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10/15/2012 1
Electronic InstrumentationExperiment 3
•Part A: Making an Inductor•Part B: Measurement of Inductance•Part C: Simulation of a Transformer•Part D: Making a Transformer
10/15/2012 Electronic Instrumentation 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/allie/labmanu
als/1271l1sp03.doc
Vs
L
Ideal Inductor
C
Ideal Capacitor
R
Ideal Resistor
10/15/2012 Electronic Instrumentation 3
Review RLC and Resonance
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Voltage Transfer Function
Frequency KHz
Vx/V
s
Vx=VRVx=VLVx=VC
http://ecow.engr.w isc.edu/cgi-
bin/getbig/ece/271 /allie/labmanuals/1 271l1sp03.doc
10/15/2012 Electronic Instrumentation 4
Inductors & Transformers
How do transformers work?
How to make an inductor?
How to measure inductance?
How to make a transformer?
?
10/15/2012 Electronic Instrumentation 5
Part A
Inductors Review
Calculating Inductance
Calculating Resistance
10/15/2012 Electronic Instrumentation 6
Inductors-Review
General form of I-V relationship
For steady-state sine wave excitation
V L dIdt
V j LI Z j LL
10/15/2012 Electronic Instrumentation 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.
10/15/2012 Electronic Instrumentation 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
Henriesd
rNL c )( 22
0
10/15/2012 Electronic Instrumentation 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
10/15/2012 Electronic Instrumentation 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
10/15/2012 Electronic Instrumentation 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 themetal detector at the right
2)8
ln(2 w
cc r
rrNL Coil
Length (d)
Form Diameter=2rc
10/15/2012 Electronic Instrumentation 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 byl is the wire lengthA is the wire cross sectional area (rw
2)
is the wire conductivity
AlR
10/15/2012 Electronic Instrumentation 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
10/15/2012 Electronic Instrumentation 14
Wire Resistance
Using the Megaconverter
at http://www.megaconverter.com/Mega2/(see course website)
10/15/2012 Electronic Instrumentation 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.
R1
47
C11u
L1
1
2
R2
0
C21u
V1
FREQ = 1kHzVAMPL = 0.2VOFF = 0
AC = .2
10/15/2012 Electronic Instrumentation 16
In Class Problem #1
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, ω0
?
R1
47
C11u
L1
1
2
R2
0
C21u
V1
FREQ = 1kHzVAMPL = 0.2VOFF = 0
AC = .2
LCf
LC
211
00
VoutVin
CjZ
LjZRZ
C
LR
1
10/15/2012 Electronic Instrumentation 17
Determining Inductance
Reminder—The parallel combination of L and C goes to infinity at resonance. (Assuming R2 is small.)
Zj L j C
j L j C
j LLC||
1
1 1 2
LCf
LC
211
00 VoutVin R1
47
C11u
L1
1
2
R2
0
C21u
V1
FREQ = 1kHzVAMPL = 0.2VOFF = 0
AC = .2
10/15/2012 Electronic Instrumentation 18
Determining Inductance
1,,
)1(1
1
0
000
2
||
||
LjLjHresonanceat
smallHHLjLCR
LjH
ZRZ
H
LOHI
10/15/2012 Electronic Instrumentation 19
R1
47
C11u
L1
1
2
R2
0
C21u
V1
FREQ = 1kHzVAMPL = 0.2VOFF = 0
AC = .2
V
V
Frequency
100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C1:1)
0V
100mV
200mV
300mV
10/15/2012 Electronic Instrumentation 20
Even 1 ohm of resistance in the coil can spoil this response somewhat
Frequency
100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C1:1)
0V
100mV
200mV
300mV
Coil Resistance small
F r e q u e n c y
1 0 0 H z 1 . 0 K H z 1 0 K H z 1 0 0 K H z 1 . 0 M H zV ( V 1 : + ) V ( C 1 : 1 )
0 V
1 0 0 m V
2 0 0 m V
3 0 0 m V
Coil resistance of a few Ohms
Coil resistance small
10/15/2012 Electronic Instrumentation 21
Part C
Examples of Transformers
Transformer Equations
10/15/2012 Electronic Instrumentation 22
Transformers
Cylinders (solenoids)
Toroids
10/15/2012 Electronic Instrumentation 23
Transformer Equations
2aRZ
II
LL
VV
NNa L
inL
S
S
L
S
L
S
L
Symbol for transformer
10/15/2012 Electronic Instrumentation 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: M2=k2 LS LL
10/15/2012 Electronic Instrumentation 25
Transformers
Assumption 1: Both Inductor Coils must have similar properties: same coil radius, same core material, and same length.
