Loss in Transmission Line Corners

The behavior of the transmission line loss changes depending on the process corner.

The per-unit length resistance, inductance, conductance, and capacitance are denoted by the symbols R, L, G, and C, respectively. These parameters are frequency dependent. The characteristic impedance of the transmission line is:

$$\text{Characteristic impedance, }{Z}_0=\frac{\sqrt{R+j\omega L}}{\sqrt{G+j\omega C}}$$

In the loss-loss limit, the characteristic impedance reduces to:

$$Z_0=\sqrt{\frac{L}{C}}$$

The propagation constant is:

$$\text{Propagation constant, }\gamma =\alpha +j\beta =\sqrt{(R+j\omega L)}\sqrt{(G+j\omega C)}$$

For low-loss transmission lines, the equation simplifies to:

$$\begin{array}{c}\gamma =\alpha +j\beta =\sqrt{(R+j\omega L)}\sqrt{(G+j\omega C)}\\ =j\omega \sqrt{LC}\sqrt{\left(1+\frac{R}{j\omega L}\right)}\sqrt{\left(1+\frac{G}{j\omega C}\right)}\\ \approx j\omega \sqrt{LC}\left(1+\frac{R}{2j\omega L}\right)\left(1+\frac{G}{2j\omega C}\right)\\ \approx \frac{\sqrt{LC}}{2}\left(\frac{R}{L}+\frac{G}{C}\right)+j\omega \sqrt{LC}\end{array}$$

So, for low-loss transmission lines:

$$\begin{array}{l}\text{Propagation loss, }\alpha \approx \frac{\sqrt{LC}}{2}\left(\frac{R}{L}+\frac{G}{C}\right)\\ \text{Propagation delay, }\beta \approx j\omega \sqrt{LC}\end{array}$$

If there is no reflection, then the voltage at a point l along the transmission line is:

$$V(l)=V_0{e}^{-\gamma l}$$

In this equation, the β portion of the propagation constant γ corresponds to a delay of $$t_{pd}\equiv \sqrt{LC}l$$.

The α portion of the propagation constant γ gives a loss in dB which equals to:

$$\begin{array}{c}\text{Propagation loss, }\alpha =20{\log }_{10}{e}^{-\alpha l}\\ =-8.6859\alpha l\\ =-8.6859\frac{t_{pd}}{2}\left(\frac{R}{L}+\frac{G}{C}\right)\end{array}$$

From the equation of propagation loss, it is evident that the loss is proportional to propagation delay. This makes sense intuitively: the more time the signal spends in the transmission line, the more power can be dissipated.

The other thing to notice is that the portion of the loss due to the dielectric is G/C, which is proportional to the loss tangent. As a result, if C is increased, you need to increase G by the same amount to keep the loss tangent the same.

When defining process corners, both characteristic impedance Z₀ and propagation delay t_pd are varied. For the fast corner, Z₀ is increased (usually by a factor of 1.1) and t_pd is decreased (usually by a factor of 0.9). For the slow corner, the factors are reversed.

You can derive L and C from Z₀ and t_pd:

$$\begin{array}{l}L=t_{pd}{Z}_{{}_0}\\ C=\frac{t_{pd}}{Z_{{}_0}}\end{array}$$

From these expressions of L and C, you can see that for both the fast and slow corners, L is multiplied by 0.99 (and so changes very little). For the fast corner, C is multiplied by 9/11, decreasing nearly 20%. For the slow corner, C is multiplied by 11/9, increasing nearly 20%. To keep the loss tangent constant, G needs to change by the same amount

The loss depends linearly on t_pd. So for the slow corner, the loss in dB increases by 10%. For the fast corner, the loss decreases by the same amount.

If there is reflection in the transmission line, changing Z₀ changes the reflection coefficient, and a mismatch in impedance adds ripple to the loss versus frequency curve.