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AC Circuits: Impedance, Power, and Resonance in Series and Parallel RLC Circuits, Study notes of Physics

Class notes on ac circuits, covering topics such as impedance, power, and resonance in series and parallel rlc circuits. Formulas and calculations for impedance, average power, and resonance frequency. It is intended for students studying electrical engineering or physics.

Typology: Study notes

Pre 2010

Uploaded on 08/04/2009

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CPHY 122
Class Notes 17
Instructor: H. L. Neal
1 Impedance in AC Circuits
The impedance is deโ€ฆned as
Z=Vma x
Ima x
2 Power in AC Circuits
The instantaneous power is
P(t) = Vs(t)I(t)
=Vma x Ima x sin (!t) sin (!t ๎˜€๎˜ž):
The important quantity is the average power
Pav (t) = 1
TZT
0
P(t)dt;
where the period of one cycle is
T=2๎˜™
!:
We may write
Pav (t) = Vm ax Im ax
!
2๎˜™Z2๎˜™
!
0
sin (!t) sin (!t ๎˜€๎˜ž)dt
=Vma x Ima x
!
2๎˜™Z2๎˜™
!
0
sin (!t) [sin (!t) cos ๎˜ž๎˜€cos (!t) sin ๎˜ž]dt:
We have
!
2๎˜™Z2๎˜™
!
0
sin2(!t)dt =1
2;
!
2๎˜™Z2๎˜™
!
0
sin (!t) cos (!t)dt = 0;
so that
Pav (t) = Im ax Vm ax cos ๎˜ž:
2.1 Series RLC Circuit
From the previous Class Notes
๎˜ž=tan๎˜€1 !L ๎˜€(!C )๎˜€1
R!:
1
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CPHY 122

Class Notes 17

Instructor: H. L. Neal

1 Impedance in AC Circuits

The impedance is deร–ned as

Z = Vm ax Im ax

2 Power in AC Circuits

The instantaneous power is

P (t) = Vs (t) I (t) = Vm ax Im ax sin (!t) sin (!t ) :

The important quantity is the average power

Pav (t) =

T

Z T

0

P (t) dt;

where the period of one cycle is

T =

We may write

Pav (t) = Vm ax Im ax

Z 2 !

0

sin (!t) sin (!t ) dt

= Vm ax Im ax

Z 2 !

0

sin (!t) [sin (!t) cos  cos (!t) sin ] dt:

We have

! 2 

Z 2 !

0

sin^2 (!t) dt =

Z 2 !

0

sin (!t) cos (!t) dt = 0 ;

so that Pav (t) = Im ax Vm ax cos :

2.1 Series RLC Circuit

From the previous Class Notes

 = tan^1 !L^ ^ (!C)

1 R

cos  = r R R^2 +

!L (!C)^1

sin  =

!L (!C)^1

r R^2 +

!L (!C)^1

Im ax = Vm ax Z ; where

Z =

r R^2 +

!L (!C)^1

The average power is

Pav (t) = Im ax Vm ax cos  = (Im ax )^2 Z cos  = (Im ax )^2 R:

2.2 Parallel RLC Circuit

From the Quiz 8 solution

tan  =

!L

!C

R;

(Im ax )^2 = (Vm ax )^2

R^2

!L

!C

cos () = q^1 =R (1=R)^2 +

!L ^ !C

Z

R

where

Z =

R^2

!L

!C

 2 #^1 =^2

The average power is

Pav (t) = Im ax Vm ax cos 

= (Vm ax )^2 Z cos 

= (Vm ax )^2 R

3 Resonance in AC Circuits

The average power is

P (t) = Vs (t) I (t) = Vm ax Im ax sin (!t) sin (!t ) :