IV3 Energy of Magnetic Field

1
IV3 Energy of Magnetic Field
2
Main Topics

• Transformers
• Energy of Magnetic Field
• Energy Density of Magnetic Field
• An RC Circuit
• An RL Circuit
• An RLC Circuit - Oscilations

3
The Transformer I

• Transformer is a device with (usually) two or
  more coils sharing the same flux. The coil to
  which the input voltage is connected is called
  primary and the other(s) are secondary.
• Transformers are mostly used to convert voltages
  or to adjust (match) internal resistances.

4
The Transformer II

• Let us illustrate the principle of functioning of
  a transformer on a simple type with two coils
  with N1 and N2 loops. We shall further suppose
  that there is negligible current in the secondary
  coil.
• Since both lops share the same magnetic flux, in
  each loop of each coil the same EMF ?1 is
  induced
• ?1 - d?m1/dt

5
The Transformer III

• If we connect a voltage V1 to the primary coil,
  the magnetization in the core will grow until the
  counter EMF induced in this coil is equal to the
  input voltage
• V1 N1?1
• The voltage in the secondary coil is also
  proportional to its number of loops
• V2 N2?1

6
The Transformer IV

• So voltages in both coils are proportional to
  their number of loops
• V1/N1 V2/N2
• It is more difficult to understand the case when
  the secondary coil is loaded and, of course, even
  much more difficult to design a good transformer
  with high efficiency. In big transformers it can
  be close to 1!

7
The Transformer V

• Suppose, our transformer has efficiency close to
  1.
• In this case, currents are inversely proportional
  to the number of loops in each coil and
  resistances are proportional to their squares
• P V1I1 V2N1I1/N2 V2I2
• I1N1 I2N2
• R1/N12 R2/N22

8
Energy of Magnetic Field I

• An inductance opposes changes in current. That
  means it is necessary to do work to reach a
  certain current in a coil. This work is
  transferred into the potential energy of magnetic
  field and the field starts to return it at the
  moment we want to decrease the current.
• If a current I flows through a coil and we want
  to increase it, we have to deliver power
  proportional to the rate of the change we want to
  reach.

9
Energy of Magnetic Field II

• In other words we have to do work at a certain
  rate to move charges against the field of the
  induced EMF
• P I? ILdI/dt ?
• dW Pdt LIdI
• To find the work to reach current I, we
  integrate
• W LI2/2

10
Energy Density of Magnetic Field I

• Similarly as in the case of a charged capacitor
  the energy here is distributed in the field, now
  of course magnetic field.
• If the field is uniform, as in a solenoid, it is
  easy to find the density of energy
• We already know formulas for L and B
• L ?0N2A/l
• B ?0NI/l ? I Bl/?0N

11
Energy Density of Magnetic Field II

• Al is the inside volume of the solenoid, where we
  expect (most of) the field,
• can be attributed to the energy density of the
  magnetic field.
• This definition is valid in generally in every
  point of even non-uniform magnetic field.

12
RC, RL, LC and RLC Circuits

• Often not only static but also kinetic processes
  are important. So we have to find out how
  quantities depend on time when charging or
  discharging a capacitor or a coil.
• We shall se that circuits with LC will show a new
  effect un-dumped or dumped oscillations.

13
RC Circuits I

• Lets have a capacitor C charged to a voltage Vc0
  and at time t 0 we start to discharge it by a
  resistor R.
• At any instant the capacitor can be considered as
  a power source and the Kirchhoffs loop (or
  Ohms) law is valid
• I(t) Vc(t)/R
• This leads to differential equation.

14
RC Circuits II

• All quantities Q, V and I decrease exponentially
  with the time-constant ? RC
• Now, lets connect the same resistor and
  capacitor serially to a power supply V0. At any
  instant we find from the second Kirchhoffs law
• I(t)R Vc(t) V0
• a little more difficult differential equation.

15
RC Circuits III

• Now Q and V exponentially saturate while I
  exponentially decreases as in the previous case.
  The change of all quantities can be again
  described using the time-constant ? RC.

16
RL Circuits I

• A similar situation will be if we replace the
  capacitor in the previous circuit by a coil L.
• When the current grows the sign of the induced
  EMF on the coil will be the same as on the
  resistor and we can again use the second
  Kirchhofs law
• RI(t) LdI/dt V0
• This is again a similar differential equation.

