Electronic Instrumentation - Rensselaer Polytechnic …homepages.rpi.edu/~sawyes/exp03.pdf10/15/2012 Electronic Instrumentation 7 Determining Inductance Calculate it from dimensions

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

http://ecow.engr.wisc.edu/cgi-bin/getbig/ece/271/allie/labmanuals/1271l1sp03.doc




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







Inductors & Transformers

How do transformers work?

How to make an inductor?

How to measure inductance?

How to make a transformer?

?


Part A

Inductors Review

Calculating Inductance

Calculating Resistance


Inductors-Review

General form of I-V relationship

For steady-state sine wave excitation

V L dIdt

V j LI Z j LL


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.


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


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


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


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


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


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


Wire Resistance

Using the Megaconverter

at http://www.megaconverter.com/Mega2/(see course website)

http://www.megaconverter.com/Mega2/


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


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


AC = .2

LCf

LC

211

00

VoutVin

CjZ

LjZRZ

C

LR

1



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


AC = .2



1,,

)1(1

1

0

000

2

||

||

LjLjHresonanceat

smallHHLjLCR

LjH

ZRZ

H

LOHI


R1

47

C11u

L1

1

2

R2

0

C21u

V1


AC = .2

V

V

Frequency

100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C1:1)

0V

100mV

200mV

300mV


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


Part C

Examples of Transformers

Transformer Equations


Transformers

Cylinders (solenoids)

Toroids



2aRZ

II

LL

VV

NNa L

inL

S

S

L

S

L

S

L

Symbol for transformer


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


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


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


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


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


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


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


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


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

http://hibp.ecse.rpi.edu/~connor/education/transformer_notes.pdf



2aRZ

II

LL

VV

NNa L

inL

S

S

L

S

L

S

L


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


Part D

Step-up and Step-down transformers

Build a transformer


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.


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


Some Interesting Inductors

Induction Heating



Induction Heating in Aerospace

http://www.ameritherm.com/videoindex.html



Induction Forming



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



GE Genura

Light


Some Interesting Transformers

A huge range in sizes


Household Power

7200V transformed to 240V for household use


Wall Warts

Transformer

Documents

Electronic Instrumentation - Rensselaer Polytechnic …homepages.rpi.edu/~sawyes/exp03.pdf10/15/2012 Electronic Instrumentation 7 Determining Inductance Calculate it from dimensions