alternating currents

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alternating currents electromagnetic waves
PHY232 Spring 2007 Jon Pumplin http//www.pa.msu
.edu/pumplin/PHY232 (Ppt courtesy of Remco
Zegers)
2
Question
I

• At t0, the switch is closed. After that
• a) the current slowly increases from I 0 to
  I V/R
• b) the current slowly decreases from I V/R
  to I 0
• c) the current is a constant I V/R

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Answer
I

• At t0, the switch is closed. After that
• a) the current slowly increases from I0 to
  IV/R
• b) the current slowly decreases from IV/R to
  I0
• c) the current is a constant IV/R

4
Alternating current circuits
R
R
I
I
V
V

• Previously, we look at DC circuits the voltage
  delivered by the source is constant, as on the
  left.
• Now, we look at AC circuits, in which case the
  source is sinusoidal. A
  is used in circuits to denote this.

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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) V0
  sin?t V0 sin(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

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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
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power consumption in an AC circuit

• We already know for DC
• P V I V2/R I2 R
• For AC circuits with a single resistor
• P(t) V(t) I(t) V0 I0 (sin?t)2
• Average power consumption
• Pave Vrms Irms V2rms/R I2rms R
• where
• Vrms Vmax/?2)
• Irms Imax/?2

Vrms
IR(A) VR(V)
Irms
P(W)
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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).
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AC circuit with a single capacitor
C
I
I (A)
V(t)V0sin?t
Vc V0sin?t Qc CVc C V0 sin?t Ic ?Qc/?t
? C V0 cos?t So, the current peaks ahead 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.
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Capacitive circuit - continued
C
I
I (A)
V(t) V0 sin?t
Note Imax ? C V0 For a resistor we have I
V0/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 like a resistance in this circuit.
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Power consumption in a capacitive circuit
There is no power consumption in a purely
capacitive circuit Energy (1/2 C V2) gets
stored when the (absolute) voltage over
the capacitor is increasing, and released when it
is decreasing.
Pave 0 for a purely capacitive circuit
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AC circuit with a single inductor
L
I
I (A)
V(t) V0 sin?t
VL V0 sin?t L ?I/?t I -(V0/(?L)) cos?t (no
proof here you need calculus) 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.
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Inductive circuit - continued
L
IL(A)
I
I(A)
V(t) V0 sin?t
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
reactance Units of XL are Ohms. The inductive
reactance acts as a resistance in this circuit.
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Power consumption in an inductive circuit
There is no power consumption in a purely
inductive circuit Energy (1/2 L I2) gets stored
when the (absolute) current through the inductor
is increasing, and released when it is decreasing.
Pave 0 for a purely inductive circuit
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Reactance

• The inductive reactance (and capacitive
  reactance) are like the resistance of a normal
  resistor, in that you can calculate the current,
  given the voltage, using I V/XL (or I V/XC
  ).
• This works for the Maximum values, or for the
  RMS average values.
• But I and V are out of phase, so the maxima
  occur at different times.

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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 I I0 sin(?t-?)
• 2) The voltage over all three components is equal
  to the source voltage at any point in time V(t)
  V0 sin(?t)

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An LRC circuit
L
C
R
I
VR
VC
I
VL
V(t)V0sin?t

• For the resistor VR I R and VR and I are in
  phase
• For the capacitor Vc I Xc (Vc lags I by
  900)
• For the inductor VL I XL (VL leads I by 900)
• at any instant VLVcVRV0 sin(?t). But the
  maximum values of VLVcVR do NOT add up to V0
  because they have their maxima at different
  times.

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impedance
L
C
R
I
V(t)V0sin?t

• Define X XL-Xc reactance of RLC circuit
• Define Z ?R2(XL-Xc)2 ?R2X2 impedance
  of RLC cir
• Then Vtot I Z looks like Ohms law!

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Resonance

• 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

In this case, XL
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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 I2rms R

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

• questions
• what is the angular frequency of the system?what
  are the inductive and capacitive reactances?
• what is the impedance, what is the phase angle ?
• what is the maximum current and peak voltages
  over each element
• compare the algebraic sum of peak voltages with
  V0. Does this make sense?
• what are the instantaneous voltages and rms
  voltages over each element?
• what is power consumed by each element and total
  power consumption

V(t)V0sin?t
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answers
Given R250 Ohm L0.6 H C3.5 ?F f60 Hz V0150 V

• a) angular frequency ? of the system?
• ?2?f2?60377 rad/s
• b) Reactances?
• XC1/?C1/(377 x 3.5x10-6)758 Ohm
• XL ?L377x0.6226 Ohm
• c) Impedance and phase angle
• Z?R2(XL-Xc)2?2502(226-758)2588 Ohm
• ?tan-1(XL-XC)/R)tan-1(226-758)/250-64.80
  (or 1.13 rad)
• d) Maximum current and maximum component
  voltages
• ImaxVmax/Z150/5880.255 A
• VRImaxR0.255x25063.8 V
• VCImaxXC0.255x758193 V
• VLImaxXL0.255x26657.6 V
• Sum VRVCVL314 V. This is larger than the
  maximum voltage delivered by the source (150 V).
  This makes sense because the relevant sum is not
  algebraic each of the voltages are vectors with
  different phases.

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answers

• ImaxVmax/Z0.255 A
• VRImaxR63.8 V
• VCImaxXC193 V
• VLImaxXL57.6 V
• ?-64.80 (or 1.13 rad)
• Vtot150 V

• f) instantaneous voltages over each element (Vtot
  has 0 phase)?
• start with the driving voltage VV0sin?tVtot
• VR(t)63.8sin(?t1.13) (note the phase relative
  to Vtot)
• VC(t)193sin(?t-0.44) phase angle
  1.13-?/2-0.44
• VL(t)57.6sin(?t2.7) phase angle 1.13?/22.7
  
• rms voltages over each element?
• VR,rms63.8/?245.1 V
• VC,rms193/?2136 V
• VL,rms57.6/?240.7 V

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answers

• g) power consumed by each element and total power
  consumed?
• PCPL0 no energy is consumed by the capacitor or
  inductor
• PRIrms2R(Imax/?2)2R0.2552R/20.2552250/2)8.13
  W
• or PRVrms2/R(45.1)2/2508.13 W (dont use
  VrmsV0/?2!!)
• or PRVrmsIrmscos?(150/?2)(0.255/?2)cos(-64.80)
  8.13 W
• total power consumedpower consumed by resistor!

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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 VtotI Z
• 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)

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Question

• The sum of maximum voltages over the resistor,
  capacitor and inductor in an LRC circuit cannot
  be higher than the maximum voltage delivered by
  the source since it violates Kirchhoffs 2nd rule
  (sum of voltage gains equals the sum of voltage
  drops).
• a) true
• b) false

answer false The maximum voltages in each
component are not achieved at the same time!
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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 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

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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
? ?(1/LC)690 rad/s f ?/2?110 Hz
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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

answer c) the resonance frequency only depends on
L and C
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transformers
transformers are used to convert voltages to
lower/higher levels
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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
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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?

1. Vs(Ns/Np)Vp or Ns(Vs/Vp)Np 120 windings
2. VpIpVsIs so IsVpIp/Vs8.33 A

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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

answer high voltage If the voltage is high, the
current is low If the current is low, the voltage
drop over the power line (with resistance R) is
low, and thus the power dissipated in the line
(?V2/RI2R) also low
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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

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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)
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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 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
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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

dipole antenna
I
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producing the electric field wave
antenna
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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
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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?

1. yes
2. no

answer no there is no varying current and hence
no wave can be made.
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cf ?