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1 CHAPTER 22 ALTERNATING CURRENT Alternating Currents Phasors Reactance Impedance Power in AC Circuits R-L-C Circuits Resonance

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• 1

CHAPTER 22

ALTERNATING CURRENT

Alternating Currents

Phasors

Reactance

Impedance

Power in AC Circuits

R-L-C Circuits

Resonance

• 2

For Resistor

Consider voltage that is sinusoidal.

=

Note that V is the maximum value of voltage and v varies from 0

to a maximum of V.

Also, a current that is sinusoidal.

=

Note that is the maximum value of voltage and varies from 0 to a maximum of .

• 3

Fig 22.1

Can represent using phasors. Phasors are just tools to help

visualize.

• 4

The

phasor, the vector from the origin to the curved line, has the

length equal to the magnitude of the current or voltage or

whatever the phasor represents.

The projection on the horizontal axis represents the

instantaneous value of the quantity.

• 5

AVERAGE VALUES

Values for the ac quantities used to make calculations of averages

are called root-mean-square values.

They can be obtained from the max values, the amplitude, by:

= 2

= 2

• 6

• 7

Resistor in ac circuit with current

=

Instantaneous voltage across resistor is

= =

=

cos

• 8

• 9

• 10

Inductor in ac Circuit with current =

=

• 11

=

Current changes rapidly - Voltage large

Current doesnt change - Voltage Zero

• 12

• 13

It can be shown that

(Calculus by differentiating, ! " )

= = ( + &)

Where & is the phase angle. For inductor that is 900.

Inductive Reactance,

Defined as ( =

By analogy voltage for inductor = (

Now Capacitor in ac Circuit with current =

• 14

• 15

)* = +* Therefore

)*t = 1. qt = 1. = 1.

Thus a sinusoidal curve as seen in Fig. 22.8.

• 16

Thus the voltage curve lags the current curve by 900.

And we have (can be derived by calculus)

0 = 1.

Or

0 = 1. cos( 903)

Or 0 = 0cos( 903)

Capacitive reactance

(0 = 1.

And 0 = (0

• 17

• 18

IMPEDANCE

Impedance of a circuit is the sum of the resistance and reactance

in the circuit. The symbol for impedance is Z.

4 = 56 + (6

4 = 56 + (( (0)6

4 = 76 + 8 1.96

• 19

Consider

(Across Resistor)

What is reactance of the capacitor?

(0 = 1. = 1:2500 ([email protected]) = 80

What is the current in the circuit?

= = 1.2 :2500 300

= [email protected] :2500

• 20

What is the impedance?

4 = 76 +8 1.96

4 = 76 8 1.96

4 = 5(300)6 (80)6 = 290

• 21

Power in ac circuits

Remember this about power for pure resistance

IJKL = 12 = 12 12 = 12 12 =

But with capacitors or inductors

MN)O = PQRSTUVW = RQ SQTUVW = SXYVRXYVTUVW

cosW is called The Power Factor

Also IZ[