S
L
S
L
LLa
NNalet 2
2
220
220
)(
)(
S
L
cS
cL
S
L
NN
drN
drN
LL
10/15/2012 Electronic Instrumentation 26
Transformers
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
IS
IL
j LI j MIS L
j LI j MIL S
IS ILNote Current Direction
10/15/2012 Electronic Instrumentation 27
Transformers
Sum of primary voltages must equal the source
Sum of secondary voltages must equal zeroV R I j L I j MIS S S S S L
0 R I j L I j MIL L L L S
10/15/2012 Electronic Instrumentation 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
LjZ
0 R I j L I j MIL L L L S
j L I j MIL L S 2222SLL IMIL
LS
L
LM
II
10/15/2012 Electronic Instrumentation 29
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
LS LL
= LS LL
and
Therefore,
S
L
LLa
aLLs
LLL
LM
II
LL
LS
LS
L 1
10/15/2012 Electronic Instrumentation 30
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]
StotalinL RZZZS
Z VI R j L L L
R j LINS
SS S
S L
L L
2
V R I j L I j MIS S S S S L 0 R I j L I j MIL L L L S
10/15/2012 Electronic Instrumentation 31
Transformers
Find a common denominator and simplify
By Assumption 2, RL
is small compared to the impedance of the transformer, so
LL
LSIN RLj
RLjZ
2aR
LRLZ L
L
LSIN
10/15/2012 Electronic Instrumentation 32
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 equationshttp://hibp.ecse.rpi.edu/~connor/education/transformer_notes.pdf
N V N VS L L S
10/15/2012 Electronic Instrumentation 33
Transformer Equations
2aRZ
II
LL
VV
NNa L
inL
S
S
L
S
L
S
L
10/15/2012 Electronic Instrumentation 34
In Class Problem #2Ns :NL
Vs VLVGEN
aN LN S
V LV S
Z inR L
a2VGEN
=120VRL
=20 ΩNL
=1NS
=12
Is
Vs VLZin
1. Find VL if RS
~02.
Find VL if Rs
= 1 k Ω
Hint: Is VGEN = VS
? Under what conditions is this not true? How would you find VS
? Need Zin
10/15/2012 Electronic Instrumentation 35
Part D
Step-up and Step-down transformers
Build a transformer
10/15/2012 Electronic Instrumentation 36
Step-up and Step-down TransformersStep-up Transformer
12
12
12
12
LL
IIVVNN
Step-down Transformer
12
12
12
12
LL
IIVVNN
Note that power (P=VI) is conserved in both cases.
10/15/2012 Electronic Instrumentation 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 ωL >> R?
S
L
S
L
VV
NNa
10/15/2012 Electronic Instrumentation 38
Some Interesting Inductors
Induction Heating
10/15/2012 Electronic Instrumentation 39
Some Interesting Inductors
Induction Heating in Aerospace
10/15/2012 Electronic Instrumentation 40
Some Interesting Inductors
Induction Forming
10/15/2012 Electronic Instrumentation 41
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!)
Primary
Coil
Secondary
Coil
10/15/2012 Electronic Instrumentation 42
Some Interesting Inductors
GE Genura
Light
10/15/2012 Electronic Instrumentation 43
Some Interesting Transformers
A huge range in sizes
10/15/2012 Electronic Instrumentation 44
Household Power
7200V transformed to 240V for household use
10/15/2012 Electronic Instrumentation 45
Wall Warts
Transformer