Lecture Outline

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LC Circuits
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Today...

• Oscillating voltage and current
• Transformers
• Qualitative descriptions
• LC circuits (ideal inductor)
• LC circuits (L with finite R)
• Quantitative descriptions
• LC circuits (ideal inductor)
• Frequency of oscillations
• Energy conservation

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Oscillating Current and Voltage
Q. What does mean??
A. It is an A.C. voltage source. Output voltage
appears at the terminals and is sinusoidal in
time with an angular frequency w.
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Transformers

• AC voltages can be stepped up or stepped down by
  the use of transformers.

• The AC current in the primary circuit creates a
  time-varying magnetic field in the iron

• This induces an emf on the secondary windings due
  to the mutual inductance of the two sets of
  coils.

• The iron is used to maximize the mutual
  inductance. We assume that the entire flux
  produced by each turn of the primary is trapped
  in the iron. (Recall from magnetism lab how the
  ferromagnet sucks in the B-field.)

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Ideal Transformers (no load)

• The primary circuit is just an AC voltage source
  in series with an inductor. The change in flux
  produced in each turn is given by

• Therefore,
• N2 gt N1 Þ secondary V2 is larger than primary V1
  (step-up)
• N1 gt N2 Þ secondary V2 is smaller than primary
  V1 (step-down)
• Note no load means no current in secondary.
  The primary current, termed the magnetizing
  current is small!

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Ideal Transformer (with a load)

• Changing flux produced by primary coil induces an
  emf in secondary. When we connect a resistive
  load to secondary coil, emf in secondary ?
  current I2 in secondary

• Power is dissipated only in the load resistor R.

Where did this power come from? It could come
only from the voltage source in the primary
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Preflight 18
An ideal transformer has N1 4, N2 6. Side 1
is connected to a generator with
e Vmax sin ( w t )
2) What is the maximum EMF on side 2?
a) V2max 2e1/3 b) V2max e1 c) V2max 3e1/2
1
2
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Lecture 18, ACT 1

• The primary coil of an ideal transformer is
  connected to an AC voltage source as shown. There
  are 50 turns in the primary and 200 turns in the
  secondary.
• If V1 120 V, what is the potential drop across
  the resistor R ?

• If 960 W are dissipated in the resistor R, what
  is the current in the primary ?

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Lecture 18, ACT 1

• The primary coil of an ideal transformer is
  connected to an AC voltage source as shown. There
  are 50 turns in the primary and 200 turns in the
  secondary.
• If V1 120 V, what is the potential drop across
  the resistor R ?

The ratio of turns, (N2/N1) (200/50) 4 The
ratio of secondary voltage to primary voltage is
equal to the ratio of turns, (V2/V1) (N2/N1)
Therefore, V2 480 V
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Lecture 18, ACT 1

• The primary coil of an ideal transformer is
  connected to an AC voltage source as shown. There
  are 50 turns in the primary and 200 turns in the
  secondary.
• If V1 120 V, what is the potential drop across
  the resistor R ?

The ratio of turns, (N2/N1) (200/50) 4

The ratio (V2/V1)
(N2/N1). Therefore, V2 480 V

• If 960 W are dissipated in the resistor R, what
  is the current in the primary ?

Energy is conserved960 W should be produced in
the primary P1 V1 I1 implies that 960W / 120V
8 A
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An ideal transformer steps down the voltage in
the secondary circuit. The number of loops on
each side is unknown.
4) Compare the currents in the primary and
secondary circuits

• I1 lt I2
• I1 I2
• I1 gt I2

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Whats Next?
Where are we going?

• Oscillating circuits
• radio, TV, cell phone, ultrasound, clocks,
  computers, GPS

• Why and how do oscillations occur in circuits
  containing capacitors and inductors?
• naturally occurring, not driven for now
• stored energy
• capacitive lt-gt inductive

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Energy in the Electric and Magnetic Fields
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LC Circuits

• Consider the RC and LC series circuits shown

• Suppose that the circuits are formed at t0 with
  the capacitor charged to value Q.

There is a qualitative difference in the time
development of the currents produced in these two
cases. Why??

• Consider from point of view of energy!
• In the RC circuit, any current developed will
  cause energy to be dissipated in the resistor.
• In the LC circuit, there is NO mechanism for
  energy dissipation energy can be stored both in
  the capacitor and the inductor!

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RC/LC Circuits
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LC Oscillations(qualitative)
Þ
ß
Ý
Ü
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Alternate way to draw
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LC Oscillations(qualitative)
These voltages are opposite, since the cap and
ind are traversed in opposite directions
dI
dt
0

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Lecture 18, Act 2

• At t0, the capacitor in the LC circuit shown has
  a total charge Q0. At t t1, the capacitor is
  uncharged.
• What is the value of VabVb-Va, the voltage
  across the inductor at time t1?

