Frǝd's Ham Radio Scratchpad - W6BSD¶
SWR along the line animation: vswr_wave_0.html
What about SWR¶
Mathematically we calculate SWR by determining the reflection coeficient gamma ($\Gamma$) which is a function of the load impedance ($Z_L$) and the source impedance ($Z_0$). These 2 impedances are complex number.
$$ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$or
$$ \Gamma = \frac{V_{rev}}{V_{fwd}} $$or
$$ \Gamma = \frac{VSWR+1}{VSWR-1} $$VSWR is them calculated using $\Gamma$ in the following equation:
$$ VSWR = \frac{1+|\Gamma|}{1-|\Gamma|} $$In our example above we have the forward voltage is 20V, and the reverse voltage is 15V:
$$ \begin{align} \Gamma &= \frac{10}{20} && = 0.5 \\ \nonumber VSWR &= \frac{1+0.5}{1-0.5} && = 3 \end{align} $$We can calculate the SWR from the Return Loss: $$ VSWR = \frac{10^{\frac{RL}{20}}+1}{10^{\frac{RL}{20}}-1} $$
The following equation is especially useful when using a directional power meter. It allows you to calculate the VSWR from the forward and reverse power.
$$ VSWR = \frac{1+\sqrt{\frac{P_{ref}}{P_{fwd}}}}{1-\sqrt{\frac{P_{ref}}{P_{fwd}}}} $$The return loss ($R_L$) expressed in $dB$ is normally calculated as follows: $$ R_L = -10 \log_{10}{\left(\frac{P_{ref}}{P_{fwd}}\right)} $$
It is them possible to calculate the Return loss ($R_L$) from the VSWR using the equation: $$ R_L = -20 \log_{10}{\left(\frac{VSWR+1}{VSWR-1}\right)} $$
Q Factor of an inductor¶
$$ Q=\frac{X_L}{R} $$or
$$ \begin{align} Q &= \frac{2{\pi}fL}{R}\\ \nonumber &= \frac{2{\pi} \times 14.15 \times {10^6} \times 220 \times {10^{-9}}}{.05} \\ \nonumber &= 391 \end{align} $$LC Q factor equations¶
Series LC¶
$$ Q = \frac{1}{R} \times \sqrt{\frac{L}{C}} $$Parallel LC¶
$$ Q = R\times\sqrt{\frac{C}{L}} $$Power: dBm -> Watt¶
$$ \begin{align} P_{mW} &= 10^{\left(\frac{P_{dBm}}{10}\right)} \\ \nonumber P_{W} &= 10^{-3} \times 10^{\left(\frac{P_{dBm}}{10}\right)} \end{align} $$Power: Watt -> dBm¶
$$ \begin{align} P_{dBm} &= 10 \times \log_{10}(P_{W} \times 10^{-3}) \\ \nonumber &= 10 \times \log_{10}(P_{W}) + 30 \end{align} $$Volt -> dBm¶
$$ P_{dBm} = 10 \times \log_{10}\left(\frac{V_{rms}*1000}{Z}\right) $$dBm -> Voltage¶
$$ V = {\sqrt {R \times {\frac {10^{\frac{P_{dBm}}{10}}}{1000}}}} $$Antenna Tuning¶
To determine the target length of the antenna elements:
- Measure the current length and the resonant frequency
- Apply the following formula to determine the new length
Example¶
- Length of an element 19.37m
- Frequency measured = 3.6MHz
- Target Frequency = 3.75MHZ
- The new length for each element is 18.6 meter