Coupled Inductors
Ideal Transformer
Transformer Circuit
Power Transfer
Impedance Matching
ECSE-2010
Lecture 24.1
sawyes@rpi.edu
COUPLED INDUCTORS
+
v1
L1
−
L2
+
v2
−
If Magnetic Field is Time Varying => Creates Voltages
www.rpi.edu/~sawyes
2
COUPLED INDUCTORS
Magnetic Fields of Inductors Link Each Other
sawyes@rpi.edu
www.rpi.edu/~sawyes
3
Consider 2 Inductors, L1, L2:
Inductors can be in Same Circuit, or
in Different Circuits but Close to
Each Other:
Current Flowing in L1 Creates a
Magnetic Field B1 which can Link L2:
Current Flowing in L2 Creates a
Magnetic Field B2 which can Link L1:
Referred to as Coupled Inductors
sawyes@rpi.edu
www.rpi.edu/~sawyes
4
1
COUPLED INDUCTORS
COUPLED INDUCTORS
If Magnetic Fields Change with
Time =>Create Voltages in the
Coupled Inductors L1, L2:
Polarity of the Induced Voltages
Depends on Which Way the
Inductors are Wound:
We Keep Track of the Windings by
Using the Dot Convention:
sawyes@rpi.edu
www.rpi.edu/~sawyes
Most Common Application of
Coupled Inductors is in Circuits
Operating in the AC Steady State:
Good News: Can Use to Make
Transformers
Bad News: Happens Even When We
Do Not Want It To => Mutual
Inductance
We Will Look at Both
5
sawyes@rpi.edu
www.rpi.edu/~sawyes
6
TRANSFORMERS
Thomas, Rosa, Toussaint, The Analysis and Design of Linear
Circuits, 7th edition.
sawyes@rpi.edu
www.rpi.edu/~sawyes
7
Transformers are Easy to Make:
Transformers Can Be Used to
Easily Change AC Voltages:
One of the Key Reasons why
World Runs on AC, not DC:
Can Model Most “Real”
Transformers with “Ideal”
Transformers:
sawyes@rpi.edu
www.rpi.edu/~sawyes
8
2
IDEAL TRANSFORMER
i1
1:N
N1:N 2
IDEAL TRANSFORMER
• •
+
i1
i2
+
−
−
Primary
Secondary
Winding
N 2 Turns
Dots indicate direction of windings
Winding
N1 Turns
www.rpi.edu/~sawyes
1:N
−
Same Side
or Opposite
Sides
Primary
9
•
Secondary
Winding
N 2 Turns
Winding
N1 Turns
sawyes@rpi.edu
N=
N2
N1
Turns Ratio
−
−
Dots
on Opposite
Sides
www.rpi.edu/~sawyes
i1
10
1:N
• •
i2
+
+
v2
v1
v2
−
−
−
+
Dots
on Same
Sides
Transformer "Rules":
i
v 2 = N v1
i2 = 1
N
p 2 = v 2i 2 = p1 = v1i1 ⇒ No Energy Loss
For an Ideal Tranformer
there is No Energy Loss from
Primary to Secondary
www.rpi.edu/~sawyes
+
v2
Dots
are on
i2
v1
sawyes@rpi.edu
•
IDEAL TRANSFORMER
• •
+
i2
v1
IDEAL TRANSFORMER
i1
1:N
+
Turns Ratio
v2
v1
sawyes@rpi.edu
N
N= 2
N1
11
sawyes@rpi.edu
www.rpi.edu/~sawyes
12
3
IDEAL TRANSFORMER
i1
IDEAL TRANSFORMER
i1
i2
1:N
•
+
+
v1
v2
−
−
•
on Opposite
Sides
Transformer "Rules":
i
v 2 = − N v1
i2 = − 1
N
OR: Treat N as a Negative Number
sawyes@rpi.edu
13
TRANSFORMER CIRCUIT
+
Vs
Zs
−
I1
v2
−
−
sawyes@rpi.edu
www.rpi.edu/~sawyes
14
I1
I2
1:N
+
V1
+
V2
−
−
ZL
+
VL
Zs
+
Vs
−
−
I2
1:N
+
V1
+
V2
−
−
ZL
+
VL
−
2 Choices for the Equivalent Circuit
Refer Secondary Circuit to the Primary
OR
Refer Primary Circuit to the Secondary
with an Equivalent Circuit that has No Transformer!
