Download Experiment 3 - Rensselaer Polytechnic Institute

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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
Z L  jL
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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 )
L
Henries
d
2
2
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}
rw
2
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.
l
 Resistance of a wire is given by
R
sA
l is the wire length
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
VOFF = 0
VAMPL = 0.2
FREQ = 1kHz
AC = .2
1
C2
1u
C1
1u
L1
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
47
R2
V1
VOFF = 0
VAMPL = 0.2
FREQ = 1kHz
AC = .2
1
C2
1u
C1
1u
L1
0  1
LC
f0  1
2 LC
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|| 

2
jL   1 jC 1   LC


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Determining Inductance
H
Z||
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
R2
V
V1
VOFF = 0
VAMPL = 0.2
FREQ = 1kHz
AC = .2
1
C2
1u
C1
1u
L1
2
0
300mV
200mV
100mV
0V
100Hz
V(V1:+)
1.0KHz
10KHz
100KHz
1.0MHz
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
Coil resistance small
100mV
0V
100Hz
V(V1:+)
1.0KHz
10KHz
100KHz
1.0MHz
V(C1:1)
Frequency
300mV
200mV
Coil resistance of a few Ohms
100mV
0V
100Hz
V(V1:+)
1.0KHz
10KHz
100KHz
1.0MHz
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
a



N S VS
LS I L
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RL
Z in  2
a
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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: 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 L2 rc )
2
LL
N
d
L


LS (  0 N S2 rc 2 ) N S2
d
2
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NL
let a 
NS
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 a
LL
LS
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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

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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)
We know
M2=k2
Therefore,
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
LS LL= LS LL and a 
IL M


I S LL
LS LL

LL
Electronic Instrumentation
LL
LS
Ls 1

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
 It can be determined from the loop equations
S
• 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]
Z IN
V
 S
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2

L
L
S
L
 RS  jLS 
IS
 RL  jLL 
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Transformers

Find a common denominator and simplify
Z IN

jLS RL

jLL  RL
By Assumption 2, RL is small compared to the
impedance of the transformer, so
Z IN
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LS RL RL

 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 S VL  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
a



N S VS
LS I L
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RL
Z in  2
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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