Download Chapter 21

Review on R, L, C and RLC in AC circuits

An AC power source is a sinusoidal voltage source which is
v
described as
v  Vmax sin t 

Vmax
We study the current i as a function of time when a resistor,
inductor, capacitor or the serial connection of the three, to the
AC voltage source.
When resistance is involved,
i t 
v  Vmax sin t 
Assuming AC voltage source:
Laws to
apply
R
Ohm’s Law
C
RLC
Kirchhoff’s
Loop
Kirchhoff’s
Loop
Kirchhoff’s
Loop
In phase or
Vmax
 0
R
i  I max sin t  2  v
I max 
Vmax
XL
, X L  L
i  I max sin t  2 
I max 
Vmax
XC
1
, XC 
C
leads, or
  2
i leads, or
   2
i  I max sin t   
I max 
Vmax
X  XC
tan   L
R
Z
Z  R2   X L  X C 
Phasor diag.
Power
pav  Vrms  I rms
i  I max sin t 
I max 
L
Phase w/  v
Current i
2
Vmax  2Vrms
I max  2I rms
No power is
consumed.
No power is
consumed.
To be
discussed
today
Power discussion of RLC in an AC circuit
I did not discuss power with L or C in AC circuit.
Because there was no power consumed. With R
comes into the play, we have to examine the
power issue.
i  I max sin t   
The power delivered from the AC voltage
source is:
v  Vmax sin t 
p  v  i  Vmax sin t   I max sin t   
 Vrms  I rms 2 sin t  sin t   
 Vrms  I rms  cos    cos  2t    
Vrms  2Vmax , I max  2I rms
2 sin  sin   cos      cos    
The (more useful) average power is:
T
T


1
1
pav   pdt   Vrms  I rms  cos    cos  2t     dt
T0
T0
T
T

1
 Vrms  I rms cos     dt   cos  2t   dt 
T 0
0

 Vrms  I rms cos  
The final result:
pav  Vrms  I rms cos  
Power and phase
The average power delivered from the AC
voltage source to the RLC circuit is:
pav  Vrms  I rms cos  
This power depends on the phase angle.
Circuits involving heavy motors (heavy
inductive load) usually have capacitors to shift
the phase to improve the power delivery
efficiency.
i  I max sin t   
v  Vmax sin t 
Resonance in an AC Circuit
A series RLC circuit, R=0.01Ω, L= 4.34mH,
C=1.00mF. ΔVmax=150 V. Complete the
following table.
i  I max sin t   
v  Vmax sin t 
 (rad/sec)
440
460
480
500
520
I max (A)
pav (W)
Formula needed:
I max 
Vmax
Z
Z  R2   X L  X C 
X L  L
pav  Vrms  I rms cos  
XC 
1
C
2
See the
spreadsheet
tan  
X L  XC
R
Resonance in an AC Circuit
From these formulas:
I max 
Vmax
Z
Z  R2   X L  X C 
2
i  I max sin t   
1
1
, or  2 
When  L 
C
LC
We have
X L  X C and Z min  R
v  Vmax sin t 
Under this condition,
V
I max  max reaches maximum with a given Vmax
Z
X  XC
tan   L
 0,  =0 and cos   1
R
This frequency is called the resonant frequency:
0 
1
LC
Power also resonates
The average power is
pav  Vrms  I rms cos   
2
Vrms
Z
cos  
Now I want to express the power as a function
of the angular frequency
2
Z
Vrms
XL- XC
pav 
cos  
Z


2
Vrms
R
Z2
cos   
2
Vrms
R
1 

R  L 
C 

2
Vrms
R 2
R
Z
R
2
2


1 

R 2 2  L2   2 

LC 

2
Vrms
R 2

R 2 2  L2  2  02

2
multiply  2
2
So when   0 the power reaches a maximum: resonates.
PLAY
ACTIVE FIGURE
How narrow (good) is the resonance:
the Q (quality) factor
The sharpness of the resonance curve is
usually described by a dimensionless parameter
known as the quality factor, Q
0 0 L
Q


R
Δω is the half-power width: width of the average
power curve, measured between the two values
of ω for which pav has half its maximum value.
Because R usually comes from the resistance of
the wire that is used to construct the inductor,
one tries to design the inductor in such a way
that it maximizes the L, and minimizes the R.