Can Then Use AC Steady State Circuit Analysis
www.rpi.edu/~sawyes
v1
TRANSFORMER CIRCUIT
To do Circuit Analysis, Want to Replace this Circuit
sawyes@rpi.edu
+
Uses for Transformers:
"Step Up" or "Step Down" Voltages
Isolate Load from Source
Impedance Matching
www.rpi.edu/~sawyes
AC Steady State
• •
+
Dots
i2
1:N
15
sawyes@rpi.edu
www.rpi.edu/~sawyes
16
4
IDEAL TRANSFORMERS
IDEAL TRANSFORMERS
Circuit Analysis with
Transformers:
Two Choices for Finding the
Equivalent Circuit:
Want to Replace Ideal Transformer
with a Circuit that Does NOT
Contain an Ideal Transformer so
We Can Use our Regular AC Steady
State Techniques for Circuit
Analysis
sawyes@rpi.edu
www.rpi.edu/~sawyes
Refer Secondary Circuit to Primary
Refer Primary Circuit to Secondary
Can Use Either; Choose the Referral
Method that is Easiest for the
Particular Problem
17
REFERRAL TO PRIMARY
I1
Zs
+
Vs
+
V1
−
−
sawyes@rpi.edu
I1
I2
1:N
+
V2
−
ZL
+
VL
+
Vs
−
−
V1
I1
Find Equivalent Impedance Seen
Looking IN to the Primary
www.rpi.edu/~sawyes
18
REFERRAL TO PRIMARY
Zs
+
V1
−
Z1
I2
1:N
+
V2
−
ZL
+
VL
−
V2
I1 = N I 2
N
V
V N V2 1
Z
= L2
Z1 = 1 = 2
=
2
N
I1
N I2
I2 N
V1 =
Z1 =
sawyes@rpi.edu
www.rpi.edu/~sawyes
19
sawyes@rpi.edu
www.rpi.edu/~sawyes
20
5
REFERRAL TO SECONDARY
REFERRAL TO PRIMARY
I1
I1
Zs
Zs
+
Vs
+
V1
−
−
+
Vs
ZL
N2
−
www.rpi.edu/~sawyes
21
REFERRAL TO SECONDARY
I1
+
Vs
Zs
−
1:N
V1 = V s
www.rpi.edu/~sawyes
+
V2
−
−
+
VL
ZL
−
V oc ; ZT
sawyes@rpi.edu
www.rpi.edu/~sawyes
22
REFERRAL TO SECONDARY
I2 = 0
I1
Zs
+
V1
+
V oc
−
−
+
Dead Network V1
−
ZL
It
1:N
+
V2
−
+
Vt
−
I1
V
ZT = t
N
It
Vt
NV1
2 V1
2
=> ZT =
=
=N
= N Zs
It
− I1 N
−I1
V 2 = V t = NV1 I t = −I 2 = −
I 2 = 0 => I1 = NI 2 = 0
sawyes@rpi.edu
+
V1
Find Thevenin Equivalent Circuit Seen
Looking BACK IN to the Secondary
Equivalent to Basic Transformer Circuit
Can Now Do AC Steady State Circuit Analysis
sawyes@rpi.edu
I2
1:N
=> V oc = NV1 = NVs
23
sawyes@rpi.edu
www.rpi.edu/~sawyes
24
6
REFERRAL METHODS
REFERRAL TO SECONDARY
I2
Can Replace Ideal Transformer
and Secondary Circuit with
ZL/N2 in the Primary Circuit:
N 2 Zs
+
NV s
ZL
−
+
V2 = VL
Referral to Primary
−
Can Replace Primary Circuit and
Ideal Transformer with N Vs, N2
Zs in the Secondary Circuit:
Equivalent to Basic Transformer Circuit
Referral to Secondary
Can Now Do AC Steady State Circuit Analysis
sawyes@rpi.edu
www.rpi.edu/~sawyes
25
REFERRAL METHODS
sawyes@rpi.edu
www.rpi.edu/~sawyes
26
POWER TRANSFER
Can Refer to Primary OR Refer
to Secondary => Choose the
Easiest for Particular Problem:
Zs
Vs
Zs = R s + jX s
ZL
Z L = R L + jX L
For Maximum Power to ZL, Choose ZL = Z*s
=> R L = R s and X L = −X s
sawyes@rpi.edu
www.rpi.edu/~sawyes
27
sawyes@rpi.edu
www.rpi.edu/~sawyes
28
7
IMPEDANCE MATCHING
Example
For PMAX to ZL => Want ZL = Zs*
For Fixed ZL, Helps to Use Transformer
Make ZL / N2 = Zs*
Provides an Additional Knob
Most Power Amps use a Transformer to
Couple the Output to the Speakers
Provides both Isolation and Impedance
Matching
Let’s Look at this with an Example
sawyes@rpi.edu
www.rpi.edu/~sawyes
29
Example
50 + jX Ω
12/ 00
Choose Values for X and N to Maximize Power to Load
sawyes@rpi.edu
www.rpi.edu/~sawyes
30
Example
Refer to Primary
Refer to Primary
2 + j3 Ω
50 + jX Ω
N2
12/00
www.rpi.edu/~sawyes
2 + j3 Ω
50 + jX Ω
N2
12/00
For Maximum Power, Choose ZL = Z*s
50
X
=> 2 = 2 and 2 = −3
N
N
sawyes@rpi.edu
1: N
2 + j3 Ω
For Maximum Power to Load
X
X
50
=
= −3 => X = −75 Ω
= 2 => N = 5
N 2 25
N2
31
sawyes@rpi.edu
www.rpi.edu/~sawyes
32
8