RF Mismatch Loss, Error and Uncertainty

For maximum transfer of the power from any RF source component, the total input impedance of the component connected to the source must be equal to the conjugate of the source impedance. In practice, there is always some degree of impedance mismatch between the source and connected component. The power loss due to the impedance mismatch is the mismatch loss. Mismatch loss in transmission line theory is the amount of power expressed in decibels that will not be available on the output due to impedance mismatches and reflections.

The impedance mismatch is expressed in terms of reflection coefficient $\Gamma$, voltage standing wave ratio (VSWR) and return loss (RL). These quantities are interrelated:

$$\Gamma=\frac{Z-Z_0}{Z+Z_0}$$

$$|\Gamma|=\frac{VSWR-1}{VSWR+1}$$

$$RL=-20\log_{10}|\Gamma|$$

If there are two components in cascade, the resultant mismatch loss is not only due to the mismatches from the individual components, but also from the reflections from each component and their combination. The overall mismatch loss cannot be calculated by just adding up the individual loss contributions from each component. Depending on how the multiple reflections combine, the overall system loss may be lower or higher than the sum of the mismatch loss from each component. The uncertainty associated in the evaluation of the mismatch error is known as the mismatch uncertainty and expressed in dB.

Transducer Power Transfer Efficiency

For two cascaded RF components, defined as source and load, given:

Source with output reflection coefficient $\Gamma_S$
Load with input reflection coefficient $\Gamma_L$
Source and Load are connected

Let $a$ be the incident wave toward the load and $b=\Gamma_La$ be the reflected wave from the load, then the net power delivered to the load is:

$$P_L=|a|^2-|b|^2=|a|^2(1-|\Gamma_L|^2)$$

But $a$ is not necessarily equal to the wave that the source would deliver under matched conditions. It is reduced due to reflection from the load and mismatch at the source.

$a$ can be written as a geometric series due to multiple reflections between source and load:

$$a=a_S[1+\Gamma_S\Gamma_L+(\Gamma_S\Gamma_L)^2+...]=\frac{a_S}{1-\Gamma_S\Gamma_L}$$

where $a_S$ is the wave provided by the source.

The delivered power is:

$$P_L=|a|^2(1-|\Gamma_L|^2)=|a_S|^2\cdot \frac{1-|\Gamma_L|^2}{|1-\Gamma_S\Gamma_L|^2}$$

The maximum available power from the source is the power the source would deliver if the load was conjugately matched to it:

$$P_{avail}=|a_S|^2\cdot \frac{1-|\Gamma_S|^2}{|1-\Gamma_S\Gamma_S^*|^2}=\frac{|a_S|^2}{1-|\Gamma_S|^2}$$

Therefore, the transducer power efficiency is:

$$T=\frac{P_L}{P_{avail}}=|a_S|^2\cdot \frac{1-|\Gamma_L|^2}{|1-\Gamma_S\Gamma_L|^2}\cdot \frac{1-|\Gamma_S|^2}{|a_S|^2}$$

$$T=\frac{(1-|\Gamma_S|^2)(1-|\Gamma_L|^2)}{|1-\Gamma_S\Gamma_L|^2}$$

The mismatch loss is:

$$ML=\frac{1}{T}\Rightarrow ML_{dB}=-10\log_{10}T$$

Mismatch Loss and Uncertainty

In practical cascade analysis, the phase difference between source and load reflection coefficients $\Gamma_S$ and $\Gamma_L$ is usually unknown, unless a full S-parameter or EM simulation is available.

The minimum and maximum power transfer efficiencies are:

$$T_{min}=\frac{(1-|\Gamma_S|^2)(1-|\Gamma_L|^2)}{(1+|\Gamma_S||\Gamma_L|)^2}$$

$$T_{max}=\frac{(1-|\Gamma_S|^2)(1-|\Gamma_L|^2)}{(1-|\Gamma_S||\Gamma_L|)^2}$$

Assuming the phase between reflections is unknown and uniformly distributed over $(0,2\pi)$, the typical-case or mean power transfer efficiency can be estimated.

Let:

$\gamma_S=|\Gamma_S|$
$\gamma_L=|\Gamma_L|$
$\theta=\angle(\Gamma_S\Gamma_L)$, the unknown phase combination

Then:

$$T(\theta)=\frac{(1-\gamma_S^2)(1-\gamma_L^2)}{1-2\gamma_S\gamma_L\cos(\theta)+\gamma_S^2\gamma_L^2}$$

The mean power transfer efficiency is given by:

$$\bar{T}=\frac{1}{2\pi}\int_0^{2\pi}T(\theta)d\theta=\frac{(1-\gamma_S^2)(1-\gamma_L^2)}{1-\gamma_S^2\gamma_L^2}$$

Therefore,

$$\begin{cases}T_{min}=\frac{(1-\gamma_S^2)(1-\gamma_L^2)}{(1+\gamma_S\gamma_L)^2} \\ \\ T_{typical}=\frac{(1-\gamma_S^2)(1-\gamma_L^2)}{1-\gamma_S^2\gamma_L^2} \\ \\ T_{max}=\frac{(1-\gamma_S^2)(1-\gamma_L^2)}{(1-\gamma_S\gamma_L)^2}\end{cases}$$

The minimum and maximum mismatch losses are:

$$\begin{cases}ML_{dB,min}=-20\log_{10}T_{max} \\ \\ ML_{dB, typical}=-20\log_{10}T_{typical} \\ \\ ML_{dB,max}=-20\log_{10}T_{min}\end{cases}$$

The uncertainty limits in the mismatch error is given in decibels (dB) by,

$$20\log_{10}(1\pm \gamma_S\gamma_L)$$

The power magnitude ripple in dB can be estimated as:

$$ripple_{mag}=10\log_{10}\frac{1+\gamma_S\gamma_L}{1-\gamma_S\gamma_L}$$

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RF Mismatch Loss, Error and Uncertainty

Transducer Power Transfer Efficiency

Mismatch Loss and Uncertainty

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