Power in a mismatched transmission line

This article presents a derivation of the power at a point in a transmission line in terms of ρ (the magnitude of the complex reflection coefficient Γ) and Forward Power and Reflected Power as might be indicated by a Directional Wattmeter. Mismatch Loss is also explained.

Power

We start by deriving the apparent power Pa in terms of transmission line parameters.

Apparent power at a point on a transmission line is given by the expression

$P=\frac{VI^*}{2}$

Γ is the complex reflection coefficient.

$Let\: \Gamma=a + j b$

ρ is the magnitude of Γ.

$\rho^2=\left |\Gamma\right |^2=a^2+b^2$

Now solving Pa in terms of the above

$V=(1+\Gamma) V_f$

$V=(1+a + j b)V_f$

$I=(1-\Gamma) I_f$

$I=(1-a - j b)I_f$

$Since \: If=\frac{V_F}{Z_0}$

$I=(1-a - j b)\frac{V_f}{Z_0}$

$P_a=\frac{(1+a + j b)V_f \cdot (1-a - j b)(\frac{V_f}{Z_0})^*}{2}$

$P_a=\frac{\left | V_f \right |^2}{2Z_0^*}((1+a)(1-a)+(1-a)j b +(1+a)j b -b^2)$

In the case where Z0 is real, then real power Pr can be simplified to

$P_a=\frac{\left | V_f \right |^2}{2Z_0}((1-(a^2+ b^2))$

$P_a=\frac{\left | V_f \right |^2}{2Z_0}(1-\rho^2)$

$Since \: \left |V_r \right |=\left | \Gamma \right | \cdot \left | V_f \right | =\rho \left | V_f \right |$

$P_r=\frac{\left |V_f \right |^2}{2 Z_0}-\frac{\left | V_r \right |^2}{2 Z_0}=P_{fwd}-P_{ref}$

Pa has both reactive and real components. We are only interested in the real component of power.

The quantity $$\frac{\left | V_f \right |^2}{2 Z_0}$$ is commonly known as Pfwd, and $$\frac{\left | V_r \right |^2}{2 Z_0}$$ is commonly known as Pref, but Pr=Pfwd-Pref is true only when Z0 is real.

Mismatch Loss

Mismatch Loss occurs at an interface where the load impedance is not equal to the complex conjugate of the source impedance. $$MismatchLoss=\frac{PowerAvailable}{PowerTransmitted}$$ and it can be expressed in dB $$MismatchLoss=10 log_{10}\frac{PowerAvailable}{PowerTransmitted} \: dB$$.

Practical power measurement

It is the impedance Zo for which the Directional Wattmeter is calibrated that is important, not the Zo of the transmission line external to the instrument, or even the internal line to a certain extent (neither of which are perfectly real). If the Directional Wattmeter’s sampling element is calibrated for zero Pref on a practical resistive termination at the end of a very short transmission line, the error in assuming that P=Pfwd-Pref is very small, insignificant wrt typical standard error of RF power measurement.

For example, if a Bird 43 calibrated for 50Ω is inserted in a 75Ω line, Pfwd reads 100W and Pref reads 10W, the power (ie rate at which energy flows past that point) is 90W.

Obviously, as Pref approaches Pfwd, the standard error of the calculated power increases, and the technique is of limited use in extreme VSWR cases.

Conclusions

• A Directional Wattmeter can be used to determine power in a transmission line:
• if the calibration impedance is real (purely resistive);
• even if the impedance at the point it is sampling is not equal to its calibration impedance;
• a known resistive load is not necessary; and
• power equals Forward Power less Reflected Power.