Physics 272
April 8
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Spring 14 - von Doetinchem - 218
L-C in parallel
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Follow a similar idea
like in the easy case
with L and C in series
→ use Kirchhoff's law
Phys272 - Spring 14 - von Doetinchem - 219
L-C in parallel
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We assume that the system is oscillating with a
special frequency
Differential equation systems with the oscillations
can be solved with an exponential approach
Phys272 - Spring 14 - von Doetinchem - 220
L-C in parallel
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Now we have system of non-linear equations
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Let's write it down in matrix form:
Phys272 - Spring 14 - von Doetinchem - 221
L-C in parallel
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No general solution exists, only for a special choice
of  when the determinant of the matrix is zero:
Phys272 - Spring 14 - von Doetinchem - 222
L-C in parallel
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If the term under the square root is smaller than
zero the system is oscillating
This is driven by the choice of L,R,C
Phys272 - Spring 14 - von Doetinchem - 223
L-C in parallel
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Right after closing the
switch:
Phys272 - Spring 14 - von Doetinchem - 224
L-C in parallel
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Use the following substitutions:
Technically any linear combination of sine and cosine
solutions to our problem are allowed before taking the initial
conditions into account
For charge in the capacitor we know that the charge is 0 after
closing the switch (t=0):
Phys272 - Spring 14 - von Doetinchem - 225
L-C in parallel
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Using our initial condition:
Use this result to calculate the current in the
inductor (Kirchhoff's loop rule):
Phys272 - Spring 14 - von Doetinchem - 226
Review
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Mutual inductance
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Self-inductance
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Magnetic field energy
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RL circuit
Phys272 - Spring 14 - von Doetinchem - 227
Review
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LC circuit
LRC circuit is a realistic approach where inductors
and wire have a non-zero resistance
The resistance in this case is similar to friction in
mechanics
Phys272 - Spring 14 - von Doetinchem - 228
Alternating current
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Electric power distribution uses alternating current
(AC)
Transformer can easily be used to step voltage up
and down
High voltages with low currents are used for longdistance power
transmission to keep
i2R losses small
Phys272 - Spring 14 - von Doetinchem - 230
Root-Mean-Square (rms) values
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Averaging a sinusoidal current is not very useful
→ average value is 0
Rectified average current is the average of the
absolute current |I cost|:
Another way of describing the alternating current is
the root-mean-square value:
Phys272 - Spring 14 - von Doetinchem - 231
Resistor in an AC circuit
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Current and voltage have both the same
dependence on cosine:
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When current is at maximum → voltage is at maximum
–
Current and voltage amplitudes are related in the same
way as in a DC circuit
Phys272 - Spring 14 - von Doetinchem - 232
Resistor in an AC circuit
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Current and voltage have both the same
dependence on cosine:
–
When current is at maximum → voltage is at maximum
–
Current and voltage amplitudes are related in the same
way as in a DC circuit
Phys272 - Spring 14 - von Doetinchem - 233
Inductor in an AC circuit
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Ideal inductor
with zero
resistance
Potential difference is not caused by dissipation of energy in
wire, but by self-induced emf
Voltage across the conductor is proportional to rate change
Phys272 - Spring 14 - von Doetinchem - 234
Inductor in an AC circuit
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Voltage peaks occur a quarter cycle earlier
→ voltage leads the current by 90deg
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Inductive reactance:
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Be careful: current and voltage are out of phase
Phys272 - Spring 14 - von Doetinchem - 235
The meaning of inductive reactance
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XL is description of the self-induced emf that
opposes any change in current through a conductor
More rapid variation in current increases inductive
reactance
High frequency voltages gives only small currents
compared to lower-frequency voltages
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This can be used to block high frequency noise
Phys272 - Spring 14 - von Doetinchem - 236
An inductor in an AC circuit
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Current amplitude in a pure inductor in a radio
receiver is 250A with voltage amplitude 3.6V at
frequency 1.6MHz
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What inductance is needed:
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Change of current with different frequencies:
Phys272 - Spring 14 - von Doetinchem - 237
Capacitor in an AC circuit
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Capacitor
constantly charges and discharges in AC circuit
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Current into one plate and equal current out of other plate
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Equal displacement current between plates
→ effectively we can say that alternating current is
going through the capacitor
Phys272 - Spring 14 - von Doetinchem - 238
Capacitor in an AC circuit
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Current has greatest magnitude when the voltage is rising or
falling most steeply
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Voltage lags the current by 90deg
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Capacitive reactance:
