Lecture 11 - Linear Transformer

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Lecture 11
Chapter 3: Linear transformer
Electric Circuits II
Diego Mej́ıa Giraldo
March 8, 2017
Energy in a Coupled Circuit
i1
L1
+
−
v1
+
−
v2
i2
L2
M
Let’s compute the total energy stored in the circuit:
I Maintaining i2 = 0, increase i1 from 0 to I1. Power
applied to coil 1 is p1 = v1i1. Then, energy stored in coil
1 is
w1 =
∫
p1dt =
∫ I1
0
i1L1di1 =
1
2
L1I
2
1
I Maintain i1 = I1 = constant, increase i2 from 0 to I2.
Power applied to coil 2 is
p2 = v2i2 + v
ind
1 I1 = L2i2di2/dt + M12I1di2/dt. Then,
energy stored in coil 2 is
w2 =
∫
p2dt =
∫ I2
0
(v2i2 + M12I1) di2 =
1
2
L2I
2
2 +M12I1I2.
I Total energy stored in the circuit: w = w1 + w2
w =
1
2
L1I
2
1 +
1
2
L2I
2
2 + M12I1I2
Energy in a Coupled Circuit
i1
L1
+
−
v1
+
−
v2
i2
L2
M
Let’s compute the total energy stored in the circuit:
I If we first feed coil 2 and then coil 1, we obtain
w = w1 + w2 as
w =
1
2
L1I
2
1 +
1
2
L2I
2
2 + M21I1I2
I Therefore,
M12 = M21 = M
Since dot convention affects the induced voltage polarity, so will
the energy. Energy stored can be expressed as:
w =
1
2
L1i
2
1 +
1
2
L2i
2
2 ±M21i1i2
where i1 and i2 represent the instantaneous currents.
Question: is w positive for every value of M?
Prove that M ≤
√
L1L2 guarantees w ≥ 0
Energy in a Coupled Circuit
i1
L1
+
−
v1
+
−
v2
i2
L2
M
Let’s compute the total energy stored in the circuit:
I If we first feed coil 2 and then coil 1, we obtain
w = w1 + w2 as
w =
1
2
L1I
2
1 +
1
2
L2I
2
2 + M21I1I2
I Therefore,
M12 = M21 = M
Since dot convention affects the induced voltage polarity, so will
the energy. Energy stored can be expressed as:
w =
1
2
L1i
2
1 +
1
2
L2i
2
2 ±M21i1i2
where i1 and i2 represent the instantaneous currents.
Question: is w positive for every value of M?
Prove that M ≤
√
L1L2 guarantees w ≥ 0
Energy in a Coupled Circuit
i1
L1
+
−
v1
+
−
v2
i2
L2
M
Let’s compute the total energy stored in the circuit:
I If we first feed coil 2 and then coil 1, we obtain
w = w1 + w2 as
w =
1
2
L1I
2
1 +
1
2
L2I
2
2 + M21I1I2
I Therefore,
M12 = M21 = M
Since dot convention affects the induced voltage polarity, so will
the energy. Energy stored can be expressed as:
w =
1
2
L1i
2
1 +
1
2
L2i
2
2 ±M21i1i2
where i1 and i2 represent the instantaneous currents.
Question: is w positive for every value of M?
Prove that M ≤
√
L1L2 guarantees w ≥ 0
Coupling coefficient
We know that
M ≤
√
L1L2
But, how much M must be less than
√
L1L2?
Coupling coefficient: k =
M√
L1L2
Definition
The coupling coefficient k between two coils is the degree of their
magnetic coupling. It is a constant ∈ [0, 1].
k depends on the geometric arrangement of the coils (closeness,
orientation, windings).
Coupling coefficient
We know that
M ≤
√
L1L2
But, how much M must be less than
√
L1L2?
Coupling coefficient: k =
M√
L1L2
Definition
The coupling coefficient k between two coils is the degree of their
magnetic coupling. It is a constant ∈ [0, 1].
k depends on the geometric arrangement of the coils (closeness,
orientation, windings).
Coupling coefficient
We know that
M ≤
√
L1L2
But, how much M must be less than
√
L1L2?
Coupling coefficient: k =
M√
L1L2
Definition
The coupling coefficient k between two coils is the degree of their
magnetic coupling. It is a constant ∈ [0, 1].
k depends on the geometric arrangement of the coils (closeness,
orientation, windings).
Example
Example (Practice problem 13.3 Alexander’s book)
Determine I0.
+−100 cos (2t) V
4Ω
1
8 F
2 H 1 H 2Ω
I0
1 H
Linear transformer
Definition
A transformer has generally four terminals which contains two
magnetically coupled coils.
It is linear because coils are wound on a magnetically linear material
(constant permeability): air, plastic, Bakelite, and wood.
Note: This definition has to be revised for three-phase transformers.
+−v
R1 i1
L1 L2
R2 i2
Load
M
R1 and R2 model the power losses of the transformer.
Reflected impedance
Question: What is the input impedance Zin =
V
I1
?
Example
Example (Assessment problem 9.14 Nilsson’s book)
A linear transformer couples a load consisting of a 360 Ω resistor in series with
a 0.25 H inductor to a sinusoidal voltage source, as shown. The voltage source
has an internal impedance of 184 + j0 Ω and a maximum voltage of 245.2 V,
and it is operating at 800 rad/s. The transformer parameters are R1 = 100 Ω,
L1 = 0.5 H, R2 = 40 Ω, L2 = 0.125 H, and k = 0.4. Calculate (a) the reflected
impedance; (b) the primary current; and (c) the secondary current.
+−Vs
ZS a
R1
jωL1
b
jωL2
R2
ZL
jωM
Equivalent T circuit of a linear transformer
Please, develop a mathematical model in which a linear
transformer can be replaced by a set of inductances that are not
magnetically coupled.
??