Summary: This blog post examines the operation of a typical "lumpedelement" antenna tuner (for example, an Lnetwork tuner) and its effect on forward and reflected wave interactions on a transmission line.
I also analyze at the effect of source impedance (i.e. a transmitter's impedance looking into its output port) on the transmission line waves and conclude that they are only affected by the source impedance during transient conditions, but not in steadystate. For an impedance matching network that has been "tuned" to provide a match, for example, to transform a load to 50 ohms, the source impedance has no effect on the matchingnetwork's input impedance in steadystate.
I usually don't care what the impedance of an antenna tuner is looking backwards into its output port (towards the transmitter). I adjust the tuner so that its input impedance looks like 50 ohms, and with that accomplished, I'm a happy camper.
But recently I was wondering if there was a way to calculate this impedance (assuming a load attached to the tuner's output port via a transmission line) and how this impedance would affect the forward and reflected voltages on the transmission line. And how the transmitter's output impedance (i.e. the system's source impedance) might affect the value of the tuner's output impedance.
And thus this post.
This post is a continuation of two earlier posts exploring transient and
steadystate responses in Antenna Tuners (i.e. impedance matching devices)
attached to transmission lines.
The
first post
calculates the transient and steadystate responses of an Antenna Tuner that,
for ease of analysis, consists of a single wideband transformer.
The transformer, being a wideband device, allows one to easily calculate the
system's impulse response, from which the transient and steadystate responses
are calculated using the convolution integral.
The
second post
is a continuation of the first post in which a wideband matching device is
used. In this case, though, the wideband matching device is a
quarterwave transformer. The quarterwave transformer, in this sense,
is a wideband device in that its frequency response is wide band (there is no
attenuation if the transmission line is assumed to be ideal), although it only
matches impedances over specific narrow bands of frequency.
Because of its wideband frequency response, the quarterwave transformer's
impulse response (in a simple system) is easy to determine by hand and
transient and steadystate responses for a sinusoidal stimulus easily
calculated.
This new post examines the startup and steadystate wave interactions with the more
general case of a "Lumpedelement" Antenna Tuning network. The tuner
might be an Lnetwork, a Tnetwork, a PInetwork, or some other topology 
the approach described below should work for any twoport lumpedelement impedance matching
topology.
Lumpedelement networks such as these are bandlimited networks, and an accurate impulse response is significantly more painful to calculate (because an impulse is no longer an impulse after passing through such a network).
And because of this frequencydependency, a true timedomain transient response (that includes, for example network "ringing") is difficult to calculate by hand. But a "limited" transient response that simply shows how reflections and rereflections build over time on a transmission line is easier to calculate. This is the path this post will.
The goal of these posts is to demonstrate the underlying wave
mechanics that occur on the transmission lines and at the impedance matching
devices (rather than a deepdive into a rigorous mathematical derivation), and some assumptions are
made to keep the math manageable.
These assumptions are:
 Transmission lines are assumed to be lossless.

Transmission line delays (and thus their lengths) are assumed to delay a
sinewave by multiples of 90 degrees or 180 degrees, depending upon the
application. E.g. the quarterwave transformer has a length that adds
a phaseshift of 90 degrees when comparing the phase of the signal at its
input versus the phase of the signal at its output.

Antenna Tuning networks are assumed to consist of ideal components.
E.g. wideband lossless transformers, or lossless inductors and capacitors. No resistors.

Transmitters are assumed to be ideal, consisting of a voltage source with a
source impedance Zs.
 Loads are assumed to be resistive, with no reactive component.

Directional Couplers (i.e. SWR meters) are assumed to be lumpedelement
circuits consisting of ideal components (e.g. ideal lossless transformers
and resistors without parasitics).

Unless otherwise specified, Directional Couplers are assumed to be
referenced to 50 ohms. That is, the voltage from their "Reflected
Voltage" port will be zero when the impedance connected to their Output port
is 50 + j0 ohms.
Let's look at a basic station setup consisting of a Transmitter, an SWR Meter,
an Antenna Tuner, and a long Transmission Line connecting the Tuner's output
to a load (antenna).
If the interconnects between the transmitter, SWR meter, and antenna tuner are
short (my ruleof thumb is that lengths are less than 1/20 of the wavelength of the highest
frequency), then these system elements and their interconnects can be treated
as a lumpedelement system, and lumpedelement circuit analysis techniques can be used
to calculate the voltages and currents at the various nodes of this part of
the system.
On the other hand, the transmission line portion of the system is best
analyzed using the familiar concepts of reflection and transmission
coefficients of transmissionline systems.
Goals of this Post:
One goal of this post is to show a method of calculating, and then
calculate, the following voltages:

The Forward and Reflected voltages on the transmission line and how they
change from startup to steadystate.

