and the load current:

If the load current leads the load voltage, the power factor is leading; if it lags the load voltage,the power factor is lagging.

R-C Circuit Example (4/6)

Phasor Diagram of Voltage and Current

Current leads voltage.

R-C Circuit Example (5/6)

36.87o

I~

~V120V

2.4A

The Power Triangle showing leadingpower factor.

R-C Circuit Example (6/6)

230.4 W

-172.8 VAR288.0 VA

real

imag

Calculating Complex Power (1/2)

2S P jQ I R jX 2 *I I I

*

2* 2 2

,I c jd I c jd

then I I c d I

*S P jQ I I R jX

R jXI~

+ V -~

c

jdI~

Calculating Complex Power (2/2)

V R jX I

*S P jQ V I

From now on, we use the above method to calculate complex power.

Lagging Power Factor Example (1/4)

v(t)

5

22.97 mHi

Calculate complex power directly:

Lagging Power Factor Example (2/4)

1200°

5

j8.66 I~

Power Factor:

Power Factor is Lagging

Phasor Diagram of Voltage and Current

Lagging Power Factor Example (3/4)

60o

1200° V

12-60° A

imag

real

Power Triangle for lagging Power Factor

Lagging Power Factor Example (4/4)

1440

VA

720 W

1247 VAR's

imag

real

Phasor Power Example (1/4)

480V(rms)

0.5 j2

40

j30

-j150

+

V

-

~

I~

480 07.94 20.08

(56.75 20.75)

VI A

j

(40 30)( 150)(56.25 18.75)

40 30 150p

j jZ j

j j

Phasor Power Example (2/4)

3.550.949 94.9%

3.74PF or Lagging

*

(471 1.65 )(7.94 20.08 )

3.55 1.18

RLCS V I

V A

kW j kVAR

(56.25 18.75) 471 1.65V I j

Phasor Power Example (3/4)

*

(471 1.65 )(9.42 38.52 )

3.55 2.66

RL RLS V I

V A

kW j kVAR

471 1.659.42 38.52

(40 30)RL

VI A

j

Capacitor VAR’s:

2 24711.48

150c

VQ kVAR

Phasor Power Example (4/4)

3.55 kW

2.66 kVAR

1.18 kVAR 4.44 k

VA

3.74 kVA

1.48 kVAR}

P

Q

Impedance and Admittance

jXRZ

jBGZ

Y 1

22 XRR

G

22 XRX

B

Impedance

Admittance

Conductance

Susceptance