Transcript Lecture 21: Alternating Current Circuits and EM Waves

Chapter 21: Alternating Current
Circuits and EM Waves
Resistors in an AC Circuits
Homework assignment : 22,25,32,42,63
 AC circuits
• An AC circuit consists of combinations of
circuit elements and an AC generator or an
AC source, which provides the alternating
current. The alternating current can be
expressed by :
v  Vmax sin 2ft  Vmax sin t
• The current and the voltage reach their
maximum at the same time; they are said
to be in phase.
• The power, which is energy dissipated in
a resistor per unit time is:
P  i 2 R where i is the instaneous
current at the resistor
Resistors in an AC Circuits

Rms current and voltage
• rms current
i 

I rms 
I max
 0.707 I max
2
2
av
1 2
I max
2
2
Pav  I rms
R
• rms voltage
Vrms 
Vmax
 0.707Vmax
2
voltage difference at a resitor
VR ,rms  I rms R
VR ,max  I max R
Capacitors in an AC Circuits

Capacitors in an AC circuit
• Phase difference between iC and vC
At t=0, there is no charge at the
capacitor and the current is free
to flow at i=Imax.
Then the current starts to decrease.
When the direction of the current
is reversed, the amount of the
charge at the capacitor starts to
decrease.
I
Q
t
Q  CV
The voltage across a capacitor
always lags the current by 90o.
Capacitors in an AC Circuits

Capacitors in an AC circuit
• Reactance and VC,rms vs. Irms
Reactance: impeding effect of a
capacitor
XC 
1
1

2fC C
VC ,rms  I rms X C
XC in W
f in Hz
C in F
capacitive
reactance
Inductors in an AC Circuits

Inductors in an AC circuit
• Phase difference between iL and vL
The changing current output of the
generator produces a back emf that
impedes the current in the circuit.
The magnitude of the back emf is:
I
vL  L
t
The voltage across an inductor
always leads the current by 90o.
The effective resistance of coil in an
AC circuit is measured by the inductive
reactance XL:
X L  2fL  L
VL ,rms  I rms X L
inductive reactance
XL in W
f in Hz
L in H
a: iL/t maximum vL max.
b: iL/t zero vL zero
RLC Series Circuits

A simple RLC series circuit
• Current in the circuit
i  I max sin 2ft  I max sin t
• Phase differences
The instantaneous voltage vR is in
phase with the instantaneous current i.
The instantaneous voltage vL leads
the current by 90o.
The instantaneous voltage vC lags
the current by 90o.
RLC Series Circuits

A simple RLC series circuit
• Phasors
It is convenient to treat a voltage
across each element in a RLC circuit
as a rotating vector (phasor) as shown
in the phasor diagram on the right.
v  Vmax sin( 2ft   )  Vmax sin( t   )
• Phasor diagram
VL  VL,max , VR  VR ,max
VC  VC ,max
Vmax  VR2  (VL  VC ) 2
tan  
VL  VC
 VR
RLC Series Circuits

A simple RLC series circuit
• Impedance
Vmax  VR2  (VL  VC ) 2  I max R 2  ( X L  X C ) 2
Z  R2  ( X L  X C )2
Vmax  I max Z
tan  
impedance
in form of Ohm’s law
X L  XC
R
Note that quantities with
subscript “max” is related
with those with “rms”, all
the results in this slide are also applicable to quantities with subscript “rms”
RLC Series Circuits

Impedances and phase angles
RLC Series Circuits
 Filters : Example
Vout
Vout  IR  
~
Vout


R2  X C
R
R 
2
Ex.: C = 1 μf, R = 1Ω
1
C

2
2
1

1
 
0 2

1
0 
RC
High-pass filter
1
"transmission"
R
0.8
0.6
High-pass filter
0.4
0.2
0
0.E+00
1.E+06
2.E+06
3.E+06
4.E+06
(Angular) frequency, om ega
5.E+06
6.E+06
Note: this is ω, f 

2
RLC Series Circuits
 Filters (cont’d)
Vout
~
~
ω=0
No current
Vout ≈ 0
ω=∞
Capacitor ~ wire
Vout ≈ ε
Vout
Vout
ω = ∞ No current
Vout ≈ 0
High pass filter
0
Vout

Lowpass filter
ω = 0 Inductor ~ wire
Vout ≈ ε
ω = 0 No current because of capacitor
~
ω = ∞ No current because of inductor
(Conceptual sketch only)

