**3. Magnetic field coupling with resonance**

Compared with electromagnetic induction, magnetic resonance coupling utilizes resonance with the series capacitor in primary and secondary sides to approach high efficiency and high power (**Figure 5**). Based on Kirchhoff's voltage law, Eqs. (18) and (19) can be obtained as below [16]:

$$\frac{1}{\text{joC}\_1} \mathbf{I}\_1 + \mathbf{R}\_1 \mathbf{I}\_1 + \text{joL}\_1 \mathbf{I}\_1 + \text{joL}\_m \mathbf{I}\_2 = \mathbf{U}\_1 \tag{18}$$

$$\text{joL}\_2\text{I}\_2 + \text{R}\_2\text{I}\_2 + \frac{\text{1}}{\text{jo}\text{C}\_2}\text{I}\_2 + \text{R}\_\text{L}\text{I}\_2 + \text{jo}\text{L}\_\text{m}\text{I}\_1 = \text{0} \tag{19}$$

I2 can be expressed as Eq. (20), according to Eq. (18).

**Figure 5.** *Circuit topology of magnetic field coupling with resonance.*

$$\mathbf{I}\_2 = \frac{-\text{joL}\_{\text{m}}}{\text{joL}\_2 + \frac{1}{\text{joL}\_2} + \text{R}\_2 + \text{R}\_L} \mathbf{I}\_1 \tag{20}$$

The relation between I1 and U1 can be expressed as Eq. (21), according to Eqs. (18) and (20).

$$\mathbf{I}\_{1} = \left\{ \frac{\mathbf{R}\_{2} + \mathbf{R}\_{L} + \mathbf{j}\left(\alpha \mathbf{L}\_{2} - \frac{1}{\alpha \mathbf{C}\_{2}}\right)}{\left[\left(\mathbf{R}\_{1} + \mathbf{j}\left(\alpha \mathbf{L}\_{1} - \frac{1}{\alpha \mathbf{C}\_{1}}\right)\right)\left[\mathbf{R}\_{2} + \mathbf{R}\_{L} + \mathbf{j}\left(\alpha \mathbf{L}\_{2} - \frac{1}{\alpha \mathbf{C}\_{2}}\right)\right] + \alpha^{2} \mathbf{L}\_{m}^{2}} \right] \mathbf{U}\_{1} \tag{21}$$

The relation between I2 and U1 can be expressed as Eq. (22), according to Eqs. (20) and (21).

$$\mathbf{I}\_{2} = -\left\{ \frac{\text{joL}\_{\text{m}}}{\left[ \left( \text{R}\_{1} + \text{j} \left( \text{a} \text{L}\_{1} - \frac{1}{\text{a} \text{C}\_{1}} \right) \right) \left[ \text{R}\_{2} + \text{R}\_{L} + \text{j} \left( \text{a} \text{L}\_{2} - \frac{1}{\text{a} \text{C}\_{2}} \right) \right] + \text{a}^{2} \text{L}\_{\text{m}}^{2} \right]} \right\} \mathbf{U}\_{1} \tag{22}$$

The relation between I1 and U1 can be expressed as Eq. (23), according to Eqs. (21) and (22).

$$\frac{\mathbf{I}\_1}{\mathbf{I}\_2} = -\frac{\mathbf{R}\_2 + \mathbf{R}\_L + \text{jo}\left(\text{oL}\_2 - \frac{1}{\text{oC}\_2}\right)}{\text{jo}\,\text{L}\_m} \tag{23}$$

When the operating and resonant frequencies are equal on primary and secondary sides, I1 and I2 can be expressed as (25) and (26), according to Eqs. (21), (22), and (24).

$$\alpha\_0 = \alpha\_1 = \alpha\_2 = \frac{1}{\sqrt{\mathcal{L}\_1 \mathcal{C}\_1}} = \frac{1}{\sqrt{\mathcal{L}\_2 \mathcal{C}\_2}} \tag{24}$$

$$\mathbf{I}\_{1} = \frac{\mathbf{R}\_{2} + \mathbf{R}\_{\mathrm{L}}}{\mathbf{R}\_{1}(\mathbf{R}\_{2} + \mathbf{R}\_{\mathrm{L}}) + \mathrm{o}^{2}\mathrm{L}\_{\mathrm{m}}^{-2}} \,\mathrm{U}\_{1} \tag{25}$$

$$\mathbf{I}\_2 = -\frac{\text{joL}\_{\text{m}}}{\mathbf{R}\_1(\mathbf{R}\_2 + \mathbf{R}\_L) + \text{o}^2 \mathbf{L}\_{\text{m}}^2} \mathbf{U}\_1 \tag{26}$$

The ratio of I1 and I2 can be rewritten as Eq. (27), according to Eqs. (25) and (26).

$$\frac{\mathbf{I}\_1}{\mathbf{I}\_2} = -\frac{\mathbf{R}\_2 + \mathbf{R}\_L}{\text{joL}\_m} \tag{27}$$

To further understand the input impedance and power efficiency of the magnetic field coupling with the resonance circuit, a T-type equivalent circuit is used as **Figure 6**.

$$\mathbf{V\_{Lm2}} = \mathbf{j} \mathbf{o} \mathbf{L\_m} \mathbf{I\_1} \tag{28}$$

$$\mathbf{V\_{Lm2}} = j \alpha \mathbf{L\_m} \frac{\mathbf{R\_2} + \mathbf{R\_L}}{\mathbf{R\_1}(\mathbf{R\_2} + \mathbf{R\_L}) + \alpha^2 \mathbf{L\_m}^2} \mathbf{U\_1} \tag{29}$$

Zin2 can be obtained as Eq. (30), according to Eqs. (26) and (29).

