Physics 272
April 3
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 - 192
Summary
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Flux through coil 2 is directly proportional to current in coil 1 if
both coils are in vacuum
Mutual inductance only depends on geometry of coils and the
orientation of the coils to each other
Mutual inductance is also the same for the case when coil 2 carries
the current and induces a current in coil 1:
Careful:
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only time varying currents induce emf in second coil
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Induced emf is directly proportional to rate of change of the current
Phys272 - Spring 14 - von Doetinchem - 193
Summary
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Self-induction effect, elf-inductance L is defined as:
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current changes in circuit → magnetic flux changes
self-induced emf after applying Faraday's law
Phys272 - Spring 14 - von Doetinchem - 194
Summary
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Energy is stored in magnetic field in inductor like
energy is stored in the electric field of a capacitor
Magnetic energy density of a magnetic field:
Phys272 - Spring 14 - von Doetinchem - 195
The R-L circuit
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Self-induced emf prevents rapid changes in circuit
The greater the rate change of current
→ the greater the self-induced emf
Inductors in circuits:
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Use Kirchhoff's loop rule
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When we go through an inductor in the same direction as
the current → voltage drop (-Ldi/dt)
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When we go through an inductor in the opposite direction
to the current → voltage increase (Ldi/dt)
Phys272 - Spring 14 - von Doetinchem - 196
The R-L circuit
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Closed S1: current growth based
on inductor di/dt limits the rate
change
Closing S1 after some time time t:
Phys272 - Spring 14 - von Doetinchem - 197
The R-L circuit
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At the instant when S1 is closed:
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No current drop over resistor, i=0:
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The greater L, the slower the increase
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Current increases over time and
approaches final current → di/dt goes
to zero
Final current does not depend on
inductor
Phys272 - Spring 14 - von Doetinchem - 198
The R-L circuit
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Solve differential equation
Phys272 - Spring 14 - von Doetinchem - 199
The R-L circuit
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Solve differential equation
For long times → current change
gets slower
Phys272 - Spring 14 - von Doetinchem - 200
Current decay in an R-L circuit
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If S1 is closed for a while
→ current reached final value
→ then close S2 and S1 at the
same time
Current goes to zero following an
exponential decrease
Energy stored in inductor slows
down current decrease:
Phys272 - Spring 14 - von Doetinchem - 201
R-L circuit
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Time constant for R-L circuit:
small L current increases/decreases faster
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Compare to R-C circuit:
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Energy consideration:
Phys272 - Spring 14 - von Doetinchem - 202
Example
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Sensitive electronics with R=175 operates at 36mA
Critical: current should not rise above 4.9mA within the
first 58s after turn on
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Use an inductor in series with component
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Required emf:
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Calculate required L from current increase formula in R-L
circuit:
Phys272 - Spring 14 - von Doetinchem - 203
The L-C circuit
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Start with a charged capacitor
Capacitor discharges through
inductor
Capacitor potential equals
induced emf
Current increases
Increasing current in inductor establishes magnetic
field
→ energy stored in electric field of capacitor is
transferred into magnetic field in inductor
Once the capacitor is fully discharged the current in
the circuit does not stop instantly
Phys272 - Spring 14 - von Doetinchem - 207
The L-C circuit
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Instead it continues in the same
direction for a little bit
Capacitor is charging in opposite
direction → current drops
At the same time the magnetic
field in the inductor becomes
weaker
Inductor provides emf in direction of current and supports
charging of capacitor in opposite direction
Capacitor is fully charged in opposite direction
→ no current flow, no magnetic field in inductor, no
opposing emf
Process starts the other way around
→ Oscillation
Phys272 - Spring 14 - von Doetinchem - 208
Electrical osciallations in an L-C circuit
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Formal way:
This is the same type of equation describing
mechanical harmonic oscillations
Phys272 - Spring 14 - von Doetinchem - 209
Electrical osciallations in an L-C circuit
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Solution:
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Current in LC circuit:
Phys272 - Spring 14 - von Doetinchem - 210
Energy in an L-C circuit
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Energy conservation gives same result:
Phys272 - Spring 14 - von Doetinchem - 211
Example
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A 300V has charged a 25F capacitor connected to
a 10mH inductor
Capacitor charge after 1.2ms:
Phys272 - Spring 14 - von Doetinchem - 212
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 - Spring 14 - von Doetinchem - 215
Analyzing an L-R-C series circuit
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LRC circuit is oscillating (underdamped case):
Phys272 - Spring 14 - von Doetinchem - 216
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 - Spring 14 - von Doetinchem - 217
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 - 218