Coupled Resonators

Topics

• Mutual inductance of two shielded-loop resonators

• Effects of distance and alignment on mutual inductance

• Critical coupling distance

• Full-wave bridge rectifiers

Fig. 39 Coupled Shielded-Loop Resonator with Load

Calculating Mutual Inductance

Fig. 40 Mutual Inductance for Axially Aligned Coupled Loops

\[ \begin{align*} M = \mu \sqrt{r_1 r_2}\bigg((\dfrac{2}{k} - k)\ K(k) - \dfrac{2}{k}E(k) \bigg) \end{align*} \]

where \(K(k)\) is the complete elliptical integral with radius \(r_1\) and \(E(k)\) is the complete elliptical integral with radius \(r_2\).

\[\begin{split} \begin{align*} K(k) &= \int_{0}^{\frac{\pi}{2}} \dfrac{d \beta}{\sqrt{1-k^2 sin^2 \beta}} \\[0.5em] E(k) &= \int_{0}^{\frac{\pi}{2}} d\beta \sqrt{1-k^2 sin^2 \beta} \\[0.5em] k &= \sqrt{\dfrac{4r_1 r_2}{(r_1 + r_2)^2 + d^2}} \end{align*} \end{split}\]

Input Impedance of Coupled Shielded-Loop Resonators

At resonance, the input impedance to the coupled shielded-loop simplifies to

\[\begin{split} \begin{align*} Z_{in} &= \dfrac{v_x}{i_1} = R_1 + j\bigg(\omega L_1 - \dfrac{1}{\omega C_1} \bigg) + \dfrac{(\omega M)^2}{R_2 + R_L + j\bigg(\omega L_2 - \dfrac{1}{\omega C_2} \bigg)} \bigg|_{\ \omega \ = \ \omega_0} \\[0.5em] Z_{in}(\omega_0) &= R_{in}(\omega_0) = R_1 + \dfrac{(\omega_0 M)^2}{R_2 + R_L} \end{align*} \end{split}\]

Feedline Effects

The load impedance \(Z_L\) is designed to the characteristic impedance \(Z_0\) of the transmission line to eliminate feedline effect and reduce the circuit model.

\[Z_L = Z_0 = 50 \Omega\]

Fig. 41 Impedance Simplification

Weak, Critical and Strong Coupling

Isolated loop resonance occurs when

\[\omega = \omega_0 = \dfrac{1}{\sqrt{LC}} \]

Coupled loop resonance also occurs when

\[\bigg(\omega L - \dfrac{1}{\omega C} \bigg)^2 = (\omega M)^2 - (R + R_L)^2 \]

Solving this quadratic equation results in multiple solutions depending on the mutual inductance as a function of distance.

Strong coupling

\[\omega M > R + R_L\]

• Odd mode solution: frequency is slightly less than \(\omega_0\) and the currents in the coupled loops are 180° out-of-phase

• Even mode solution: frequency is slightly greater than \(\omega_0\) and the currents in the coupled loops are in-phase

• Resonant frequency: \(\omega_0\) the current in the load loop leads that of the source loop by approximately 90°

Critical coupling

\[\omega M = R + R_L\]

The even and odd mode frequencies merge into the resonant frequency \(\omega_0\)

Weak coupling

\[\omega M < R + R_L\]

In weak coupling, there is only one resonant frequency \(\omega_0\)

Power Transfer

The power transfer efficiency of a wireless power transfer system is the ratio of power delivered to the load \(P_L\) and the power available from the source \(P_A\).

\[P_A = \dfrac{|V_S|^2}{8Re\{Z_S\}}\]

• \(Z_S\) characteristic impedance of the voltage source where \(Re\{Z_S\} = R_S\)

• \(V_S\) peak voltage of the source

• \(P_A\) power available from the source if the source impedance is terminated in an impedance equal to its complex conjugate

For a system operating at critical or weak coupling, the power transfer efficiency is

\[\eta = \dfrac{4R_L^2(\omega_0 M)^2}{\bigg((R+R_L)^2 + (\omega_0 M)^2\bigg)^2}\]

For a system operating at strong coupling (even or odd), the power transfer efficiency is

\[\eta = \dfrac{R_L^2}{(R+R_L)^2}\]

For matched source and load impedances (\(50 \Omega\)), the power transfer efficiency is

\[\eta = |S_{21}|^2\]

Voltage Rectification

Fig. 42 Full Wave Bridge Rectifier

In this circuit, the negative cycle of the input AC signal is not blocked. Instead it is converted to a positive value for the RC filter to “smooth”

Given metrics since vector analyzer has maximum output power of 0 dBm.

	Input Return Loss (dB)	Input Impedance (\(\Omega\))
Rectifier at \(\omega = \omega_0\)	\(-0.486\)	\(9.221 - j119.4\)

Measurements

5 cm loop and \(\omega = \omega_0\)

Distance (cm)	\(\Gamma_{in}' (mU)\)	\(S_{21} (dB)\)	\(R_{in}\)	\(M (calc.)\)	\(\eta (calc.)\)
2
4
6
8
10
12
14
16

9 cm loop and \(\omega = \omega_0\)

Distance (cm)	\(\Gamma_{in}' (mU)\)	\(S_{21} (dB)\)	\(R_{in}\)	\(M (calc.)\)	\(\eta (calc.)\)
2
4
6
8
10
12
14
16