1. Magnetically Coupled Networks

2. Magnetically Coupled Networks
A new four-terminal element, the transformer, is introduced in this chapter
A transformer is composed of two closely spaced inductors, that is, two or more magnetically coupled coils
primary side is connected to the source
secondary side is connected to the load

4. Dot Convention
dot convention: dots are placed beside each coil (inductor) so that if the currents are entering (or leaving) both dotted terminals, then the fluxes add
right hand rule says that curling the fingers (of the right hand) around the coil in the direction of the current gives the direction of the magnetic flux based on the direction of the thumb
We need dots on the schematic to know how the coils are physically oriented out one another

6. Mutually Coupled Coils
The following equations define the coupling between the two inductors assuming that each respective current enters the dot side which is also the positive voltage side where L1 and L2 are the self-inductances of the coils (inductors), and M is the mutual inductance between the two coils
i1(t) + –
v1(t) L1
i2(t)
v2(t) L2 M

7. EXAMPLE
Write a set of mesh-current equations that describe the circuit shown in terms of the currents i1, i2, and i3

8. The second mesh equation is
(In the following set of mesh-current equations, voltage drops appear as positive quantities on the right-hand side of each equation.)
Summing the voltages around the first mesh yields
The second mesh equation is
The third mesh equation is

9. Mutually Coupled Coils (AC)
The frequency domain model of the coupled circuit is essentially identical to that of the time domain

10. Source Input Impedance Linear Transformer
The source sees an input impedance, Zi, that is the sum of the primary impedance, and a reflected impedance, ZR, due to the secondary (load) side L1 L2 M ZL + – VS Z

11. THE LINEAR TRANSFORMER
source
load ZS ZL I1 I2 R1 R2
jωL1
jωL2 Vs a b c d
jωM
source
Load

14. REFLECTED IMPEDANCE

15. EXAMPLE The parameters of a certain linear transformer are R1 = 200 Ω,
R2 = 100 Ω, L1 = 9 H, L2 = 4 H, and k = 0.5. The transformer
couples an impedance consisting of an 800 Ω resistor in series with a 1 µF capacitor to a sinusoidal voltage source. The 300 V (rms) source has an internal impedance of j100 Ω and a frequency of 400 rad/s.
a) Construct a frequency-domain equivalent circuit of the system.
b) Calculate the self-impedance of the primary circuit.
c) Calculate the self-impedance of the secondary circuit.
d) Calculate the impedance reflected into the primary winding.
e) Calculate the scaling factor for the reflected impedance.
f) Calculate the impedance seen looking into the primary terminals of the transformer.
g) Calculate the rms value of the primary and secondary current.
h) Calculate the rms value of the voltage at the terminals of the load and source.
i) Calculate the average power delivered to the 800 Ω resistor.
j) What percentage of the average power delivered to the transformer is delivered to the load?

16. S 0 L U T I 0 N
a) frequency-domain equivalent circuit of the system

17. c). The self-impedance of the secondary circuit.
d). The impedance reflected into the primary winding.
e). The scaling factor by which Z22* is reflected is 8/9

18. f). The impedance seen looking into the primary terminals of the transformer is the impedance of primary winding plus the reflected impedance, thus
g). Calculate I1 and I2

20. j). The average power delivered to the transformer is
Therefore

21. Energy Analysis
An energy analysis of the mutually coupled inductors provides an expression for the instantaneous stored energy
The sign is positive (+) if currents are both entering (or leaving) the dots; sign is negative (-) if currents are otherwise

23. Quantifying the Coupling
The mutual inductance, M, is in the range
The coefficient of coupling (k) between two inductors is defined as
for k > 0.5, inductors are said to be tightly coupled
for k  0.5, coils are considered to be loosely coupled

24. DRILL EXERCISE
The self-inductances of the coils shown are L1 = 5 mH and L2 = 33.8 mH. If the coefficient of coupling is 0.96, calculate the energy stored in the system in millijoules when (a) i1 = 10 A, i2 = 5 A; (b) i1 = -10 A, i2 = -5 A; (c) i1 = -10 A, i2 = 5 A; and (d) i1 = 10 A, i2 = -5 A.