Physics 272
November 13
Fall 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Fall 14 - von Doetinchem - 44
The L-R-C series circuit
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LC circuit is an idealization
of the real world
Every real circuit has a
non-zero resistance value
Resistance in a circuit can
be regarded to in a similar
way as friction in a mechanical setup
In comparison to LC circuit: inductor stores less
energy than initially stored in capacitor due to i2R
losses in resistor
Phys272 - Fall 14 - von Doetinchem - 45
The L-R-C series circuit
http://www.falstad.com/circuit/e-lrc.html
Phys272 - Fall 14 - von Doetinchem - 46
Analyzing an L-R-C series circuit
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LRC circuit is oscillating (underdamped case):
Phys272 - Fall 14 - von Doetinchem - 47
Analyzing an L-R-C series circuit
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LRC circuit is oscillating (underdamped case):
exponential
envelope
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When R becomes too large
→ system no longer oscillates
→ starts when value under the square root
becomes negative
Phys272 - Fall 14 - von Doetinchem - 48
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 - Fall 14 - von Doetinchem - 50
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 - Fall 14 - von Doetinchem - 51
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
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Current and voltage amplitudes are
related in the same way as in a DC
circuit
Phys272 - Fall 14 - von Doetinchem - 52
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 - Fall 14 - von Doetinchem - 53
Inductor in an AC circuit
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Voltage peaks occur a quarter cycle
earlier
→ voltage leads the current by
90deg
Inductive reactance:
Be careful: current and voltage
are out of phase
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 give only small currents compared to
lower-frequency voltages
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This can be used to block high frequency noise
Phys272 - Fall 14 - von Doetinchem - 54
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 - Fall 14 - von Doetinchem - 55
Capacitor in an AC circuit
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Voltage lags the current by 90deg
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Capacitive reactance:
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Current has greatest magnitude
when the voltage is rising or falling
most steeply
Also here: voltage and current are
out of phase
With smaller frequency the capacitive reactance
becomes higher
Capacitors tend to pass high frequency current and
to block low frequencies (opposite to inductors)
Phys272 - Fall 14 - von Doetinchem - 56
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 - Fall 14 - von Doetinchem - 57
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 - Fall 14 - von Doetinchem - 58
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
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Resistance does not depend on the frequency
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Inductive and capacitive reactances depend on frequency
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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
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voltage across capacitor becomes zero (no charge build up)
Phys272 - Fall 14 - von Doetinchem - 59
Additional Material
Phys272 - Fall 14 - von Doetinchem - 63
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 - Fall 14 - von Doetinchem - 64
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 - Fall 14 - von Doetinchem - 65
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 - Fall 14 - von Doetinchem - 66
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 - Fall 14 - von Doetinchem - 67
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 - Fall 14 - von Doetinchem - 68
L-C in parallel
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Right after closing the
switch:
Phys272 - Fall 14 - von Doetinchem - 69
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 - Fall 14 - von Doetinchem - 70
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 - Fall 14 - von Doetinchem - 71
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 - Fall 14 - von Doetinchem - 72
Crossover network for loudspeaker
Phys272 - Fall 14 - von Doetinchem - 73