Inductive wireless power transfer - lossless coils with primary and secondary circuits tuned to same resonant frequency

Prepared by Dr. Aaron Scher
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Introduction

Here we present the design equations for an inductive wireless power system consisting of lossless coils with primary and secondary circuits tuned to the same resonant frequency.

System description and design equations

Consider two resonant RLC circuits magnetically coupled as shown in Figure 1. Both circuits resonate at the same frequency. The coils and capacitors are assumed to be lossless. The generator is connected to the primary coil (coil #1) and is modelled as an ideal voltage source Vg in series with an internal resistance Rg. The load has resistance RL and is connected to the secondary coil (coil #2). The (real) power available from the source is Pavg=(Vg)^2/(8*Rg). The goal of this circuit is to deliver this power to the load. The condition for maximum power transfer is shown in the figure. Click here for a derivation of the design equations.

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Figure 1. Lossless resonant circuits tuned to same frequency and condition for maximum power transfer.

Example 1

Problem Setup

In this example, we will design a simple inductive wireless power system. The source resistance is 5 Ohms and the load resistance is 5 Ohms. The inductors L1 = L2 = 150 uH. The frequency of operation is 50 kHz. What should be the value of the capacitors? What should be the coupling coefficient?

Solution

Spice simulation

Figure 2 shows the Spice schematic. Figure 3 shows plots of gain G of the system (G = Pload/Pgenerator) over frequency for three different values of the coupling coefficients k. Figure 4 demonstrates how component tolerances could easily degrade performance in this system. Here we plot the gain G of the system for k= 0.1061 (critical coefficient of coupling), and vary the inductance of circuit 2 (L2), while leaving everything else the same.

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Figure 2. Spice screenshot for Example 1.
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Figure 3. Power gain versus frequency for different values of k.
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Figure 4. The resonant frequencies of circuits 1 and 2 must be exactly matched for total power transfer.

Example 2

Problem Setup

In this example, we will design another inductive wireless power transfer system. The source resistance is 50 Ohms and the load resistance is 50 Ohms. The inductors L1 = L2 = 150 uH. The frequency of operation is 50 kHz. The coupling coefficient is specified to be k = 0.07.

Solution using tapped-capacitor impedance transformer network

This problem is different than Example 1, because not only are the source and load resistances given, but also the coupling coefficient k. To solve this problem, we are going to have to add additional impedance matching circuitry on both the input and output side of the basic inductive wireless power transfer system shown in Figure 1. Let us ignore for a moment that the source and load resistances are specified to be 50 Ohms, and find the proper load/source resistance R' that corresponds to k = 0.07. Assuming both circuits 1 and 2 are equal, such that Q1=Q2=Q= ω0L/R', where L = 150 uH, we can solve for R':

k=1/sqrt(Q1*Q2)=1/Q=R'/(Lω0) → R'=kLω0 → R'=3.2987 Ohms

Figure 5 shows the prototype wireless power transfer system with both source and load resistances equal to R'=3.2987 Ohms. In this figure we also demonstrate the use of the "tapped capacitor" impedance transforming network for transforming R' into the required 50 Ohms. This tapped capacitor transformation is accomplished using the series RC to partial parallel RC transformation presented in Series-to-parallel impedance transformation. Figure 6 shows the Spice schematic of our final circuit and Figure 7 shows the gain (simulated in Spice).

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Figure 5. Transforming the source and load resistances using tapped-capacitors for Example 2
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Figure 6. Spice screenshot for network shown in Figure 5.
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Figure 7. Spice simulation results for Figure 6: power gain versus frequency

Solution using magnetically-coupled inductor transformer network

There are many alternative options to the tapped-capacitor impedance matching network presented above. Figure 9 demonstrates the use of magnetically coupled inductors for transforming R'=3.2987 Ohms into the required 50 Ohms. Figure 10 shows the Spice schematic of our final circuit, and Figure 11 shows the gain (simulated in Spice).

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Figure 9. Alternative approach to Example 2.
Transforming the source and load resistances using magnetically coupled inductors.
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Figure 10. Spice screenshot for network shown in Figure 9.
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Figure 11. Spice simulation results for Figure 10: power gain versus frequency

References

[1]F. E. Terman, Electronic and Radio Engineering, McGraw-Hill Inc., NY, 1955. See Section 3-5.