Useful SWR, Voltage, and Power Equations

Here are some equations for calculating maximum voltage and current on a transmission line when the load is mismatched.  These can be used, for example, for calculating maximum flux densities in directional-coupler current and voltage sense transformer.

First, equations for the Reflection Coefficient (Gamma) and SWR:

Equations for Forward Power:

Using the above equations, we can derive equations for Forward and Reflected Voltages on a transmission line, based upon Gamma and Power delivered to Load:

And finally, equations to calculate Vmax, Vmin, Imax, and Imin:

Let's take an example...

I want to design a directional coupler so that the flux densities of its current and voltage sense transformers don't cause the ferrite cores to overheat (see this blogpost: Calculating Directional Coupler Flux Densities) for a given maximum load power and a maximum SWR.

These flux densities are a function of |Vmax| for the voltage-sense transformer and |Imax| for the current-sense transformer. I want to calculate these values using my known values of  load power and SWR.

Let's say that the maximum power I'll deliver to my load is 100 watts to my load and that the load's  SWR is 4:1.

First, I'll calculate the magnitude of Gamma, the Reflection Coefficient:

|Γ| = (SWR - 1)/(SWR + 1) = 3/5 = 0.6

Next, I'll calculate the magnitude of the forward voltage on the transmission line, given the 100 watts  being delivered to (and dissipated by) the load, a transmission line Zo of 50 ohms, and |Γ| equal to  0.6.

Vf = ((Zo*Pload) / (1 - Γ^2))^0.5

Vf = ((50*100) / (1-(0.6^2))^0.5

Vf = 88.39 V

(Note that Γ^2 and |Γ|^2 are equivalent).

Next, using Vf , |Γ|, and Zo, I can calculate |Vmax| and |Imax|:

|Vmax| = (1 + |Γ|)*Vf = (1+0.6)*88.39 = 141.4 V

|Imax| = |Vmax|/Zo = 141.4/50 = 2.83 A

With these values I can then calculate my transformer maximum flux densities.

Other Transmission-Line Posts:

http://k6jca.blogspot.com/2021/02/antenna-tuners-transient-and-steady.html.  This post analyzes the transient and steady-state response of a simple impedance matching system consisting of a wide-band transformer.  I calculate the system's impulse response and find the time-domain response by convolving this impulse-response with a stimulus signal.

http://k6jca.blogspot.com/2021/02/the-quarter-wave-transformer-transient.html.   This post analyzes the transient and steady-state response of a Quarter-Wave Transformer impedance matching device.  I calculate the system's impulse response and find the time-domain response by convolving this impulse-response with a stimulus signal.

http://k6jca.blogspot.com/2021/03/useful-swr-voltage-and-power-equations.html.  This post lists (in an easily accessible location that I can find!) some equations that I find useful

http://k6jca.blogspot.com/2021/05/antenna-tuners-lumped-element-tuner.html.  This post analyzes the transient and steady-state reflections of a lumped-element tuner (i.e. the common antenna tuner).  I describe a method for making these calculations, and I note that the tuner's match is independent of the source impedance.

http://k6jca.blogspot.com/2021/05/lc-network-reflection-and-transmission.html.  This post describes how to calculate the "Transmission Coefficient" through a lumped-element network (and also its Reflection Coefficient) if it were inserted into a transmission line.  

http://k6jca.blogspot.com/2021/09/does-source-impedance-affect-swr.html.  This post shows mathematically that source impedance does not affect a transmission line's SWR.  This conclusion is then demonstrated with Simulink simulations.

https://k6jca.blogspot.com/2021/10/revisiting-maxwells-tutorial-concerning.html  This posts revisits Walt Maxwell's 2004  QEX rebuttal of Steven Best's 2001 3-part series on Transmission Line Wave Mechanics.  In this post I show simulation results which support Best's conclusions.

Standard Caveat:

I might have made a mistake in my designs, equations, schematics, models, etc. If anything looks confusing or wrong to you, please feel free to comment below or send me an email.

Also, I will note:

This design and any associated information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.