# Experiment 3

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```					10/3/2012                                            1

Electronic Instrumentation
Experiment 3
•Part A: Making an Inductor
•Part B: Measurement of Inductance
•Part C: Simulation of a Transformer
•Part D: Making a Transformer
Inductors & Transformers

   How do transformers work?
   How to make an inductor?


How to measure inductance?
How to make a transformer?
?
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Part A
   Inductors Review
   Calculating Inductance
   Calculating Resistance

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Inductors-Review
   General form of I-V relationship
dI
V L
dt
   For steady-state sine wave excitation

ZL  jL          V  jLI

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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.
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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
(  0 N  rc )
2       2

L                Henries
d
where this expression only holds when the length d is
very much greater than the diameter 2rc
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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
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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
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Making an Inductor

   If the coil length is much smaller than the
diameter (rw is the wire radius)
8rc
L   N rc {ln( )  2}
2

rw
Such a coil is used in the
metal detector at the right

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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
R
l is the wire length                         sA
A is the wire cross sectional area (rw2)
s is the wire conductivity

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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

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Wire Resistance
   Using the Megaconverter at
http://www.megaconverter.com/Mega2/
(see course website)

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Part B: Measuring Inductance with a
Circuit                                R1

47
R2

V1                     1
V OFF = 0
V A MPL = 0.2          C2         C1       L1
FRE Q = 1kHz           1u         1u
A C = .2

2
0

   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.

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Determining Inductance
R1
Vin                               Vout

0  1                    f0  1
47
R2

V OFF = 0
V A MPL = 0.2
V1

C2       C1
1

L1
LC                2 LC
FRE Q = 1kHz          1u       1u
A C = .2

2
0

   Reminder—The parallel combination of L and
C goes to infinity at resonance. (Assuming R2 is small.)
jL 1 jC
       
             jL
Z||                  
jL   1 jC 1   LC
2
       
       
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Determining Inductance
Z ||
H
R1  Z||
jL
H
R1(1   2 LC )  jL
jL
H HI  H LO          small
R1
jL
at resonance, 0 , H 0        1
jL

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V
R1

47
V       R2

V1                         1
V OFF = 0
V A MPL = 0.2         C2            C1         L1
FRE Q = 1kHz          1u            1u
A C = .2

2
300mV                                                     0

200mV

100mV

0V
100Hz                     1.0KHz                    10KHz                   100KHz    1.0MHz
V(V1:+)   V(C1:1)
Frequency

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   Even 1 ohm of resistance in the coil can spoil
this response somewhat
300mV

200mV

Coil Resistance small

100mV
Coil resistance small

0V
100Hz                         1.0KHz               10KHz                  100KHz         1.0MHz
V(V1:+)    V(C1:1)
Frequency

300mV

200mV

Coil resistance of a few Ohms
100mV

0V
100Hz                         1.0KHz                10KHz                   100KHz         1.0MHz
V(V1:+)     V(C1:1)
Frequency

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Part C
   Examples of Transformers
   Transformer Equations

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Transformers
   Cylinders (solenoids)

   Toroids

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Transformer Equations

Symbol for
transformer
N L VL   LL I S                               RL
a                                      Z in  2
N S VS   LS I L                               a

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Deriving Transformer Equations
   Note that a transformer has
two inductors. One is the
primary (source end) and
end): LS & LL
   The inductors work as
expected, but they also
couple to one another
through their mutual
inductance: M2=k2 LS LL

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Transformers

   Assumption 1: Both Inductor Coils must have
similar properties: same coil radius, same core
material, and same length.
(  0 N L rc )
2   2

2
LL          d         NL                 NL                    LL
                  2         let a                  a
LS (  0 N S  rc ) N S
2    2
NS                    LS
d

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Transformers                                IS               IL
Note Current
Direction

   Let the current through the primary be I S
   Let the current through the secondary be I L
   The voltage across the primary inductor is
jLI S  jMI L
   The voltage across the secondary inductor is
jLI L  jMI S

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Transformers

   Sum of primary voltages must equal the source
VS  RS I S  jLS I S  jMI L
   Sum of secondary voltages must equal zero
0  RL I L  jLL I L  jMI S

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Transformers

   Assumption 2: The transformer is designed such that
the impedances Z  jL are much larger than any
resistance in the circuit. Then, from the second loop
equation
0  RL I L  jLL I L  jMI S
jLL I L  jMI S                  L I M I
2 2
L L
2 2
S
IL M
         
I S LL
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Transformers
   k is the coupling coefficient
• If k=1, there is perfect coupling.
• k is usually a little less than 1 in a good transformer.
   Assumption 3: Assume perfect coupling (k=1)
LL
We know           M2=k2   LS LL= LS LL and a 
LS

Therefore,                 IL M            LS LL            Ls 1
                                      
I S LL           LL              LL a
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Transformers
 The input impedance of the primary winding
reflects the load impedance. Z L  Z in  Z total  RS
S

 It can be determined from the loop equations
• 1] VS  RS I S  jLS I S  jMI L
• 2] 0  RL I L  jLL I L  jMI S
   Divide by 1] IS. Substitute 2] and M into 1]


VS
 RS  jLS 
 2 LS LL
Z IN          IS                          RL  jLL 
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Transformers
   Find a common denominator and simplify
jLS RL
Z IN   
jLL  RL

   By Assumption 2, RL is small compared to the
impedance of the transformer, so
LS RL RL
Z IN          2
LL   a
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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
N SVL  N LVS
Note that the coil with more turns has the larger
voltage.
 Detailed derivation of transformer equations
http://hibp.ecse.rpi.edu/~connor/education/transformer_notes.pdf

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Transformer Equations

N L VL   LL I S                               RL
a                                      Z in  2
N S VS   LS I L                               a

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Part D
   Step-up and Step-down transformers
   Build a transformer

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Step-up and Step-down Transformers
Step-up Transformer                     Step-down Transformer

N 2  N1                                  N 2  N1
V2  V1                                   V2  V1
I 2  I1                                  I 2  I1
L2  L1                                       L2  L1

Note that power (P=VI) is conserved in both cases.

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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 ωL >> R?
N L VL
a    
N S VS

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Some Interesting Inductors

   Induction Heating

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Some Interesting Inductors

   Induction Heating in Aerospace
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Some Interesting Inductors

   Induction Forming

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Some Interesting Inductors
Primary
Coil

Secondary
Coil

   Coin Flipper

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Some Interesting Inductors

   GE Genura Light

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Some Interesting Transformers

   A huge range in sizes
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Some Interesting Transformers

   High Temperature Superconducting Transformer
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Household Power

   7200V transformed to 240V for household use

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Wall Warts

Transformer

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