Reflection coefficient in a microwave amplifier

Question

I have a question about the reflection coefficient in a microwave amplifier (two-port network), what is it and what is it based on?

Answer 1

what is it and what is it based on?

The reflection coefficient (\$\Gamma\$) numerically indicates how different an impedance is to a "reference" impedance. The "reference" impedance is usually 50 Ω or 75 Ω or any value used in cable signal transmission. So, if the cable is designed to have a characteristic impedance of 50 Ω we might be interested in knowing how different a load impedance might be compared to 50 Ω.

If the cable is of significant length, then we could get a destructive signal reflection. Reflection coefficient (in its simplest non-complex form) is a number between -1 and +1 that represents how different a load or source impedance is to the transmission-line characteristic impedance (usually a resistive value).

To get to the root meaning of reflection coefficient we can set up a thought experiment: -

• Assume a transmission line (t-line) of characteristic impedance \$Z_0\$
• Assume an applied voltage (\$V_F\$) at one end of the line
• The current (\$I_F\$) that (initially) flows into the t-line equals \$V_F\$ divided by \$Z_0\$

When the voltage and accompanying current reach the end of the t-line and meet \$Z_L\$ there will be a violation of ohm's law if \$Z_0\$ does not equal \$Z_L\$. We have to fix this violation. The mathematical process of fixing of this violation uncovers the meaning of the reflection coefficient.

For instance, if \$Z_L\$ > \$Z_0\$ we have to consider a mechanism that prevents the violation of ohm's law. The options are: -

• Somehow make the voltage arriving at \$Z_L\$ a bit bigger and, at the same time
• Somehow make the current arriving at \$Z_L\$ a bit smaller
• We "adjust" voltage and current in such a way so as to produce a ratio of \$Z_L\$

Or, algebraically we could say: -

$$\dfrac{V_F + \delta V_F}{I_F - \delta I_F} = Z_L$$

$$\therefore \dfrac{V_F}{I_F}\cdot \dfrac{1 + \delta}{1 - \delta} = Z_L\longrightarrow Z_0\cdot \dfrac{1 + \delta}{1 - \delta} = Z_L$$

$$\text{Hence,}\hspace{1cm}\delta Z_0 +\delta Z_L = Z_L - Z_0$$

$$\text{And,}\hspace{1cm}\delta = \dfrac{Z_L-Z_0}{Z_L+Z_0}$$

But, of course, we call \$\delta\$ by it's usual name (reflection coefficient) \$\Gamma\$. \$\delta\$ is just a device I invented to get through the thought experiment.

However, the important subtlety that prevents an ohm's law violation is the "bit" we add to voltage and the "bit" we subtract from current (\$\delta V_F\$ and \$\delta I_F\$). If we examined their ratio we would find it is \$Z_0\$. This means that they can naturally flow (together) back into the transmission line because they have the perfectly correct ratio to do so.

That is called a reflection and travels from load to source.

Clearly, if \$V_F\$ and \$I_F\$ were originally of a ratio that matched the load (\$Z_0\$) impedance (right from the start), we wouldn't need to set up the algebra that figured out how to deal with the unwanted signals and, there would be no thoughts of violating ohm's law nor talk of reflections.