The Forward and Reflected voltages that would be measured by the
lumpedelement Directional Coupler (i.e. SWR meter) inserted between the
Transmitter and the Antenna Tuner, and how these voltages change from startup to
steadystate.
Note that in a lumpedelement system there actually aren't forward and
reflected waves. But the SWR meter, itself being a lumpedelement
circuit, does not know that they do not exist  it is measuring the
voltages and currents at a single point (node) and from these deriving what the
forward and reflected voltages would be if it were connected in a 50 ohm
transmission line. (More on this topic, below).
Vf and Vr (forward and reflected) voltages derived with the equations I will present will be verified with
simulated results generated with Simulink.
Another goal is to determine, when the Antenna Tuner is tuned for a
1:1 SWR at its input (i.e. the impedance looking into the antenna tuner's input port is 50
ohms), the value of the impedance looking back into the OUT port of the antenna tuner
(towards the transmitter). This is the impedance that reflections from
the load will encounter as they travel backwards on the transmission line towards the tuner, and this is the impedance that then rereflects
these reflections back to the load.
In other words, when the system is in steadystate with the Antenna Tuner
tuned so that its input impedance is 50 + j0 ohms...
Foundation Concepts:
The following concepts form the foundation for my equations and
calculations later in this post.
1. On a Transmission Line, the voltage at any point on that line is
the sum of the Forward Voltage and the Reflected Voltage:
V(at point) = Vf(at point) + Vr(at point)
2. On a Transmission Line, when a traveling wave arrives at an
impedance discontinuity, this "incident" wave splits into two parts.
One part is reflected back (Vr), and the other part continues forward (Vf)
through the discontinuity. The amplitudes of Vr and Vf with respect to
the incident wave's voltage are such that energy is conserved.
The amplitudes of the signals traveling in the two directions can be
determined using the calculated values of the Reflection Coefficient and the
Transmission Coefficient at the discontinuity
We can use the same V(transmitted) equation for a transmission line with a
resistive load. After all, a resistor simply looks like an infinitely long
transmission line whose characteristic impedance equals the resistor's
resistance:
3. On a Transmission Line, delay can be expressed as a complex number:
On an ideal lossless transmission line with a sinusoidal waveform, we can
mathematically treat delay as e^{jφ}, where φ is the phase shift
introduced by the transmission line's delay (note: e^{jφ} = cos(φ)  j*sin(φ)).
For example, if a transmission line is one wavelength long, a sinewave signal's phase will
shift by 360 degrees (2 pi radians). And so in this case, the delay
factor would be e^{j2π} = 1 + j0.
If the transmission line were 1/4 λ long, the delay factor would be
e^{jπ/2} = 0 + j1.
4. The Principle of Superposition:
The "Principle of Superposition" is a fundamental principle of circuit
analysis which states that, in a circuit with multiple voltage and current
sources, we can calculate the voltage and current at any node in the system by
first calculating the individual contributions of each voltage or current
source to the voltage (or current) at that node (by first turning all other sources off  replacing voltage sources with shorts and current sources
with opens), and then summing the individual contributions together to get
the final value.
This same principle can be applied to a transmission line upon which
multiple reflections might exist. For example, on a mismatched
transmission line, the total reflected voltage on the line can be
represented by the sum of an infinite series of individual reflected
voltages. (More on this in a bit).
5. An ideal lumpedelement Directional Coupler can be represented by the
following circuit:
The circuit above is the Directional Coupler I will use in my Simulink
simulations.
Lumpedelement directional couplers measure current and voltage at a single
point. As such, they have no knowledge of the direction of wave
travel or even the existence of waves. But they will generate two voltages that would represent Vfwd
and Vref on a transmission line if the directional coupler were installed in
a transmission line whose characteristic impedance Zo equals the directional
coupler's "Rref".
Another way to look at this circuit's operation is to consider it to be a lumpedelement bridge circuit, and the voltage Vref will equal zero when Zload (connected
to the directional coupler's Vout port) equals 50 + j0 ohms (assuming Rref =
50 ohms).
Therefore, if this directional coupler is attached to the input port of an
antenna tuner and the tuner adjusted until Vref equals 0, then the impedance
looking into the antenna tuner's input will be 50 + j0 ohms.
Note that for large transformer turnsratios, the voltage drop across the currentsense transformer will essentially be 0, and Vin will essentially equal
Vout (which we want). We can derive the following equations for Vfwd and
Vref based upon the voltage at Vout and the load impedance attached to it. (Note that the turns ratio 'n' results in the attenuation of Vfwd and Vref and so my "ideal" directional coupler adds multiplication factors that remove this attenuation effect):
Vfwd = Vout *(Zload + Rref)/(2*Zload)
Vref = Vout *(Zload  Rref)/(2*Zload)
(More on the derivation of these equations, here: http://k6jca.blogspot.com/2015/01/notesondirectionalcouplersforhf.html )
If we know Vfwd and Vref, we can derive the Zload connected to the directional coupler's output port:
Zload = Rref * (Vfwd + Vref) / (Vfwd  Vref)
The drawing below illustrates these equations:
6. Any waveform can be represented by an infinite series of impulses, and we can examine the operation of a system by taking these impulses, one at a time, and investigating their effect on the system's voltages, and then summing their contributions:
With these definitions out of the way, let's apply these principles to an
example...
Example with a Source Impedance Equal to 50 Ohms and a 200
Ohm Load:
Let's take an example with a 200 ohm load at the end of a 50ohm