0
Vout

0
Band-pass filter

Power in an AC Circuit

Power in an AC circuit
• No power losses are associated with pure capacitors
- When the current increases in one direction in an AC circuit,
charge accumulates on the capacitor and the voltage drop
appears across it.
- At the maximum value of the voltage, the energy stored in the
1
PE

C (Vmax ) 2
capacitor is:
C
2
- When the current reverses direction, the charge leaves the
capacitor to the voltage source and the stored energy decreases.
- As long as there is no resistance, there is no energy loss.
• No power losses are associated with pure inductors
- The source must do work against the back emf of an inductor
that is carrying a current.
- At the maximum value of the current, the energy stored in the
1
PE

L( I max ) 2
inductor is:
L
2
- When the current starts to decrease, the stored energy returns
to the source as the inductor tries to maintain the current in
the cuircuit.
- As long as there is no resistance, there is no energy loss.
Power in an AC Circuit

Power in an AC circuit (cont’d)
The average power delivered by the generator is converted to
internal energy in the resistor. No power loss occurs in an ideal
capacitor or inductor.
2
Pav  I rms
R
Average power delivered to the resistor
R  VR / I rms
Pav  I rmsVR
Vrms
VR  Vrms cos 
Pav  I rmsVrms cos 
Resonance in a Series RLC Circuit

Resonance
• The current in a series RLC circuit
I rms 
Vrms

Z
Vrms
R 2  ( X L  X C )2
This current reaches the maximum when XL=XC (Z=R).
XC  X L
2f 0 L 
f0 
1
2f 0C
1
2 LC
Transformers
 Transformers
• AC voltages can be stepped up or
stepped down by the use of
transformers.
The AC current in the primary circuit
creates a time-varying magnetic field
in the iron.
E ~
This induces an emf on the secondary
windings due to the mutual inductance of
the two sets of coils.
iron
V1
V2
N
1
(primary)
N
2
(secondary)
• We assume that the entire flux produced by each turn of the primary is
trapped in the iron.
Transformers
 Ideal transformer without a load
No resistance losses
All flux contained in iron
Nothing connected on secondary
The primary circuit is just an AC voltage
source in series with an inductor. The
change in flux produced in each turn is given by:
turn
V
 1
t
N1
iron
 ~ V 1
V 2
N1
N2
(primary) (secondary)
• The change in flux per turn in the secondary
coil is the same as the change in flux per turn in the primary coil (ideal case).
The induced voltage appearing across the secondary coil is given by:
turn N 2
V2   N 2

V1
t
N1
• Therefore,
•N2 > N1 -> secondary V2 is larger than primary V1 (step-up)
•N1 > N2 -> secondary V2 is smaller than primary V1 (step-down)
• Note: “no load” means no current in secondary. The primary current,
termed “the magnetizing current” is small!
Transformers
 Ideal transformer with a load
What happens when we connect a resistive load
to the secondary coil?
 ~
iron
V1
V2
Changing flux produced by primary coil induces
an emf in secondary which produces current I2
I2 
V2
R
N1
(primary)
R
N2
(secondary)
This current produces a flux in the secondary coil
µ N2I2, which opposes the change in the original
flux -- Lenz’s law
This induced changing flux appears in the primary
circuit as well; the sense of it is to reduce the emf in
the primary, to “fight” the voltage source. However,
V1 is assumed to be a voltage source. Therefore, there
must be an increased current I1 (supplied by the voltage
source) in the primary which produces a flux µ N1I1
which exactly cancels the flux produced by I2.
I1 
N2
I2
N1
Transformers
 Ideal transformer with a load (cont’d)
Power is dissipated only in the load resistor R.
V22
Pdissipated  I R 
 V2 I 2
R
Where did this power come from?
It could come only from the voltage source in the primary:
2
2
Pgenerated  V1 I1
V1I1  V2 I 2
N2
V1
I1 V2 N1
N


 1
I 2 V1
V1
N2
N
V2 N 2 V1  N 2 


I1  I 2 2 

N1
R N1
R  N1 
2
The primary circuit has
to drive the resistance R
of the secondary.
Maxell’s Equations
 Maxwell’s equations
  Qencl
Gauss’s law
 E  dA 
Gauss’s law for
magnetism
Farady’s law
Ampere’s law
 