17
RL Circuits II

• The coil refuses immediate growth of the current.
• I starts from zero and exponentially saturates.
• The EMF on the coil (VL) starts from its maximal
  value, equal to V0, and exponentially decreases.
  When the current becomes constant, the EMF on the
  coil disappears.

18
LC Circuits I

• Qualitatively new situation appears when we
  connect a charged capacitor C to a inductance L.
• It can be expected that now the energy will
  change from the electric form to the magnetic
  form and back. We obtain un-dumped periodic
  movement.

19
LC Circuits II

• This circuit is called an LC oscillator and it
  produces, so called, electromagnetic
  oscillations.
• We can use the second Kirchhoffs law
• L dI/dt Vc 0
• This is again a differential equation but of
  higher order.

20
LC Circuits III

• What happens qualitatively
• In the beginning the capacitor is charged and it
  tries to discharge through the coil. However EMF
  equal to the voltage on the capacitor builds on
  the coil to prevent quick growth of the current.
  The current is zero in the beginning. But its
  time derivative must be non-zero, so current
  slowly grows.

21
LC Circuits IV

• The capacitor discharges which causes decreases
  the current growth and thereby also the EMF in
  the coil.
• At the point, the capacitor is discharged, the
  voltage on it and thereby the rate of the growth
  of the current as well as the voltage on the coil
  are zero. But the current is now in its maximum
  and the coil prevents it to drop instantly.

22
LC Circuits V

• The EMF on the coil will now grow in the opposite
  direction to oppose the decrease of the current.
  But anyway the EMF as well as the decrease rate
  of the current grows. The capacitor will now be
  charged it the opposite polarity.
• At the moment the capacitor is fully charged the
  current is zero and everything repeats again.

23
LC Circuits VI

• Using formulas for electric and magnetic energy
  we can find
• Energy changes as expected from electric to
  magnetic and the total is constant.

24
LRC Circuits I

• If we add a resistor the oscillations will be
  dumped. The energy will be lost by thermal loses
  on the resistor.
• The level of dumping depends on the resistance.

25
Homework

• No homework today!

26
Things to read and learn

• This lecture covers
• Chapter 26 4 29 5, 6 30 3, 4, 5, 6
• Advance reading
• Chapter 25 6, 7 29 4 30 6 31 1, 2

27
RC Circuit I

• We use definition of the current I dQ/dt and
  relation of the charge and voltage on a capacitor
  Vc Q(t)/C

• The minus sign reflects the fact that the
  capacitor is being discharged. This first order
  homogeneous differential equation can be solved
  by separating the variables.

28
RC Circuit II

• Where we define a time-constant ? RC. We can
  integrate both sides of the equation

• The integration constant can be found from the
  boundary conditions Q0 CVc0

29
RC Circuit III

• By dividing this by C and then by R we get the
  time dependence of the voltage on the capacitor
  and the current in the circuit.

30
RC Circuit IV

• We substitute for the current I dQ/dt and the
  voltage and reorganize a little

• We get a similar equation for the charge on the
  capacitor but now its the right side is not zero.
  We can solve it by solving first a homogeneous
  equation and then adding one particular solution
  e.g. final Qk CV0 .

31
RC Circuit V

• Since we have already solved the homogeneous
  equation in the previous case, we can write

The integration constant we again get from the
initial condition Q(0) 0 ? Q0 -CV0.
32
RC Circuit VI

• By dividing this by C we get the time dependence
  of the voltage on the capacitor

33
RC Circuit VII

• To get the current we have to calculate from its
  definition as the time derivative of the charge

34
RL Circuit I

• First we solve a homogeneous equation (with zero
  in the right) and add a particular solution Im
  V0/R (maximal current)

35
RL Circuit II

• This can be solved by separation of the
  variables. Using the previous, defining the
  time-constant ? L/R and adding the particular
  solution, we get

• We apply the starting conditions I(0) 0 ? I0
  -Im and we get

36
RL Circuit III

• The voltage on the coil we get from
• V LdI/dt

37
LC Circuit I

• We use definition of the current I dQ/dt and
  relation of the charge and voltage on a capacitor
  Vc Q(t)/C

• We take into account that the capacitor is
  discharged by positive current. This is
  homogeneous differential equation of the second
  order. We can guess the solution.

38
LC Circuit II

• Now we get parameters by finding the second
  derivative of the Q(t) and substituting it into
  the equation

• The solition are un-dumped harmonic oscillations.