Experiment 3

1
Experiment 3

• Part A Making an Inductor
• Part B Measurement of Inductance
• Part C Simulation of a Transformer
• Part D Making a Transformer

2
Review RLC and Resonance

• How can the transfer function be greater than 1?
• At resonance, impedance value is a minimum
• At resonance, impedance of inductor and capacitor
  cancel each other out (equal in magnitude, phase
  is opposite)
• So circuit is purely resistive at resonance
• H depends on the position of Vout

http//ecow.engr.wisc.edu/cgi-bin/getbig/ece/271/a
llie/labmanuals/1271l1sp03.doc
3
Review RLC and Resonance
http//ecow.engr.wisc.edu/cgi-bin/getbig/ece/271/a
llie/labmanuals/1271l1sp03.doc
4
Inductors Transformers

• How do transformers work?
• How to make an inductor?
• How to measure inductance?
• How to make a transformer?

?
5
Part A

• Inductors Review
• Calculating Inductance
• Calculating Resistance

6
Inductors-Review

• General form of I-V relationship
• For steady-state sine wave excitation

7
Determining Inductance

• Calculate it from dimensions and material
  properties
• Measure using commercial bridge (expensive
  device)
• Infer inductance from response of a circuit. This
  latter approach is the cheapest and usually the
  simplest to apply. Most of the time, we can
  determine circuit parameters from circuit
  performance.

8
Making an Inductor

• For a simple cylindrical inductor (called a
  solenoid), we wind N turns of wire around a
  cylindrical form. The inductance is ideally given
  by
• where this expression only holds when the
  length d is very much greater than the diameter
  2rc

9
Making an Inductor

• Note that the constant ?o 4? x 10-7 H/m is
  required to have inductance in Henries (named
  after Joseph Henry of Albany)
• For magnetic materials, we use ? instead, which
  can typically be 105 times larger for materials
  like iron
• ? is called the permeability

10
Some Typical Permeabilities

• Air 1.257x10-6 H/m
• Ferrite U M33 9.42x10-4 H/m
• Nickel 7.54x10-4 H/m
• Iron 6.28x10-3 H/m
• Ferrite T38 1.26x10-2 H/m
• Silicon GO steel 5.03x10-2 H/m
• supermalloy 1.26 H/m

11
Making an Inductor

• If the coil length is much smaller than the
  diameter (rw is the wire radius)
• Such a coil is used in the
• metal detector at the right

Form Diameter 2rc
Coil Length (d)
12
Calculating Resistance

• All wires have some finite resistance. Much of
  the time, this resistance is negligible when
  compared with other circuit components.
• Resistance of a wire is given by
• l is the wire length
• A is the wire cross sectional area (prw2)
• s is the wire conductivity

13
Some Typical Conductivities

• Silver 6.17x107 Siemens/m
• Copper 5.8x107 S/m
• Aluminum 3.72x107 S/m
• Iron 1x107 S/m
• Sea Water 5 S/m
• Fresh Water 25x10-6 S/m
• Teflon 1x10-20 S/m
• Siemen 1/ohm

14
Wire Resistance

• Using the Megaconverter at http//www.megaconverte
  r.com/Mega2/
• (see course website)

15
Part B Measuring Inductance with a Circuit

• For this circuit, a resonance should occur for
  the parallel combination of the unknown inductor
  and the known capacitor. If we find this
  frequency, we can find the inductance.

16
In Class Problem 1
Vout
Vin

• What is ZLC (assuming R2 is very small)?
• What does R2 represent?
• What is its transfer function (equation)?
• What is H at low and high frequencies?
• What is H at the resonant frequency, ?0?

17
Determining Inductance
Vout
Vin

• ReminderThe parallel combination of L and C goes
  to infinity at resonance. (Assuming R2 is small.)

18
Determining Inductance
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(No Transcript)
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• Even 1 ohm of resistance in the coil can spoil
  this response somewhat

Coil resistance small
Coil resistance of a few Ohms
21
Part C

• Examples of Transformers
• Transformer Equations

22
Transformers

• Cylinders (solenoids)
• Toroids

23
Transformer Equations
Symbol for transformer
24
Deriving Transformer Equations

• Note that a transformer has two inductors. One is
  the primary (source end) and one is the secondary
  (load end) LS LL
• The inductors work as expected, but they also
  couple to one another through their mutual
  inductance M2k2 LS LL

25
Transformers

• Assumption 1 Both Inductor Coils must have
  similar properties same coil radius, same core
  material, and same length.

