alternating currents

1
alternating currents electromagnetic waves

• PHY232
• Remco Zegers
• zegers_at_nscl.msu.edu
• Room W109 cyclotron building
• http//www.nscl.msu.edu/zegers/phy232.html

2
Alternating current circuits
R
R
I
I
V
V

• previously, we look at DC circuits (the voltage
  delivered by the source was constant).
• Now, we look at AC circuits, in which case the
  source is sinusoidal. A
  is used in circuits to denote the difference

3
A circuit with a resistor
R
IR(A)
I
V010 V R2 Ohm ?1 rad/s
V(t)V0sin?t

• The voltage over the resistor is the same as the
  voltage delivered by the source
  VR(t)V0sin?tV0sin(2?ft)
• The current through the resistor is IR(t) V0/R
  sin?t
• Since V(t) and I(t) have the same behavior as a
  function of time, they are said to be in phase.
• V0 is the maximum voltage
• V(t) is the instantaneous voltage
• ? is the angular frequency ?2?f f frequency
  (Hz)
• SET YOUR CALCULATOR TO RADIANS WHERE NECESSARY

4
lon-capa

• you should now do problem 1 from set 7.

5
rms currents/voltages

• To understand energy consumption by the circuit,
  it doesnt matter what the sign of the
  current/voltage is. We need the absolute average
  currents and voltages (root-mean-square values)
• VrmsVmax/?2
• IrmsImax/?2
• The following hold
• VrmsIrmsR
• VmaxImaxR

IR(A)
Vrms
IR(A) VR(V)
Irms
6
power consumption by an AC circuit

• We already saw (DC)
• PVIV2/RI2R
• For AC circuits with a single resistor
• P(t)V(t)I(t)V0I0sin2?t
• The average power consumption
• PaveVrmsIrmsV2rms/RI2rmsR
• Pave(Vmax/?2)( Imax/?2) ImaxVmax/2

Vrms
IR(A) VR(V)
Irms
P(W)
7
vector representation
V0
V
??t
time (s)
-V0
The voltage or current as a function of time can
be described by the projection of a vector
rotating with constant angular velocity on one
of the axes (x or y).
8
phasors
I(t), V(t) are in phase, so point in the same
direction
IR(A)
??t
I(t) V(t)
t
The instantaneous current and voltage over R are
the projections on the t-axis (horizontal axis)
of vectors rotating with ang. frequency ?. The
length of the vectors indicate the maximum
current or voltage.
9
question
t
V(t) I(t)

• Given a phasor diagram for a single resistor in
  circuit.
• If the voltage scale is V and current scale
  Ampere,
• then the resistor has a resistance
• lt 1 Ohm
• gt 1 Ohm
• 1 Ohm

10
A circuit with a single capacitor
C
I
IC(A)
V(t)V0sin?t
Vc V0sin?t QcCVcCV0sin?t Ic?Qc/?t ?CV0cos?t
?CV0sin(?t?/2) So, the current peaks ahead in
time (earlier) of the voltage There is a
difference in phase of ?/2 (900)
why? When there is not much charge on the
capacitor it readily accepts more and current
easily flows. However, the E-field and potential
between the plates increase and consequently it
becomes more difficult for current to flow
and the current decreases. If the potential over
C is maximum, the current is zero.
11
phasor diagram for capacitive circuit
IC(A)
??t
I(t) V(t)
t
Note Imax ?CV0 For a resistor we have IV0/R
so 1/?C is similar to R And we write
IV/Xc with Xc 1/?C the capacitive
reactance Units of Xc are Ohms. The capacitive
reactance acts as a resistance in this circuit.
12
power consumption in a capacitive circuit
There is no power consumption in a purely
capacitive circuit Energy (1/2CV2) gets stored
when the (absolute) voltage over the capacitor is
increasing, and released when it is decreasing.
Pave 0 for a purely capacitive circuit
13
question
?
??t
I(t) V(t)
t

• The angle ? between the current vector and
  voltage
• vector in a phasor diagram for a capacitive
  circuit is
• 00
• 450
• 900
• 1800

