6.1.1 Analytical Inductance and Resistance Calculation

The analytical expression for the inductance $L$ is obtained from the magnetic energy [102]

$$E_m = \int_{\mathcal{V}}\frac{B^2}{2\mu} \mathrm{d}V = \frac{I^2... ...}V = \frac{1}{I^2}\sum_{i=1}^3\int_{\mathcal{V}_i}\frac{B^2}{\mu} \mathrm{d}V.$$ (6.3)

It must be distinguished between three regions: the inner conductor $\mathcal{V}_1$ , the dielectric between the conductors $\mathcal{V}_2$ , and the outer conductor $\mathcal{V}_3$ .

For $\mathcal{V}_1 \vert r\in[0;a]$ the magnetic flux $B$ and the integral are given by

$$B = \mu\frac{Ir}{2\pi{}a^2} \mathrm{and} \int_{\mathcal{V}... ...m{d}V = \int_0^a\mu\frac{I^2r^2}{4\pi^2a^4}l2\pi{}r dr = \mu\frac{I^2l}{8\pi}.$$ (6.4)

For $\mathcal{V}_2 \vert r\in[a;b]$

$$B = \mu\frac{I}{2\pi{}r} \mathrm{and} \int_{\mathcal{V}_2}... ...a^b\mu\frac{I^2}{4\pi^2r^2}l2\pi{}r dr = \mu\frac{I^2l}{2\pi} \ln\frac{b}{a}.$$ (6.5)

The current $I$ flows along the inner conductor. The same current returns along the outer conductor flowing in the opposite direction. Similarly to (6.4), for $\mathcal{V}_3$ where $r\in[b;c]$ , only the current through the circle inside the integration loop must be considered

$$I - \frac{I\left(\pi{}r^2 - \pi{}b^2\right)}{\pi{}c^2 - \pi{}b^2}... ...2} \mathrm{and} B = \frac{\mu{}I}{2\pi{}r} \frac{c^2 - r^2}{c^2 - b^2}.$$ (6.6)

$$\begin{split}\int_{\mathcal{V}_3}\frac{B^2}{\mu} \mathrm{d}V... ...eft(c^2 - b^2\right)\left(3c^2 - b^2\right)\right]. \end{split}$$ (6.7)

Now the inductance can be obtained from the integral over the entire domain

$$L = \frac{\mu{}l}{2\pi{}} \left\{\frac{1}{4\pi} + \ln\frac{b}{a} ... ...{b} - \frac{1}{4}\left(c^2 - b^2\right)\left(3c^2 - b^2\right)\right] \right\}.$$ (6.8)

The resistance $R$ of the conductors is given by

$$R = R_{in} + R_{out}, \mathrm{where} R_{in} = \frac{l}{\ga... ...2} \mathrm{and} R_{out} = \frac{l}{\gamma\pi\left(c^2 - b^2\right)}.$$ (6.9)

Equations (6.8) and (6.9) are obtained assuming a constant current density distribution in the conductors. This is true only for low frequencies. The distinction between low and high frequency is in terms of the skin effect. Thus, whether an operating frequency is considered as low or high depends also on the dimensions of the geometries, not only on the frequency itself. At high frequencies for which skin effect is not negligible $L$ and $R$ are modified to read [102]

$$L = \frac{\mu{}l}{2\pi} \left\{ \frac{\delta}{2a} + \ln\frac{b}{a... ...(2\frac{d}{\delta}\right) - \cos\left(2\frac{d}{\delta}\right)\right]} \right\}$$ (6.10)

$$R = \frac{a}{2\delta}R_{in} + \frac{d\left[\sinh\left(2\frac{d}{\... ...t(2\frac{d}{\delta}\right) - \cos\left(2\frac{d}{\delta}\right)\right]}R_{out},$$ (6.11)

where $R_{in}$ and $R_{out}$ are taken from (6.9), $d$ is the thickness of the outer conductor ($d = c - b$ ), and the skin depth $\delta$ is given by the expression

$$\delta = \sqrt{\frac{2}{\mu\gamma\omega}}.$$ (6.12)

Notice that (6.10) and (6.11) are valid only, if $d$ is reasonably small compared to $b$ ($d{\ll}b$ ).