26
Transformers
IS
IL
Note Current Direction

• Let the current through the primary be
• Let the current through the secondary be
• The voltage across the primary inductor is
• The voltage across the secondary inductor is

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Transformers

• Sum of primary voltages must equal the source
• Sum of secondary voltages must equal zero

28
Transformers

• Assumption 2 The transformer is designed such
  that the impedances are much
  larger than any resistance in the circuit. Then,
  from the second loop equation

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Transformers

• k is the coupling coefficient
• If k1, there is perfect coupling.
• k is usually a little less than 1 in a good
  transformer.
• Assumption 3 Assume perfect coupling (k1)
• We know M2k2 LS LL LS LL and
• Therefore,

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Transformers

• The input impedance of the primary winding
  reflects the load impedance.
• It can be determined from the loop equations
• 1
• 2
• Divide by 1 IS. Substitute 2 and M into 1

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Transformers

• Find a common denominator and simplify
• By Assumption 2, RL is small compared to the
  impedance of the transformer, so

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Transformers

• It can also be shown that the voltages across the
  primary and secondary terminals of the
  transformer are related by
• Note that the coil with more turns has the
  larger voltage.
• Detailed derivation of transformer equations
• http//hibp.ecse.rpi.edu/connor/education/transfo
  rmer_notes.pdf

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Transformer Equations
34
In Class Problem 2
NsNL
VGEN120V RL20 O NL1 NS12
Vs
VL
VGEN

• 1. Find VL if RS0
• Find VL if Rs 1 k O
• Hint Is VGEN VS? Under what conditions is this
  not true? How would you find VS? Need Zin

Is
Vs
VL
Zin
35
Part D

• Step-up and Step-down transformers
• Build a transformer

36
Step-up and Step-down Transformers

• Step-up Transformer

Step-down Transformer
Note that power (PVI) is conserved in both
cases.
37
Build a Transformer

• Wind secondary coil directly over primary coil
• Try for half the number of turns
• At what frequencies does it work as expected with
  respect to voltage? When is ?L gtgt R?

38
Some Interesting Inductors

• Induction Heating

39
Some Interesting Inductors

• Induction Heating in Aerospace

40
Some Interesting Inductors

• Induction Forming

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Some Interesting Inductors

• Coin Flipper
• Flash camera circuits charge 6 capacitors
• Large current in primary coil
• Large current induced in coin (larger by ratio of
  turns)
• Current in coin creates electromagnet of opposite
  polarity (Repel!)

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Some Interesting Inductors

• GE Genura Light

43
Some Interesting Transformers

• A huge range in sizes

44
Household Power

• 7200V transformed to 240V for household use

45
Wall Warts
Transformer
46
In Class Problem 1
Vout
Vin

• What is ZLC (assuming R2 is very small)?
• What does R2 represent?
• What is its transfer function (equation)?
• What is H at low and high frequencies?
• What is H at the resonant frequency, ?0?

47
In Class Problem 1
Vout
Vin
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In Class Problem 1
Vout
Vin
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In Class Problem 1

• What is H at low frequencies?
• This can be determined from the magnitude so
• Remember these steps!
• Take the lowest power in the numerator and
  denominator of the equation
• Write down x1, y1, x2, and y2 (can do this in
  your head)
• Use the magnitude equation (square and square
  root numerator and denominator)
• Simplify (this is the answer if approaching 0)
• Take limit as it approaches 0 (this is the answer
  at 0)

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In Class Problem 1

• What is H at low frequencies?
• Step 1 Take lowest power in numerator and
  denominator
• Step 2 Find x1, y1, x2, y2

51
In Class Problem 1

• What is H at low frequencies?
• Step 3 Use the magnitude equation
• Step 4 Simplify
• Step 5 Take limit as ? approaches 0
• Answer is that it becomes very small or 0

52
In Class Problem 1

• What is H at high frequencies?
• This can be determined from the magnitude so
• Remember these steps!
• Take the highest power in the numerator and
  denominator of the equation
• Write down x1, y1, x2, and y2 (can do this in
  your head)
• Use the magnitude equation (square and square
  root numerator and denominator)
• Simplify (this is the answer if approaching 8)
• Take limit as it approaches 0 (this is the answer
  at 8)

53
In Class Problem 1

• What is H at high frequencies?
• Step 1 Take highest power in numerator and
  denominator
• Step 2 Find x1, y1, x2, y2

54
In Class Problem 1

• What is H at high frequencies?
• Step 3 Use the magnitude equation
• Step 4 Simplify
• Step 5 Take limit as ? approaches 8
• Answer is that it becomes very small or 0

55
In Class Problem 1

• What is H at the resonant frequency, ?0?

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In Class Problem 2
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In Class Problem 2