transmission line that is one wavelength long (taking its velocity factor
into account).
Defining the Circuit Model:
I will represent the transmitter as a voltage source with a series source
impedance Zs. In this example I will set Zs to 50 + j0 ohms.
The transmission line's characteristic impedance will be defined to be 50 ohms, and the line lossless. Because the transmission line is one wavelength long, the load impedance
presented by the input of the line to the output port of the Antenna Tuner is also, conveniently, 200
ohms.
The Transmitter, SWR Meter, and Antenna Tuner are assumed to be connected
together with very short interconnects, so I will combine these three
elements together and treat them as a single lumpedelement network.
I will represent the antenna tuner with an Lnetwork that will transform the 200
+ j0 ohms at its output port to 50 + j0 ohms looking into its input port.
In the illustration, below, I've removed the directional coupler (in this example considered to be
ideal) because it has no effect on system voltages and currents (but I will present
equations that allow the calculation of the Vf and Vr it would generate, given the voltage at a node into which it would be inserted and the load impedance it
would see at that node.)
Within the lumpedelement network there are two nodes whose voltages I will
use for calculations. One is node 'A' (at the Lnetwork's input), and
the other is node 'b' (at the Lnetwork's output, which is also the
transmission line's input). A directional coupler could be inserted into the circuit at either of these two nodes to determine either the forward and reflected voltages at the input end of the transmission line (node B) or the "equivalent" forward and reflected voltages at node A.
The Transmission Line's Reflection and Transmission Coefficients are shown,
below.
The illustration below shows how the Reflection Coefficient looking into the Output port of the
Lnetwork was calculated:
SimSmith can be used as a quick check of this value:
Calculating the Model's Voltages and Wave Reflections:
1. Startup:
Let's say that at time t = 0 a sinewave of amplitude 2 volts is started. From a lumpedelement circuit analysis perspective, the
voltages in the lumpedelement network can be calculated.
First, note that the output of the Lnetwork is attached to the transmission
line. At startup the impedance that the Lnetwork sees at this output (labeled below
as "Z_TLin") is Zo, because there are not yet any reflections on the
transmission line.
Given this load impedance, the voltages at the two nodes ('a' and 'b') can be calculated, and from these we can calculated the "equivalent" Vfwd
and Vref voltages that ideal "Tandemmatch" lumpedelement directional
couplers would generate if placed at either node 'a' or node 'b' (the latter
being the Transmission line's input).
(Note: after startup, as reflections on the transmission line arrive
back at the Lnetwork's output, the impedance seen at the transmission
line's input by the Lnetwork will change. The equation to calculate
this new impedance is shown at the bottom of the illustration, above).
2. Reflections on the Transmission Line:
For a sinewave entering a transmission line, we can take a point on that sinewave and follow it as it travels down the line to the load and then reflects back towards the source.
If the impedance at the source end of the line is also mismatched from Zo, this point will rereflect and head back towards the load.
These reflections and rereflections of this point on the sinewave will continue indefinitely over time, with each reflection becoming smaller (assuming a source or load impedance with some resistive element, because I've defined the transmission line to be lossless). And the amplitude of each new reflection can be calculated using a Lattice Diagram.
So, at any time 't', the forward voltage or the reflected voltage on a transmission line can be calculated by summing all of the contributions either from all of the reflections traveling in the forward direction (plus the voltage currently being sent by the source), or from all of the reflections traveling in the reverse direction (from the load).
At startup, given the initial Vfwd on the transmission line (calculated with
the lumpedelement equations above), the resulting amplitude and phase
of the reflections and rereflections can be calculated using the equations shown in the Lattice
Diagram, below.
Here's the Lattice Diagram by itself:
Note that equations in the lattice diagram are easily determined and are
based solely upon the calculated Reflection coefficients at either end of
the Transmission line and also the Transmission coefficient at the load.
Note that the e^{jφ} factor is simply the phaseshift of the
signal due to the transmission line's delay. In this example in which the
transmission line is one wavelength long, the delay is 360 degrees, and so the delay factor e^{j2π} equals 1 + j0. In other words, it has no effect
Note that the reflections described by the lattice diagram continue
forever. Thus, when summed, they are infinite series.
Never the less, these infinite series converge to values (see
http://k6jca.blogspot.com/2021/02/antennatunerstransientandsteady.html for the math describing how to determine the equations to calculate the values
to which they converge).
For the Lattice Diagram, above, the voltages converge to the following values:
Regarding Vf2 in the equation above, it represents the sum of
all
rereflections (reflected from the Output port of the impedance matching
network) traveling towards the load. Thus it excludes the very first
forward signal (called Vf1) which represents the current signal from the
generator, passing through the matching network, to the transmission line.
3: Procedure to Perform Calculations
I calculate the various voltages (nodal and transmissionline) in time
increments of Td, where Td is the period of the 10 MHz signal. The
Transmission Line in this example is ideal (thus the velocity factor equals
1), and so Td is also the amount of time it takes a "point" on the sinewave
to travel from one end of the onewavelength long transmission line to the
other end (and so a roundtrip on the transmission line is 2*Td).
Below is the procedure I used to calculate the system's voltages over time.