14
A circuit with a single inductor
L
IL(A)
I
V(t)V0sin?t
VL V0sin?tL?I/?t IL-V0/(?L)cos?t V0 /(?L
)sin(?t-?/2) (no proof here you need
calculus) So, the current peaks later in time
than the voltage There is a difference in phase
of ?/2 (900)
why? As the potential over the inductor rises,
the magnetic flux produces a current that
opposes the original current. The voltage across
the inductor peaks when the current is just
beginning to rise, due to this tug of war.
15
phasor diagram for inductive circuit
IL(A)
??t
t
I(t) V(t)
Note Imax V0/(?L) For a resistor we have
IV0/R so ?L is similar to R And we write
IV/XL with XL ?L the inductive reactance Units
of XL are Ohms. The inductive reactance acts as
a resistance in this circuit.
16
power consumption in an inductive circuit
There is no power consumption in a purely
inductive circuit Energy (1/2LI2) gets stored
when the (absolute) current through the inductor
is increasing, and released when it is decreasing.
Pave 0 for a purely inductive circuit
17
question

• The inductive reactance (and capacitive
  reactance) are just like the resistance of a
  normal resistor, I.e. if I know the inductive
  reactance, I can calculate the current at any
  time given the voltage using IV/XL.
• a) True
• b) False

18
Combining the three the LRC circuit
L
C
R
I
V(t)V0sin?t

• Things to keep in mind when analyzing this
  system
• 1) The current in the system has the same value
  everywhere II0sin(?t-?)
• 2) The voltage over all three components is equal
  to the source voltage at any point in time
  V(t)V0sin(?t)

19
An LRC circuit
VR
I
I
?
Vtot
VR
VL
VC
??t
t
VL
VC

• For the resistor VRIRR and VR and IRI are in
  phase
• For the capacitor VcIXc and Vc lags IcI by 900
• For the inductor VLIXL and VL leads ILI by 900
• at any instant VLVcVRV0sin(?t), that is the
  total voltage Vtot is the vector addition of the
  three individual components

20
impedance
VR
VR
Vtot
VL
VL
t
t
vector sum VLVC
VC
VC

• Vtot VLVcVR (vectors)
• Vtot?VR2(VL-VC)2
• ? (IR)2(IXL-IXC)2I?R2(XL-Xc)2
• define XXL-Xc reactance of an RLC circuit
• define Z?R2(XL-Xc)2 ?R2X2 impedance of
  RLC circ.
• VtotIZ IVtot/Z looks like Ohms law

21
phase angle
VR
VR
I
?
Vtot
?
Vtot
VL
VL
??t
t
t
vector sum VLVC
VC
VC

• The current I and the voltage Vtot are out of
  phase by an angle ?. This angle can be calculated
  with
• tan?opposite/adjacent(VL -Vc )/VRX/R

22
question

• If the maximum voltage over the capacitor equals
  the maximum voltage over the inductor, the
  difference in phase between the voltage over the
  whole circuit and the voltage over the resistor
  is
• a) 00
• b)450
• c)900
• d)1800

23
power consumption by an LRC circuit

• Even though the capacitor and inductor do not
  consume energy on the average, they affect the
  power consumption since the phase between current
  and voltage is modified.
• PI2rmsRIrmsVR
• VRVrmscos? (since cos?VR/Vtot)
• So PVrmsIrmscos?
• cos? power factor of a circuit

VR
?
Vtot
VL
t
VLVC
VC
24
lon-capa

• you should now do problem 4 from LON-CAPA 7

25
example
Given R250 Ohm L0.6 H C3.5 ?F f60 Hz V0150 V
L
C
R
I

• questions
• a) what is the angular frequency of the system?
• b) what are the inductive and capacitive
  reactances?
• c) what is the impedance, what is the phase angle
  ?
• d) what is the maximum current and peak voltages
  over each element
• Compare the algebraic sum of peak voltages with
  V0. Does this make sense?
• e) make the phasor diagram. Include
  I,VL,VC,VR,Vtot, ?. Assume VR is in the first
  quadrant.
• f) what are the instantaneous voltages and rms
  voltages over each element. Consider Vtot to have
  zero phase.
• g) power consumed by each element and total power
  consumption

