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alternating currents & electromagnetic waves
PHY232Remco Zegerszegers@nscl.msu.eduRoom W109 – cyclotron buildinghttp://www.nscl.msu.edu/~zegers/phy232.html
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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Alternating current circuits
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
I
R
V
I
V
R
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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A circuit with a resistor
The voltage over the resistor is the same as the voltage delivered by the source: VR(t)=V0sint=V0sin(2ft)
The current through the resistor is: IR(t)= V0/R sint 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; =2f f: frequency (Hz)
SET YOUR CALCULATOR TO RADIANS WHERE NECESSARY
I
V(t)=V0sint
R
I R(A
)
V0=10 VR=2 Ohm=1 rad/s
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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lon-capa
you should now do problem 1 from set 7.
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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rms currents/voltages To understand energy
consumption by the circuit, it doesn’t matter what the sign of the current/voltage is. We need the absolute average currents and voltages (root-mean-square values) :
Vrms=Vmax/2
Irms=Imax/2
The following hold: Vrms=IrmsR
Vmax=ImaxR
I R(A
)|I
R|(A
) |V
R|(
V)
Vrms
Irms
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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power consumption by an AC circuit
We already saw (DC): P=VI=V2/R=I2R
For AC circuits with a single resistor:
P(t)=V(t)*I(t)=V0I0sin2t
The average power consumption:
Pave=Vrms*Irms=V2rms/R=I2
rmsR
Pave=(Vmax/2)( Imax/2)= ImaxVmax/2
|IR
|(A
) |V
R|(
V)
Vrms
Irms
P(W
)
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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vector representation
time (s)
V0
-V0
V
The voltage or current as a function of time can bedescribed by the projection of a vector rotating with constant angular velocity on one of the axes (x or y).
=t
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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phasorsI R
(A)
I(t) V(t) 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.
I(t)=V(t)=0
t
I(t)=5A V(t)=10 Vt V(t)=-10V I(t)=-5A t
I(t), V(t) are in phase, sopoint in the same direction
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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question
V(t) I(t) 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 resistancea) < 1 Ohmb) > 1 Ohmc) 1 Ohm
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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A circuit with a single capacitor
I
V(t)=V0sint
C
Vc= V0sintQc=CVc=CV0sintIc=Qc/t= CV0cost= CV0sin(t+/2)So, the current peaks ahead in time (earlier) of the voltageThere is a difference in phase of /2 (900)
I C(A
)
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 andthe current decreases. If the potential over C is maximum, the current is zero.
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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phasor diagram for capacitive circuit
I(t) V(t) t
=tI C(A
)
Note: Imax= CV0
For a resistor we have I=V0/R so ‘1/C’ is similar to ‘R’
And we write: I=V/Xc with Xc= 1/C the capacitive reactanceUnits of Xc are Ohms. The capacitive reactance acts as a resistancein this circuit.
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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 thecapacitor is increasing, and released when it is decreasing.
Pave = 0 for a purely capacitive circuit
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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question
I(t) V(t) t
=t
The angle between the current vector and voltagevector in a phasor diagram for a capacitive circuit isa) 00
b) 450
c) 900
d) 1800
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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A circuit with a single inductor
I
V(t)=V0sint
L
VL= V0sint=LI/tIL=-V0/(L)cost= V0 /(L )sin(t-/2)(no proof here: you need calculus…)So, the current peaks later in time than the voltageThere is a difference in phase of /2 (900)
I L(A
)
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.
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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phasor diagram for inductive circuit
I(t) V(t) t
=tI L(A
)
Note: Imax= V0/(L)
For a resistor we have I=V0/R so ‘L’ is similar to ‘R’
And we write: I=V/XL with XL= L the inductive reactanceUnits of XL are Ohms. The inductive reactance acts as a resistancein this circuit.
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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 theinductor is increasing, and released when it is decreasing.
Pave = 0 for a purely inductive circuit
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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 I=V/XL.