Calculate the reflection and transmission coefficients Γ_{1},
Γ_{2},, and T_{ld}.

At startup (time 't' = 0) calculate impedances Za and Zb, and voltages Va
and Vb, using the lumpedelement equations and impedances, with Z_TLin
equal to 50 ohms.

Given Va, Vb, and Z_TLin (= 50 ohms at startup), calculate the
"equivalent" forward and reflected voltages that lumpedelement
Directional Couplers would generate: Vf_LCin, Vr_LCin, and Vf_TLin
using the lumpedelement equations (Vr_TLin will be 0 as there are no reflections from the load yet).
 Set Vf1 equal to Vf_TLin. Vf1 represents the forward voltage on the transmission line generated at that moment by the source.

Create three values for keeping track of running sums of Transmission Line
Forward and Reflected voltages and the Load voltage: Vf, Vr, and
Vload. For time t = 0, Set Vf(t) equal to Vf1 and set Vr(t) and Vload(t) to 0 (there are no reflections yet, and no signal has yet arrived at the load)

Increment time t by Td (t = t + Td). Calculate Vld(t) using the
Lattice diagram equation for this time increment.
 Add Vld(t) to the Vload running sum: Vload(t) = Vload(t1) + Vld(t).

Increment time t by Td (t = t + Td). Calculate Vr_TLin(t) and Vf_TLin(t) using Lattice diagram equations for this time increment. Vr_TLin represents the reflection from the load when it arrives back at the transmission line input (i.e. the line's "source" end) and Vf_TLin represents the rereflection of the Vr_TLin signal off of the impedance discontinuity at the transmission line's input as it starts its return journey on the line back towards the load.

