AC Circuits
V0 sin t
VR  IR
• An AC circuit is made up with components.
•
•
•
•
Power source
Resistors
Capacitor
Inductors
R
VC  IX C
C
X C  1 C
VL  IX L
L
X L  L
• Kirchhoff’s laws apply just like DC.
• Special case for phase
RLC Circuits
• An RLC circuit (or LCR circuit or CRL circuit or RCL circuit) is an electrical
circuit consisting of a resistor, an inductor, and a capacitor, connected in
series or in parallel.
• The RLC part of the name is due to those letters being the usual electrical
symbols for resistance, inductance and capacitance respectively.
• The circuit forms a harmonic oscillator for current and will resonate in a
similar way as an LC circuit will. The main difference that the presence of
the resistor makes is that any oscillation induced in the circuit will die
away over time if it is not kept going by a source.
• This effect of the resistor is called damping. The presence of the resistance
also reduces the peak resonant frequency somewhat. Some resistance is
unavoidable in real circuits, even if a resistor is not specifically included as
a component. An ideal, pure LC circuit is an abstraction for the purpose of
theory.
IMPEDANCE AND THE PHASOR DIAGRAM
Resistive Elements
• For purely resistive circuit v and i were in
phase, and the magnitude:
• In phasor form,
FIG. 15.1 Resistive ac circuit.
IMPEDANCE AND THE PHASOR DIAGRAM
Resistive Elements
FIG. 15.5 Waveforms for Example 15.2.
FIG. 15.4 Example
15.2.
IMPEDANCE AND THE PHASOR DIAGRAM
Inductive Reactance
• for the pure inductor,
the voltage leads the
current by 90° and that
the reactance of the
coil XL is determined by
ψL.
FIG. 15.9 Waveforms for Example
15.3.
FIG. 15.8 Example
15.3.
IMPEDANCE AND THE PHASOR DIAGRAM
Capacitive Reactance
• for the pure capacitor, the current leads the
voltage by 90° and that the reactance of the
capacitor XC is determined by 1/ψC.
FIG. 15.17 Waveforms for Example
15.6.
FIG. 15.16 Example
15.6.
Inductors - how do they work?
R
V0
L
Start with no current in the circuit.
When the battery is connected, the
inductor is resistant to the flow of
current.
Gradually the current increases to the
fixed value V0/R, meaning that the
voltage across the inductor goes to
zero.
dI
VL  L
dt
In reality the inductor has a finite
resistance since it is a long wire so it
will then be more like a pair of series
resistances.
Inductors - time constant L/R
VR
V0
VL
Again the behavior of an inductor is seen
by analysis with Kirchoff’s laws. Suppose
we start with no current.
dI
V0  VR  VL  IR  L
dt
then
I
V0 
 Rt 
1

exp
 


R
 L 
and
Rt 

 Rt 
VL  V0 exp   VR  V0 1  exp   
 L
 L 

There is a fundamental time scale set by L/R, which has units of seconds
(=Henry/Ohm)
Mathematical analysis of a series LRC
circuit - bandpass filter
Vin
L
Z L  iL
1
iC
i

C
ZC 
C
R
First find the total impedance of the
circuit
1 

Z  R  i  L 

C 

Using a voltage divider
Vout

Vin
R
1 

R  i  L 

C 

Vout
The phase shift goes from 90°to 90°.
   tan
1
R
1 


L



C 

Mathematical analysis of a series LRC
circuit - bandpass filter (2)
The magnitude of the gain, Av, is
Vin
Av 
L
C
R
Vout
Vout

Vin
R
1 

R 2   L 


C


2
Note that for high frequencies L is
dominant and the gain is R/ L or small. At
low frequencies the gain is  RC because
the impedance of the capacitor is
dominant. At 2 = 1/LC the gain is one
(assuming ideal components).
Series RLC
i
v
• A series RLC circuit can be made
from each component.
• One loop
• Same current everywhere
R
L
C
• Reactances are used for the
capacitors and inductors.
• The combination of resistances
and reactances in a circuit is
called impedance.
Vector Map
• Phase shifts are present in AC
circuits.
• +90° for inductors
• -90° for capacitors
VL=IXL
VR=IR
VC=IXC
• These can be treated as if on the
y-axis.
• 2 D vector
• Phasor diagram
Vector Sum
• The current is the same in the loop.
• Phasor diagram for impedance
• A vector sum gives the total impedance.
XL
XC
XL
Z
R
R
XC
Vector Sum
• The total impedance is the magnitude of Z.
XC
XL
• The phase between the current and voltage is
the angle  between Z and the x-axis.
Z

R
Z  R2  X L  X C 
2
1 

2
Z  R   L 

C 

2
X L  XC
tan  
R
1


L


C
  arctan 
R









Phase Changes
• The phase shift is different in each component.
Power Factor
• Power loss in an AC circuit depends on the
instantaneous voltage and current.
• Applies to impedance
• The cosine of the phase angle is the power
factor.
p  vi  i 2 Z cos 
2
Prms  Vrms I rms  I rms
Z cos 
P
I 0 Z cos 
2
0
t