 B  dA  0
0
 
d B
d  
 E  ds   dt   dt  B  dA
 
d E
d  
 B  ds  0 ( I   0 dt )encl  0 ( I   0 dt  E  dA)encl
EM Waves by an Antenna
 Oscillating electric dipole
First consider static electric field produced by
an electric dipole as shown in Figs.
(a) Positive (negative) charge at the top (bottom)
(b) Negative (positive) charge at the top (bottom)
Now then imagine these two charge are moving
up and down and exchange their position at every
half-period. Then between the two cases there is
a situation like as shown in Fig. below:
What is the electric filed
in the blank area?
EM Waves by an Antenna
 Oscillating electric dipole (cont’d)
Since we don’t assume that change propagate instantly once new position
is reached the blank represents what has to happen to the fields in meantime.
We learned that E field lines can’t cross and they need to be continuous except
at charges. Therefore a plausible guess is as shown in the right figure.
EM Waves by an Antenna
 Oscillating electric dipole (cont’d)
What actually happens to the fields based on a precise calculate is shown in
Fig. Magnetic fields are also formed. When there is electric current, magnetic
field is produced. If the current is in a straight wire circular magnetic field is
generated. Its magnitude is inversely proportional to the distance from the
current.
EM Waves by an Antenna
 Oscillating electric dipole (cont’d)
What actually happens to the fields based on a precise calculate is shown in Fig.
E
B
B is perpendicular to E
EM Waves by an Antenna
 Oscillating electric dipole (cont’d)
This is an animation of radiation of EM wave by an oscillating electric dipole
as a function of time.
EM Waves by an Antenna
 Oscillating electric dipole (cont’d)
A qualitative summary of the observation of this example is:
1) The E and B fields are always at right angles to each other.
2) The propagation of the fields, i.e., their direction of travel away from the
oscillating dipole, is perpendicular to the direction in which the fields
point at any given position in space.
3) In a location far from the dipole, the electric field appears to form closed
loops which are not connected to either charge. This is, of course, always
true for any B field. Thus, far from the dipole, we find that the E and B
fields are traveling independent of the charges. They propagate away from
the dipole and spread out through space.
In general it can be proved that accelerating electric charges give rise to
electromagnetic waves.
EM Waves by an Antenna
 Dipole antenna
V(t)=Vocos(t)
• time t=0
+
+
-
B
-
• time t=/
one half cycle later
X
B
+
+
At a location far away from the source of the EM wave, the wave
becomes plane wave.
EM Waves by an Antenna
 Dipole antenna (cont’d)
+
+
-
x
z
y
Properties of EM Waves
 Plane EM wave
y
x
z
 Speed of light and EM wave in vacuum
c
E
c
B
1
0 0
 2.99792 10 m/s
8
Speed of light
Light is an EM wave!
0  4 10 7 N  s 2 / C 2
 0  8.85419 10 12 C 2 /( N 2 m 2 )
Properties of EM Waves
 EM wave in matter
Maxwell’s equations for inside matter change from those in vacuum
by change 0 and 0 to  = km0 and   k0:

1


1
0 0 k mk
c

k mk
For most of dielectrics the relative permeability km is close to 1 except for
insulating ferromagnetic materials :

c

1


1
0 0 k mk
 n  k mk  k

c
k mk
Index of refraction
Properties of EM Waves
 Intensity of EM wave (average power per unit area)
EM waves carry energy.
I
Emax Bmax
20
intensity of the EM wave
Emax  cBmax  Bmax / 0 0
2
Emax
c 2
I

Bmax
20c 20
Properties of EM Waves
 Momentum carried by EM wave
Momentum carried by an EM wave:
p
U
c
Momentum transferred to an area :
if the wave is completely absorbed : p  U / c
if the wave is completely reflected : p  2U / c
p=mv-(-mv)=2mv
Measurement of radiation pressure:
Spectrum of EM Waves
 Spectrum of EM waves
c  f
c : speed of light in vacuum
f : frequency
 : wavelength
Doppler Effect for EM Waves
 Doppler effect
 u
f O  f S 1   if u  c
 c
u : relative speed of the observer
with respect to the source
c : speed of light in vacuum
fo : observed frequency, fS : emitted frequency
+ if the source and the observe are approaching each other
- if the source and the observer are receding each other
A globular cluster
receding
approaching