Add these values to the Vf, Vr, and Vload running sums: Vf(t) =
Vf (t1) + Vf_TLin(t); Vr(t) = Vr(t1) + Vr_TLin(t).

Calculate the new Z_TLin using Vf, Vr, and Rref (see lumpedelement
equations).

Using this new Z_TLin, calculate impedances Za, Zb, and voltages Va, Vb,
Vf_LCin(t), and Vr_LCin(t) using lumpedelement equations.
 Go to Step 6 and repeat.
(Note: tracking running sums is not required for the voltages Vf_LCin
and Vr_LCin because their values are based upon Z_TLin, which itself is
calculated from running sums of the Transmission line Vf and Vr voltages).
4. Equation Results:
Below are the results from the procedure described above (calculated using
MATLAB and tabulated in an Excel spreadsheet):
As a check, let's take a look at the steadystate results predicted by the following Lattice Diagram equations:
The results of these equations are:
Vr = 0.8*0.6/(1  0.6*0.6) = 0.75
Vld = 0.8*1.6/(1  0.6*0.6) = 2.0
Vf2 = 0.8*0.6*0.6/(1  0.6*0.6) = 0.45
Vf = Vf1 + Vf2 = 0.8 + 0.45 = 1.25
These results match the steadystate values shown in the Excel table, above.
5. Simulink Results:
The results derived with the above procedure can be verified with a Simulink
Simulation.
Here is the Simulink model that I used for verification:
(Note that the model for the Directional Couplers has been shown earlier
in this post).
The input is a gated sinewave. This lets me determine Vf1 and Vf2
from the simulation results. Vf1 is the forward voltage on the
Transmission line at the start of the gated sinewave. And Vf2 is
the forward voltage on the Transmission line immediately after the
gatedsine has been turned off (in other words, there is no longer a Vf1,
but the earlier rereflections are still traveling from the tunerend of the transmission line towards the load).
You can see below that the amplitude of Vf1 is 0.8 volts and Vf2 is 0.45 volts. These
sum to the steadystate value Vf value of 1.25.
(Note: the magnitude of Vf, Vf only equals Vf1 + Vf2 if the reflection coefficients Γ_{1} and Γ_{1} at the two ends of the transmission line are real, without imaginary terms, and if the transmission line is a multiple of a halfwavelength in length. Otherwise, phaseshift is introduced between the two ends of the line and it is very likely that Vf2 will not be in phase with Vf1. In this case, vector addition still works (i.e. Vf = Vf1 + Vf2), but the sum of the magnitudes of Vf1 and Vf2 no longer equals the magnitude of Vf.)
And below we can see that the directional coupler's "equivalent Vref" at
the Lnetwork's input decays to 0 in steadystate, indicating that the
Lnetwork's input impedance looks like 50 ohms in steadystate.
The resulting Simulink waveforms and their amplitudes match my calculated results.
6. Summary of Forward and Reflected Voltages:
The diagram below illustrates the steadystate Forward and Reflected voltages and power when Zs equals 50 ohms:
Example with a Source Impedance Not Equal to 50 Ohms:
Let's set Zs to 5 ohms. Here's the new circuit:
The Reflection and Transmission Coefficients at the load remain
unchanged. But the Reflection Coefficient looking into the output of
the Lnetwork (towards the source) has changed from 0.6 to the complex number 0.8995 +
j0.2621.
Note that this reflection coefficient can be found two ways:
1. Circuit analysis method (in which Vs is replaced by a short):
2. SimSmith method:
Calculated Results:
The following results were calculated using the procedure described above
(and MATLAB).
Note the steadystate magnitudes of Vf, Vr, and Vload are the same
irrespective of Zs.
Examining the calculated values in their complex form, I can derive the
following diagram showing the steadystate voltages and power on the
transmission line.
Note that Vf1 is now a complex number (compared to the value calculated when
Zs was 50 ohms). Never the less, I can still calculate Vf2 and, with Vr,
calculate the Reflection Coefficient looking into the LC network's Output
port. Specifically, I made this calculation as follows:
 Vf1 (calculated at time t = 0) = 0.1166  j0.5911

Vf (at time t = 28*Td, which is as far as I took my calculations) = 0.6199
 j1.0724.