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Lecture 18, Act 2

• At t0, the capacitor in the LC circuit shown has
  a total charge Q0. At t t1, the capacitor is
  uncharged.
• What is the value of VabVb-Va, the voltage
  across the inductor at time t1?

• Vab is the voltage across the inductor, but it
  is also (minus) the voltage across the capacitor!
• Since the charge on the capacitor is zero, the
  voltage across the capacitor is zero!

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Lecture 18, Act 2

• At t0, the capacitor in the LC circuit shown has
  a total charge Q0. At t t1, the capacitor is
  uncharged.

• What is the relation between UL1, the energy
  stored in the inductor at tt1, and UC1, the
  energy stored in the capacitor at tt1?

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Preflight 18
At time t 0 the capacitor is fully charged with
Qmax, and the current through the circuit is 0.
2) What is the potential difference across the
inductor at t 0?
a) VL 0 b) VL Qmax/C
c) VL Qmax/2C
3) What is the potential difference across the
inductor when the current is maximum?
a) VL 0 b) VL Qmax/C
c) VL Qmax/2C
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LC Oscillations(L with finite R)

• If L has finite R, then
• energy will be dissipated in R ? the oscillations
  will be damped.

R ¹ 0
R 0

• The number of oscillations is described by the
  Q of the oscillator (we will return to this in
  Lect. 20) NOTE Q here is not charge!

Umax is max energy stored in the system DU is
the energy dissipated in one cycle
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Review of Voltage DropsAcross Circuit Elements
Voltage determined by integral of current and
capacitance
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LC Oscillations(quantitative, but only for R0)

• What is the oscillation frequency ?0?
• Begin with the loop rule

• Guess solution (just harmonic oscillator!)

• where f, Q0 determined from initial
  conditions

• Procedure differentiate above form for Q and
  substitute into loop equation to find w0.
• Note Dimensional analysis ?

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LC Oscillations(quantitative)
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Lecture 18, Act 3

• At t0 the capacitor has charge Q0 the resulting
  oscillations have frequency w0. The maximum
  current in the circuit during these oscillations
  has value I0.
• What is the relation between w0 and w2, the
  frequency of oscillations when the initial charge
  2Q0?

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Lecture 18, Act 3

• At t0 the capacitor has charge Q0 the resulting
  oscillations have frequency w0. The maximum
  current in the circuit during these oscillations
  has value I0.
• What is the relation between w0 and w2, the
  frequency of oscillations when the initial charge
  2Q0?

• Q0 determines the amplitude of the oscillations
  (initial condition)
• The frequency of the oscillations is determined
  by the circuit parameters (L, C), just as the
  frequency of oscillations of a mass on a spring
  was determined by the physical parameters (k, m)!

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Lecture 18, Act 3

• At t0 the capacitor has charge Q0 the resulting
  oscillations have frequency w0. The maximum
  current in the circuit during these oscillations
  has value I0.
• What is the relation between I0 and I2, the
  maximum current in the circuit when the initial
  charge 2Q0?

• The initial charge determines the total energy
  in the circuit
• U0 Q02/2C
• The maximum current occurs when Q0!
• At this time, all the energy is in the inductor
  U 1/2 LIo2
• Therefore, doubling the initial charge
  quadruples the total
• energy.
• To quadruple the total energy, the max current
  must double!

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Preflight 18
The current in a LC circuit is a sinusoidal
oscillation, with frequency ?.
5) If the inductance of the circuit is increased,
what will happen to the frequency ??
a) increase b) decrease
c) doesnt change
6) If the capacitance of the circuit is
increased, what will happen to the frequency?
a) increase b) decrease
c) doesnt change
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LC OscillationsEnergy Check

• The other unknowns ( Q0, f ) are found from the
  initial conditions. E.g., in our original
  example we assumed initial values for the charge
  (Qi) and current (0). For these values Q0 Qi,
  f 0.

• Question Does this solution conserve energy?

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Energy Check
U
E
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Lecture 18, Act 4

• At t0 the current flowing through the circuit is
  1/2 of its maximum value.
• Which of the following plots best represents UB,
  the energy stored in the inductor as a function
  of time?

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Lecture 18, Act 4

• At t0 the current flowing through the circuit is
  1/2 of its maximum value.
• Which of the following plots best represents UB,
  the energy stored in the inductor as a function
  of time?

• The key here is to realize that the energy
  stored in the inductor is proportional to the
  CURRENT SQUARED.
• Therefore, if the current at t0 is 1/2 its
  maximum value, the energy stored in the inductor
  will be 1/4 of its maximum value!!

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Summary

• Transformers used to step up/down voltage

• Oscillating voltage and current
• Qualitative description

• Quantitative description
• Frequency of oscillations
• Energy conservation

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Appendix LCR DampingFor your interest, we do
not derive here, but only illustrate the
following behavior
Q


C
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In an LRC circuit, w depends also on R !
Q
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