Inductance and RL Circuits

1
Inductance and RL Circuits

• Mutual Inductance
• Self Inductance
• RL Circuits
• Magnetic Energy

2
Inductance

• Mutual Inductance
• On the table you will find two coils. The
  larger coil is connected to a AC power supply.
  This will apply a sinusoidal voltage given by
• V(t) V0sin(wt).
• This voltage produces a sinusoidal current in
  the large coil where
• I(t) I0sin(wt)
• The angular frequency w is 2p times the
  frequency. This will give us a time varying
  magnetic field in the large coil.
• Describe the change in the magnetic field as a
  function of time (do not use sine).

3
Inductance

• Mutual Inductance
• V(t) V0sin(wt).
• I(t) I0sin(wt)

4
Inductance

• The smaller coil is connected to the voltage
  input on the Pasco 750 Interface. Start Data
  Studio and configure the interface for Voltage
  input by selecting and dragging the analog icon
  (the plug icon on the left side) to channel A.
  For output select Scope (the oscilloscope) and
  connect to channel A by dragging the icon to
  channel A. Enlarge the window. On the right
  side of the oscilloscope display window you will
  find two controls. The left control should
  control the sensitivity of the input (vertical
  scale). Set the voltage for channel A to 0.1
  volts/div by clicking the vertical trace scale
  button. At the bottom of the window you will
  find the button to change the sweep rate. Change
  the time sweep to 10 msec/div.

5
Inductance

• Turn on the AC supply and turn the knob to full
  scale (12 volts). Place the smaller coil inside
  the larger coil with the axis of the small coil
  aliened with the axis of the large coil. You
  should see a sine wave trace on the oscilloscope
  screen. If you do not get help from the
  instructor.
• Slowly pull the small coil out of the large coil
  along the coil axis. Describe what happens to
  the voltage signal in the small coil.
• Return the coil to the center of the large coil.
  Rotate the small coil on its side by 90 degrees
  then 180 degrees. Describe what happens to the
  voltage signal.

6
Inductance

• Write the equation that describes the emf induced
  in one coil when there is a current changing with
  time in the other coil.
• This equation describes the results that you just
  observed.

large coil small coil
7
Inductance

• Repeat the experiment again where you slowly pull
  the small coil out of the large coil. Explain
  what happen using the equation and the definition
  of mutual inductance.
• Did the mutual induction change? Explain.

8
Inductance

• Lets calculate the mutual inductance for the two
  coils when the small coil is in the center of the
  large coil. What is the definition of mutual
  inductance?

Which coil should we use to calculate the
magnetic field?
The larger coil because the flux through the
smaller coil can be calculated assuming the
magnetic field is constant.
9
Inductance

• Calculate the magnetic field due to the largest
  coil.

• Calculate the flux through the smaller coil.

• Calculate the mutual inductance.

10
Inductance

• Self Inductance
• Self inductance is just like mutual inductance
  except that there is one coil instead of two.
  The self inductance comes from the current in the
  coil producing a flux through the coil and thus a
  emf in the coil. Because of the minus sign the
  induced emf is in the opposite direction from the
  applied emf that caused the initial current. It
  is thus called a back emf. In Example 32-3 the
  authors find the self inductance for a solenoid.
  Look up the equation for the self inductance for
  a solenoid and enter it here.
• Why are there no subscripts on f and I?

11
Inductance

• Calculate the self inductance for the small coil
  you have been using. The coil has 400 turns and
  a radius of 1 cm.

12
Inductance

• Applications Toothbrush Inductance Charger

13
Inductance

• Applications Transformer

High voltage transformer Low voltage transformer
14
Inductance

• Applications Inductors

We will see their use in the next lecture.
15
Inductance

• LR Circuits
• On your table you will find the EM board with
  resistors capacitors, and an inductor. Connect
  the inductor and the 470 W resistor in series
  with the Sweep/Function Generator. Use the cable
  with four leads. The two leads with the black
  and white plug are the voltage input leads and
  the two leads with the red and blue plugs are the
  output from the signal generator. The red cable
  with the white plug is the positive side or
  higher potential side. The Sweep/Function
  Generator produces different types of voltages.
  Select the square wave by pressing the square
  wave button on the upper right side. Set the
  amplitude to 2 volts and a frequency of 20 Hz.
  Connect the voltage input to the Pasco interface
  to the output of the Sweep/Function Generator.

16
Inductance

• You should still have Science Workshop open with
  the voltage input and the oscilloscope as an
  output for channel A. Set the vertical scale for
  2 volts/div and the horizontal scale to 5 ms/div.
  You should see a square wave of the screen.
  Change the amplitude on the Sweep/Function
  Generator and watch the signal change. Does it
  change the way you expect?
• Now change the frequency on the Sweep/Function
  Generator. Watch the square wave trace change.
  Does the trace change the way you expect? Can
  you relate the period measured on the screen to
  the frequency of the Function Generator? Do it.
  Measure the period and calculate the frequency.
• Does it compare with the frequency on the
  Function Generator?

17
Inductance

• Change the frequency back to 20 Hz. and connect
  the voltage probes to the 470 W resistor. The
  voltage across the resistor is proportional to
  the current. The voltage across the resistor and
  the current have the same functional form. You
  should see the trace rise and level off. This is
  what your text calls current build up in an
  inductor. This is described by the following
  equation
• where the inductance time constant t L/R.
  Discuss in your group. What will happen if you
  increased the resistance. Write your answer
  here.
• Change the resistor in the circuit two or three
  times and see if you were correct.

18
Inductance

• Now close the oscilloscope window and connect the
  graph window to channel A the voltage input. Set
  the time scale to msec and the voltage scale to
  2 volts and -2 volts. Make sure the resistor is
  the 470 W resistor and record the voltage across
  the resistor for one or two seconds. Click the
  box that rescales the graph to the data and then
  click the box that allows you to select and
  rescale. Get one full period on the screen. You
  can see current build-up and decay.
• Discuss in your group the value of the voltage
  for one and two time constants. Use this to find
  the time constant.
• Do you get the same time constant for current
  build-up and decay? Check this out.

19
Inductance

• From the time constant and the resistance find
  the self inductance for the inductor. An ideal
  inductor has zero resistance, but a real inductor
  has a resistance. It is the resistance of the
  wire that forms the inductor. This resistance is
  part of the time constant. The total resistance
  is the 470 W resistor and the 165 W resistance of
  the inductor

20
Inductance

• RL Circuit Current Build-up and Decay

21
Inductance

• RL Circuit Current Decay

-
22
Inductance

• RL Circuit Current Build-up

23
Inductance

• RL Circuit Current Build- up

24
Inductance

• RL Circuit Resistor and Inductor in Parallel

On current build-up
VL dI/dt
t t
Slope dI/dt
t 3-4 t
So most of the current goes through the resistor.
25
Inductance

• Energy in an Inductor
• Look up the energy stored in an inductor and
  write it here.
• Is the energy constant in the circuit you are
  using? Explain.
• If the maximum current in the inductor is 40
  mAmps, what is the maximum energy stored in the
  inductor?
• What is the energy in terms of the magnetic field?

U (1/2)(25mH)(40mA)2