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Dr. S. Tara Kalyani
WPT - 1
Suppose we have a coil made of, say, a spool of litz wire. Its equivalent circuit may be
drawn as a series RLC circuit. If we have a system of N coils, where the first coil is excited
by a voltage Ves , and the last coil is connected to a load, and all the coils in between are just
short-circuited, the system can be represented by
ZI = V
where Z refers to the impedance matrix of size N × N , I is the vector of mesh currents and
V is the vector of potential rises in each mesh. If the first mesh has the source voltage Vs and
all others are passive, the k th element of the voltage vector is Vk = Vs δ(k − 1). If √
Rk , Lk , Ck
are the resistance, inductance and capacitance of the k coil, and if Mmn = κmn Lm Ln is
the mutual inductance between the mth and the nth coils, we can write
Rm + j ωLm− ωCm
RL + RN + j ωLN − ωC1N
jωκmn Lm Ln
m 6= n
In the above, we assume that the N th mesh has the load resistance RL . Solving (1) we get
the mesh currents. From the current in the N th mesh, which we can write as IeL we can get
the load power as PL = |IeL |2 RL . Next let us consider two different cases. A two coil system
and a three-coil system. The circuits are drawn below.
Figure 1: (a) A two-coil WPT system (b) A three-coil WPT system
Here is the work for our UG students. For both the circuits, they should derive expressions
for the driving point admittance Yin seen by Vs . If VeS = 1∠0◦ , Ie1 = Yin . Look at its imaginary
part and see for what frequencies it is zero. Then develop expressions for PL . Find all its
maxima – global maximum and all secondary maxima, which occur at resonances. This is
the first exercise. Due: Mar 9, 2016.
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