Transcript Transformers

Lecture #25
EGR 272 – Circuit Theory II
Read: Chapter 12 in Electric Circuits, 6th Edition by Nilsson
Transformers:
Our earlier study of mutual inductance introduced the idea that the magnetic field
produced by one coil can induce a voltage across another coil. This effect is
maximized with transformers, where two coils (or sets of windings) are wound
around the same core. This allows for a direct transfer of magnetic flux.
f
i1
i2
+
+
V1
N1
turns
_
N2
turns
V2
Load
_
Primary
windings
Secondary
windings
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EGR 272 – Circuit Theory II
Lecture #25
Recall that
 = flux linkage
N = number of turns
f = magnetic flux (in Webers, Wb)
and  = Nf = Li , where L = inductance in Henries, H
d
d(Nf )

dt
dt
Also
v
So
v  N
df
dt
Since the same magnetic flux, f, flows through the windings it is also true that
the rate of change of magnetic flux, df/dt is the same, so
df
V
V
 1  2 , so
dt
N1
N2
a 
N1
V
 1
N2
V2
N1
V
 1
N2
V2
where a = turns ratio
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EGR 272 – Circuit Theory II
Lecture #25
Key transformer relationships
We just saw that
Similarly
If Z1 
V1
I1
a 
N1
V
 1
N2
V2
a 
N1
I
 2
N2
I1
and Z2 
where a = turns ratio
V2
then show that
I2
so
a 
a2 
N1
V
I
 1  2
N2
V2
I1
Z1
Z2
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Lecture #25
EGR 272 – Circuit Theory II
Transformer symbols
General Transformer
Iron-core Transformer
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Lecture #25
EGR 272 – Circuit Theory II
Examples of transformers (reference: www.allelectronics.com)
Primary: 120V
Secondary: 28V, 1.5A
Primary: 120V
Secondary: 40VCT, 0.25A
Primary: 110V
Secondary: 15V, 0.4A
Utility pole transformer
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Lecture #25
EGR 272 – Circuit Theory II
Examples of transformers (reference: www.allelectronics.com)
PC Mount Transformer
Primary: 120V
Secondary: 16VCT, 0.8A or 8V, 1.6A
Up/Down Transformer
(110V to 220V) or (220V to 110V)
Toroidal Transformer
Primary: 120V
Secondary: 8.5V and 9.4V
Variac (Variable Transformer)
Input: 110V
Output: 0 to 130V
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Lecture #25
EGR 272 – Circuit Theory II
Dot Convention
The direction of the windings in the secondary with respect to the direction of the
windings in the primary determines the polarity of the secondary voltage. However,
rather than showing the direction of the windings, dots are often placed at one end
of each winding. Then the following convention applies:
Dot convention: “A positive voltage at one dotted terminal will produce a
positive voltage at the other dotted terminal.”
Note that the relationships a 
N1
V
I
 1  2
N2
V2
I1
Imply that:
• the positive terminals of V1 and V2 are at the dotted terminals
• I1 enters the dotted terminal and I2 leaves the dotted terminal
a:1
These voltage polarities and
N1:N2
i1
i2
current directions are indicated
+
+
to the right.
V1
V2
_
_
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Lecture #25
EGR 272 – Circuit Theory II
Four Common Transformer Uses
• Change voltage levels
• Change current levels
• Change impedance levels (impedance matching)
• Isolation (to isolate the secondary load from the ground in the primary perhaps)
Examples: Show simple examples of the four transformer uses listed above.
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Lecture #25
EGR 272 – Circuit Theory II
Example: Determine the V1 and i2 in the circuit below. Find these values by:
A) Using KVL equations around each loop
1:2
+
20 mF 5
+
2
V1
5 cos(10t)
_
_
20 mH
i2
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Lecture #25
EGR 272 – Circuit Theory II
Example: Determine the V1 and i2 in the circuit below. Find these values by:
B) Reflecting the impedance of the secondary to the primary.
1:2
+
20 mF 5
+
2
V1
5 cos(10t)
_
_
20 mH
i2
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Lecture #25
EGR 272 – Circuit Theory II
Example: Determine the value of Zab below.
1:2
3:1
a
10
10
Zab
1:2
10
10
b
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Lecture #25
EGR 272 – Circuit Theory II
Example: Recall the example presented in EGR 271 where a transformer was used
to insure that maximum power was delivered to a 4 ohm speaker from the output of
an amplifier with an output resistance of 100 ohms. Illustrate and discuss.
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EGR 272 – Circuit Theory II
Lecture #25
Example: Determine the inductance, L, and the turns ratio, a, required to achieve
maximum power transfer to the secondary load.
a:1
+
62.5 pF 80 W
320
L
100cos(105t) V
_
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Lecture #25
EGR 272 – Circuit Theory II
Transformer Models:
Transformers considered in this
course have been considered to be
ideal. Although transformer
behavior is often close to ideal,
there are cases where non-ideal
behavior might be examined.
The figures to the left (reference:
Electric Power & Machinery by
Matsch) show models of a
transformer used to approximate
non-ideal behavior. The
resistances, R1 and R2, represent
the resistance of the windings and
the inductive reactances, Xl1 and
Xl2 represent leakage magnetic
flux.
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