a) True b) False
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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Combining the three: the LRC circuit
Things to keep in mind when analyzing this system:
1) The current in the system has the same value everywhere I=I0sin(t-)
2) The voltage over all three components is equal to the source voltage at any point in time: V(t)=V0sin(t)
I
V(t)=V0sint
L C R
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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An LRC circuit
For the resistor: VR=IRR and VR and IR=I are in phase
For the capacitor: Vc=IXc and Vc lags Ic=I by 900
For the inductor: VL=IXL and VL leads IL=I by 900
at any instant: VL+Vc+VR=V0sin(t), that is the total voltage Vtot is the vector addition of the three individual components
VR
I
VC
VLt
VC
I
VR
VL
Vtot
=t
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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impedance
Vtot= VL+Vc+VR (vectors) Vtot=[VR
2+(|VL|-|VC|)2]=
[ (IR)2+(IXL-IXC)2]=I[R2+(XL-Xc)2] define X=XL-Xc : reactance of an RLC circuit define Z=[R2+(XL-Xc)2]= [R2+X2] : impedance of RLC
circ. Vtot=IZ & I=Vtot/Z looks like Ohms law
t
VC
VR
VL
t
VC
VR
VL
Vtot
vector sum VL+VC
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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phase angle
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| )/VR=X/R
t
VC
I
VR
VL
Vtot
=t
t
VC
VR
VL
Vtot
vector sum VL+VC
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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.
P=I2rmsR=IrmsVR
VR=Vrmscos (since cos=VR/Vtot)
So: P=VrmsIrmscos
cos: power factor of a circuitt
VC
VR
VL
Vtot
VL+VC
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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lon-capa
you should now do problem 4 from LON-CAPA 7
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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example
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
I
V(t)=V0sint
L C R
Given:R=250 OhmL=0.6 HC=3.5 Ff=60 HzV0=150 V
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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lon-capa
you should now try problem 6 of lon-capa set 7, except for the last part
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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LRC circuits: an overview
Reactance of capacitor: Xc= 1/C
Reactance of inductor: XL= L
Current through circuit: same for all components ‘Ohms’ law for LRC circuit: Vtot=IZ
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): P=I2rmsR=IrmsVR
P=VrmsIrmscos VR=ImaxR in phase with current I, out of phase by with Vtot
VC=ImaxXC behind by 900 relative to I (and VR)
VL=ImaxXL ahead of 900 relative to I (and VR)
t
VC
I
VR
VL
Vtot
=t
vector sum VL+VC
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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 I=Vtot/Z, the current through the circuit changes
If XL=XC (I.e. 1/C= L or 2=1/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
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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example
Given:R=250 OhmL=0.6 HC=3.5 Ff=60 HzV0=150 V
Using the same given parameters as the earlier problem,what is the resonance frequency?
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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question
An LRC circuit has R=50 Ohm, L=0.5 H and C=5x10-
3 F. An AC source with Vmax=50V is used. If the resistance is replaced with one that has R=100 Ohm and the Vmax of the source is increased to 100V, the resonance frequency will:
a) increase b)decrease c) remain the same
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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loncapa
You should now try question 6, part 7 and question 5 of lon-capa set 7.
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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transformers
transformers are used to convertvoltages to lower/higher levels
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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transformers
Vp Vs
primary circuitwith Np loops incoil
secondary circuit with Ns loops in coil
iron core
If an AC current is applied to the primary circuit: Vp=-NpB/tThe magnetic flux is contained in the iron and the changing flux actsin the secondary coil also: Vs=-NsB/t
Therefore: Vs=(Ns/Np)Vp if Ns<Np then Vs<Vp
A perfect transformer is a pure inductor (no resistance), so no power loss: Pp=PS and VpIp=VsIs ; if Ns<Np then Vs<Vp and IS>Ip
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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question
a transformer is used to bring down the high-voltage deliveredby 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?
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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electromagnetic waves
James Maxwell formalized the basic equations governing electricity and magnetism ~1870:Coulomb’s lawMagnetic forceAmpere’s Law (electric currents make magnetic
fields)Faraday’s 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
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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electromagnetic waves
Maxwell found that electric and magnetic waves traveltogether through space with a velocity of 1/(00)v=1/(00)=1/(4x10-7 x 8.85x10-12)=2.998x108 m/s which is just the speed of light (c)
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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electromagnetic waves can be used to broadcast…
Consider the experiment performed by Herz (1888)
I
Herz made an RLC circuit with L=2.5 nH, C=1.0nFThe resonance frequency is = (1/LC)=6.32x108 rad/sf= /2=100 MHz. Recall that the wavelength of waves =v/f=c/f=3x108/100x106=3.0 m
wavelength: =v/f
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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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
I
dipole antenna
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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producing the electric field wave
antenna
++
++
++
----
----
--+
++
++
+--
----
----
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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producing the magnetic field wave
antenna
++
++
++
----
----
--
I
I
++
++
++
----
----
--
I
I
E and B are in phaseand E=cB withc: speed of lightThe power/m2=0.5EmaxBmax/0
The energy in the wave isshared between the E-field and the B-field
PHY232 - Remco Zegers - alternating currents and electromagnetic waves
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question
Can a single wire connected to the + and – poles of a DC battery act as a transmitter of electromagnetic waves?
a) yesb) no
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