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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.
We start by deriving the apparent power P_{a} 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 P_{a} in terms of the above
\[V=(1+\Gamma) V_f\]
\[V=(1+a + j b)V_f\]
\[I=(1\Gamma) I_f\]
\[I=(1a  j b)I_f\]
\[Since \: If=\frac{V_F}{Z_0}\]
\[I=(1a  j b)\frac{V_f}{Z_0}\]
\[P_a=\frac{(1+a + j b)V_f \cdot (1a  j b)(\frac{V_f}{Z_0})^*}{2}\]
\[P_a=\frac{\left  V_f \right ^2}{2Z_0^*}((1+a)(1a)+(1a)j b +(1+a)j b b^2)\]
In the case where Z_{0} is real, then real power P_{r} can be simplified to
\[P_a=\frac{\left  V_f \right ^2}{2Z_0}((1(a^2+ b^2))\]
\[P_a=\frac{\left  V_f \right ^2}{2Z_0}(1\rho^2)\]
\[Since \: \left V_r \right =\left  \Gamma \right  \cdot \left  V_f \right  =\rho \left  V_f \right \]
\[P_r=\frac{\left V_f \right ^2}{2 Z_0}\frac{\left  V_r \right ^2}{2 Z_0}=P_{fwd}P_{ref}\]
P_{a} 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 P_{fwd}, and \(\frac{\left  V_r \right ^2}{2 Z_0}\) is commonly known as P_{ref}, but P_{r}=P_{fwd}P_{ref} is true only when Z_{0} is real.
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 \(MismatchLoss=10 log_{10}\frac{PowerAvailable}{PowerTransmitted} \: dB\).
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=PfwdPref 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.
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