Twisted pair

The twisted pair configurations as shown in fig. 13.2 provides good low frequency shielding. Undesired signals tend to be coupled equally into eachline of the pair. A differential receiver will therefore completely cancel the interference.

Figure 13.2: twisted pair configuration

Quasi-static model

According to P. Lefferson [58] the characteristic impedance and effective dielectric constant of a twisted pair can be calculated as follows.

$$Z_L = \dfrac{Z_{F0}}{\pi\cdot\sqrt{\varepsilon_{r,eff}}}\cdot\textrm{acosh}\left(\dfrac{D}{d}\right)$$ (13.7)

$$\varepsilon_{r,eff} = \varepsilon_{r,1} + q\cdot\left(\varepsilon_r - \varepsilon_{r,1}\right)$$ (13.8)

with

$$q = 0.25 + 0.0004\cdot \theta^2 \;\;\;\; \textrm{ and } \;\;\;\; \theta = \textrm{atan}\left(T\cdot\pi\cdot D\right)$$ (13.9)

whereas $\theta$ is the pitch angle of the twist; the angle between the twisted pair's center line and the twist. It was found to be optimal for $\theta$ to be between 20and 45. $T$ denotes the twists per length. Eq. (13.9) is valid for film insulations, for the softer PTFE material it should be modified as follows.

$$q = 0.25 + 0.001\cdot \theta^2$$ (13.10)

Assuming air as dielectric around the wires yields 1's replacing $\varepsilon_{r,1}$ in eq. (13.8). The wire's total length before twisting in terms of the number of turns $N$ is

$$l = N\cdot\pi\cdot D\cdot\sqrt{1 + \dfrac{1}{\tan^2{\theta}}}$$ (13.11)

Transmission losses

The propagation constant $\gamma$ of a general transmission line is given by

$$\gamma = \sqrt{\left(R' + j\omega L'\right)\cdot \left(G' + j\omega C'\right)}$$ (13.12)

Using some transformations of the formula gives an expression with and without the angular frequency.

$$\begin{split}\gamma &= \sqrt{\left(R' + j\omega L'\right)\cdo... ...L'} + \dfrac{G'}{C'}\right)^2 + \dfrac{R'G'}{L'C'}} \end{split}$$ (13.13)

For high frequencies eq.(13.13) can be approximated to

$$\gamma \approx \sqrt{L'C'}\cdot\left(\dfrac{1}{2}\cdot\left(\dfrac{R'}{L'} + \dfrac{G'}{C'}\right) + j\omega\right)$$ (13.14)

Thus the real part of the propagation constant $\gamma$ yields

$$\alpha = Re\left\{\gamma\right\} = \sqrt{L'C'}\cdot\dfrac{1}{2}\cdot\left(\dfrac{R'}{L'} + \dfrac{G'}{C'}\right)$$ (13.15)

With

$$Z_L = \sqrt{\dfrac{L'}{C'}}$$ (13.16)

the expression in eq.(13.15) can be written as

$$\alpha = \alpha_c + \alpha_d = \dfrac{1}{2}\cdot\left(\dfrac{R'}{Z_L} + G'Z_L\right)$$ (13.17)

whereas $\alpha_c$ denotes the conductor losses and $\alpha_d$ the dielectric losses.

Conductor losses

The sheet resistance R' of a transmission line conductor is given by

$$R' = \dfrac{\rho}{A_{eff}}$$ (13.18)

whereas $\rho$ is the specific resistance of the conductor material and $A_{eff}$ the effective area of the conductor perpendicular to the propagation direction. At higher frequencies the area of the conductor is reduced by the skin effect. The skin depth is given by

$$\delta_s = \sqrt{\dfrac{\rho}{\pi\cdot f\cdot\mu}}$$ (13.19)

Thus the effective area of a single round wire yields

$$A_{eff} = \pi\cdot\left(r^2 - (r-\delta_s)^2\right)$$ (13.20)

whereas $r$ denotes the radius of the wire. This means the overall conductor attenuation constant $\alpha_c$ for a single wire gives

$$\alpha_c = \dfrac{R'}{2\cdot Z_L} = \dfrac{\rho}{2\cdot Z_L\cdot\pi\cdot\left(r^2 - (r-\delta_s)^2\right)}$$ (13.21)

Dielectric losses

The dielectric losses are determined by the dielectric loss tangent.

$$\tan{ \delta_d } = \dfrac{G'}{\omega C'} \;\;\;\; \rightarrow \;\;\;\; G' = \omega C' \cdot \tan{ \delta_d }$$ (13.22)

With

$$C' = \dfrac{1}{\omega}\cdot Im \left\{\dfrac{\gamma}{Z_L}\right\}$$ (13.23)

the equation (13.22) can be rewritten to

$$\begin{split}G' &= \dfrac{\beta}{Z_L}\cdot \tan{ \delta_d } =... ...{r,eff}}}{\lambda_0\cdot Z_L}\cdot \tan{ \delta_d } \end{split}$$ (13.24)

whereas $v_{ph}$ denotes the phase velocity, $c_0$ the speed of light, $\varepsilon_{r,eff}$ the effective dielectric constant and $\lambda_0$ the freespace wavelength. With these expressions at hand it is possible to find a formula for the dielectric losses of the transmission line.

$$\alpha_d = \dfrac{1}{2}\cdot G'Z_L = \dfrac{\pi\cdot \sqrt{\varepsilon_{r,eff}}}{\lambda_0}\cdot \tan{ \delta_d }$$ (13.25)

Overall losses of the twisted pair configuration

Transmission losses consist of conductor losses, dielectric losses as well as radiation losses. The above expressions for the conductor and dielectric losses are considered to be first order approximations. The conductor losses have been derived for a single round wire. The overall conductor losses due to the twin wires must be doubled. The dielectric losses can be used as is. Radiation losses are neglected.