The LRC series circuit

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Physics 272
April 9
Spring 2015
www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm
go.hawaii.edu/KO
Prof. Philip von Doetinchem
[email protected]
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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
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The L-R-C series circuit
http://www.falstad.com/circuit/e-lrc.html
voltage is shown in green, current in yellow
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Analyzing an L-R-C series circuit
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LRC circuit is oscillating (underdamped case):
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Analyzing an L-R-C series circuit
exponential
envelope
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LRC circuit is
oscillating
(underdamped case):
When R becomes too
large
→ system no longer
oscillates
→ starts when value
under the square root
becomes negative
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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
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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:
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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
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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
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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
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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
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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)
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Crossover network for loudspeaker
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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:
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A resistor and a capacitor in an AC circuit
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200 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
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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)
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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
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The L-R-C series circuit
Addition of amplitude voltages for alternating current:
VR, VL, VC, and V are the amplitude voltages in this case
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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/amplitude current
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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
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The meaning of impedance and phase angle
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No L:
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No C:
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No R:
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No L, C:
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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
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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
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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:
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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
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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
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Additional Material
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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
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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
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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:
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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:
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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
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L-C in parallel
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Right after closing the
switch:
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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):
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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):
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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:
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