*Wireless Power Transmission on Biomedical Applications DOI: http://dx.doi.org/10.5772/intechopen.103029*

**Figure 6.** *T-type equivalent circuit for magnetic field coupling with resonance.*

$$\mathbf{Z\_{in2}} = \frac{\mathbf{V\_{Lm2}}}{-\mathbf{I\_2}} = \frac{\text{joL}\_m \mathbf{I\_1}}{-\mathbf{I\_2}} = \mathbf{R\_2} + \mathbf{R\_L} \tag{30}$$

Then, Z2 can be further obtained as Eq. (31).

$$\mathbf{Z}\_{2} = \frac{\mathbf{V}\_{\text{Lm}1}}{\mathbf{I}\_{1}} = \frac{\text{joL}\_{\text{m}}\mathbf{I}\_{2}}{\mathbf{I}\_{1}} = \frac{\text{o}^{2}\mathbf{L}\_{\text{m}}\mathbf{I}}{\mathbf{R}\_{2} + \mathbf{R}\_{\text{L}}} \tag{31}$$

Finally, the Zin1 can be acquired, according to Eq. (32).

$$\mathbf{Z\_{in1}} = \mathbf{R\_1} + \frac{\mathbf{1}}{\text{joC}\_1} + \text{joL}\_1 + \text{Z}\_2 = \mathbf{R\_1} + \frac{\mathbf{1}}{\text{joC}\_1} + \text{joL}\_1 + \frac{\mathbf{o}^2 \text{L}\_{\text{m}}^2}{\text{R}\_2 + \text{R}\_L} \tag{32}$$

To understand the power efficiency, the ratio of energy loss for the primary-side internal resistance, secondary resistance, and load can be denoted by PR1, PR2, and PRL as Eq. (14).

$$\mathbf{P\_{R\_1}} : \mathbf{P\_{R\_2}} : \mathbf{P\_L} = \left| \mathbf{I\_1} \right|^2 \mathbf{R\_1} : \left| \mathbf{I\_2} \right|^2 \mathbf{R\_2} : \left| \mathbf{I\_2} \right|^2 \mathbf{R\_L} \tag{33}$$

The square ratio of I2 and I1 can be expressed as Eq. (34), according to Eq. (27).

$$\left|\frac{\mathbf{I}\_2}{\mathbf{I}\_1}\right|^2 = \frac{\alpha^2 \mathbf{L}\_{\text{m}}^2}{\left(\mathbf{R}\_2 + \mathbf{R}\_L\right)^2} \tag{34}$$

The power ratio can be rewritten as Eq. (35), according to Eqs. (33) and (34).

$$\mathbf{P}\_{\mathbf{R}\_{1}} : \mathbf{P}\_{\mathbf{R}\_{2}} : \mathbf{P}\_{\mathbf{L}} = \left\{ \left( \mathbf{R}\_{2} + \mathbf{R}\_{\mathbf{L}} \right)^{2} \right\} \mathbf{R}\_{1} : \left\{ \mathbf{o}^{2} \mathbf{L}\_{\mathbf{m}} \right\} \mathbf{R}\_{2} : \left\{ \mathbf{o}^{2} \mathbf{L}\_{\mathbf{m}} \right\} \mathbf{R}\_{\mathbf{L}} \tag{35}$$

Finally, the power efficiency can be obtained, according to Eq. (36).

$$\eta = \frac{\mathbf{P\_L}}{\mathbf{P\_{in}}} = \frac{\mathbf{P\_L}}{\mathbf{P\_{R\_1}} + \mathbf{P\_{R\_2}} + \mathbf{P\_L}} = \frac{\left\{\mathbf{o^2 L\_{\rm m}}^2\right\} \mathbf{R\_L}}{\left\{\left(\mathbf{R\_2} + \mathbf{R\_L}\right)^2\right\} \mathbf{R\_1} + \left\{\left<\mathbf{o^2 L\_{\rm m}}^2\right\} \mathbf{R\_2} + \left<\mathbf{o^2 L\_{\rm m}}^2\right\} \mathbf{R\_L}} \tag{36}$$


**Table 1.**

*Comparison between magnetic field coupling under either no or with resonance.*

**Table 1** summarizes the results from the previous derivation in terms of the efficiency and power of the two types of magnetic field coupling circuits. With C1 and C2 inserted on the primary and secondary sides, the power efficiency and power at Rload will increase, which is why resonance is important.