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Also here: voltage and current are out of phase
Phys272 - Spring 14 - von Doetinchem - 239
The meaning of capacitive reactance
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With smaller frequency the capacitive reactance
becomes higher
Capacitors tend to pass high frequency current and
to block low frequencies (opposite to inductors)
Phys272 - Spring 14 - von Doetinchem - 240
A resistor and a capacitor in an AC circuit
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200 resistor in series with a 5.0F capacitor
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Voltage across resistor is 1.2Vcos(2500Hz t)
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Current in circuit:
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Capacitive reactance:
Phys272 - Spring 14 - von Doetinchem - 241
A resistor and a capacitor in an AC circuit
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200W resistor in series with a 5.0mF capacitor
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Voltage across resistor is 1.2Vcos(2500Hz t)
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Voltage across capacitor
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Same current passes through resistor and capacitor, but
voltages are different in amplitude and phase
Phys272 - Spring 14 - von Doetinchem - 242
Comparing ac circuit elements
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Resistor shows no phase difference between
voltage and current
Inductors and capacitors have +/-90deg phase
differences between voltage and current
Resistance does not depend on the frequency
Inductive and capacitive reactances depend on
frequency
Phys272 - Spring 14 - von Doetinchem - 243
Comparing ac circuit elements
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For →0:
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alternating current case goes over into DC case:
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no current through capacitor
no inductive effect
For →∞:
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current in inductor goes to zero
–
voltage across capacitor becomes zero
(no charge build up)
Phys272 - Spring 14 - von Doetinchem - 244
Crossover network for loudspeaker
Phys272 - Spring 14 - von Doetinchem - 245
The L-R-C series circuit
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Instantaneous total voltage vad across all three
components is equal to the source voltage
Elements are connected in series
→ current at any instant is the same at every point
in the circuit
Phys272 - Spring 14 - von Doetinchem - 246
The L-R-C series circuit
Phys272 - Spring 14 - von Doetinchem - 247
The L-R-C series circuit
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For any network of resistors, inductors, capacitors:
impedance is defined as the ratio of:
amplitude voltage/current amplitude
Phys272 - Spring 14 - von Doetinchem - 248
The meaning of impedance and phase angle
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Impedance is similar to Ohm's law V=IR in DC
circuits
Impedance Z plays the role of R for AC circuit
Alternating currents tend to follow path of least
resistance
Important for sinusoidally varying voltages:
–
Relations also work when replacing maximum voltages
by RMS values
Phys272 - Spring 14 - von Doetinchem - 249
The meaning of impedance and phase angle
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Impedance depends on R, L, C and 
In addition to impedance the phase angle between
voltage and current is important
C
Phys272 - Spring 14 - von Doetinchem - 250
The meaning of impedance and phase angle
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L=0:
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C=0:
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R=0:
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L=0, C=0:
Phys272 - Spring 14 - von Doetinchem - 251
Example for an L-R-C circuit
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LRC series circuit with R=300, L=60mH, C=0.5F
sinusoidal voltage with amplitude voltage V=50V at
=10,000rad/s
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Impedance:
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Amplitude current and phase angle:
voltage leads current by 53deg
Phys272 - Spring 14 - von Doetinchem - 252
Example for an L-R-C circuit
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LRC series circuit with R=300, L=60mH, C=0.5F
sinusoidal voltage with amplitude voltage V=50V at
=10,000rad/s
Voltages across components:
The total voltage of 50V is
not equal to the scalar sum
of the individual voltages!
Vector sum has to be used
Phys272 - Spring 14 - von Doetinchem - 253
Power in alternating-current circuits
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Alternating currents are very important for
distributing and converting electric energy
For a particular moment in time, the power delivered
to a circuit element is:
Resistor:
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Voltage and current are in phase
→ energy is always supplied
(product of V and I always positive)
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Power is not constant, average power is:
Phys272 - Spring 14 - von Doetinchem - 254
Power in alternating-current circuits
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Power in an inductor:
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Voltage leads current by 90deg
→ average power supplied is zero (no energy transfer
over one cycle)
Power in a capacitor:
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Voltage lags the current by 90deg
→ average power supplied is zero
Phys272 - Spring 14 - von Doetinchem - 255
Power in a general AC circuit
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Power factor cos is important to determine how much current has
to be drawn for a given voltage difference
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Drawing more current is undesirable: i2R losses increase
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Ideal inductors and capacitors do not absorb net power from the line
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Lagging current can be corrected for with capacitors in parallel to
increase power factor
Phys272 - Spring 14 - von Doetinchem - 256