V(t)V0sin?t
26
lon-capa

• you should now try problem 6 of lon-capa set 7,
  except for the last part

27
LRC circuits an overview
VR
I
?
Vtot

• Reactance of capacitor Xc 1/?C
• Reactance of inductor XL ?L
• Current through circuit same for all components
• Ohms law for LRC circuit VtotIZ
• Impedance Z?R2(XL-Xc)2
• phase angle between current and source voltage
• tan?(VL -Vc )/VR(XL-Xc)/R
• Power consumed (by resistor only)
  PI2rmsRIrmsVR
• PVrmsIrmscos?
• VRImaxR in phase with current I, out of phase
  by ? with Vtot
• VCImaxXC behind by 900 relative to I (and VR)
• VLImaxXL ahead of 900 relative to I (and VR)

VL
??t
t
vector sum VLVC
VC
28
Resonances in an RLC circuit

• If we chance the (angular) frequency the
  reactances will change since
• Reactance of capacitor Xc 1/?C
• Reactance of inductor XL ?L
• Consequently, the impedance Z?R2(XL-Xc)2
  changes
• Since IVtot/Z, the current through the circuit
  changes
• If XLXC (I.e. 1/?C ?L or ?21/LC), Z is
  minimal, I is maximum)
• ? ?(1/LC) is the resonance angular frequency
• At the resonance frequency ?0 (see question on
  slide 23)

Z
I
0
?
?
29
example
Using the same given parameters as the earlier
problem, what is the resonance frequency?
Given R250 Ohm L0.6 H C3.5 ?F f60 Hz V0150 V
30
question

• An LRC circuit has R50 Ohm, L0.5 H and C5x10-3
  F. An AC source with Vmax50V is used. If the
  resistance is replaced with one that has R100
  Ohm and the Vmax of the source is increased to
  100V, the resonance frequency will
• a) increase
• b)decrease
• c) remain the same

31
loncapa

• You should now try question 6, part 7 and
  question 5 of lon-capa set 7.

32
transformers
transformers are used to convert voltages to
lower/higher levels
33
transformers
primary circuit with Np loops in coil
secondary circuit with Ns loops in coil
Vp
Vs
iron core
If an AC current is applied to the primary
circuit Vp-Np??B/?t The magnetic flux is
contained in the iron and the changing flux
acts in the secondary coil also
Vs-Ns??B/?t Therefore Vs(Ns/Np)Vp if NsltNp
then VsltVp A perfect transformer is a pure
inductor (no resistance), so no power loss
PpPS and VpIpVsIs if NsltNp then VsltVp and
ISgtIp
34
question
a transformer is used to bring down the
high-voltage delivered by a powerline (10 kV) to
120 V. If the primary coil has 10000 windings,
a) how many are there in the secondary coil? b)
If the current in the powerline is 0.1 A, what is
the maximum current at 120 V?
35
question

• Is it more economical to transmit power from the
  power station to homes at high voltage or low
  voltage?
• a) high voltage
• b) low voltage

36
electromagnetic waves

• James Maxwell formalized the basic equations
  governing electricity and magnetism 1870
• Coulombs law
• Magnetic force
• Amperes Law (electric currents make magnetic
  fields)
• Faradays law (magnetic fields make electric
  currents)
• Since changing fields electric fields produce
  magnetic fields and vice versa, he concluded
• electricity and magnetism are two aspects of the
  same phenomenon. They are unified under one set
  of laws the laws of electromagnetism

37
electromagnetic waves
Maxwell found that electric and magnetic waves
travel together through space with a velocity of
1/?(?0?0) v1/?(?0?0)1/?(4?x10-7 x
8.85x10-12)2.998x108 m/s which is just the
speed of light (c)
38
electromagnetic waves can be used to broadcast

• Consider the experiment performed by Herz (1888)

I
Herz made an RLC circuit with L2.5 nH,
C1.0nF The resonance frequency is ?
?(1/LC)6.32x108 rad/s f ?/2?100 MHz. Recall
that the wavelength of waves ?v/fc/f3x108/100x1
063.0 m
wavelength ?v/f
39
He then constructed an antenna

• charges and currents vary sinusoidally in the
  primary and secondary circuits. The charges in
  the two branches also oscillate at the same
  frequency f

dipole antenna
I
40
producing the electric field wave
antenna
41
producing the magnetic field wave
E and B are in phase and EcB with c speed of
light
The power/m20.5EmaxBmax/?0
The energy in the wave is shared between the
E-field and the B-field
antenna
42
question
Can a single wire connected to the and poles
of a DC battery act as a transmitter of
electromagnetic waves?

1. yes
2. no

43
cf?
44
lon-capa

• now try questions 2 and 7 from set 7.