Subtracting these two quantities, Vf2 = Vf  Vf1 = 0.5033  j0.4814
 Vr (at time t = 28*Td) = 0.3721  j0.6435
I can derive Γ_{1} by dividing Vf2 by Vr: Γ_{1} = Vf2/Vr = 0.8995
+ j0.2621, which is the same value calculated using circuit analysis
techniques.
(Note that because Γ_{1} now has an imaginary part, the magnitudes of Vf1 and Vf2 no longer sum to be the magnitude of Vf, as it did in the Zs = 50 ohm example, above. But Vf still equals Vf1 + Vf2, but one needs to use vector arithmetic for the calculations.)
Simulated Results:
The circuit was simulated with Simulink:
I adjusted the source amplitude from 2 to 1.09 to set the steadystate Load
voltage very close to 2.0 volts peak  this is the same Vload level for the 50 ohm Zs simulations).
Note that I've increased the length of the transmission line from 1
lambda to 2 lambda. This gives the signal a bit more time (i.e. 4
cycles) to settle down before the next reflection arrives (it takes a
bit longer for the signal to settle when Zs is 5 ohms).
The Simulation results are below. Note that an SWR meter at the
input to the LC network would still indicate an SWR of 1:1 after a
dozen or so cycles following startup:
The Transmission Line voltages Vf and Vr, and the output voltage
Vload, have the same steadystate values as they had in the Zs = 50
ohm simulation. But the transient voltage levels at startup and shutdown differ from the
50 ohm simulation. However, they are consistent with the
calculations tabulated for Zs = 5 ohms, above. So we can say
that the simulation validates the calculations performed using the
lumpedelement and transmission line models.
From these plots we can still get an idea of the
magnitude of reflection coefficient
looking into the Lnetwork's output port (we cannot get the angle because these plots don't show us the phase relationship between Vr and Vf2). Note
that Vf2 is about 0.69 volts (this is measured on the "back porch" of
Vf_TLin, just after the source turns off (and waiting a cycle or two for
the level to stabilize), while Vr (in steadystate) is about 0.744
volts.
Γ
_{1} equals Vf2/Vr = 0.69/0.74 = 0.93, which is quite
close to the magnitude of the calculated Γ
_{1} (=
0.9369)
Conclusions:
1. The Reflection Coefficient looking into the output of
the Impedance Matching network (Antenna Tuner) is a function of the
source impedance and, for a source modeled as a voltage with a series source impedance, can be calculated using basic circuit analysis
techniques.
2. This Reflection Coefficient may or may not be the Complex
Conjugate of the load impedance connected to the Output port of the
Impedance Matching network. If the source impedance is 50 ohms
(and there is no other loss in the system), then it is the complex conjugate and thus there is a Conjugate
Match (this can be quickly demonstrated with SimSmith). But for
all other possible source impedances, there is no Conjugate Match.
3. Irrespective of presence or absence of a Conjugate Match, an
SWR meter at the input of the Impedance Matching network will show an
SWR of 1:1 when the Impedance Matching network has been tuned so that
the impedance looking into the Impedance Matching network's
input port is 50 + j0 ohms. The transmitter's source resistance has no effect on this tuning.
Notes:
If you'd like to play with the Simulink models, you can find them
here: https://github.com/k6jca/AntennaTunerSimulinkModels
They were created with Simulink R2020a (version 10.1), so if you have an earlier version of Simulink, you might not be able to run them.
Other TransmissionLine Posts:
http://k6jca.blogspot.com/2021/02/antennatunerstransientandsteady.html. This post analyzes the transient and steadystate response of a simple impedance matching system consisting of a wideband transformer. I calculate the system's impulse response and find the timedomain response by convolving this impulseresponse with a stimulus signal.
http://k6jca.blogspot.com/2021/02/thequarterwavetransformertransient.html. This post analyzes the transient and steadystate response of a QuarterWave Transformer impedance matching device. I calculate the system's impulse response and find the timedomain response by convolving this impulseresponse with a stimulus signal.
http://k6jca.blogspot.com/2021/05/antennatunerslumpedelementtuner.html. This post analyzes the transient and steadystate reflections of a lumpedelement 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/lcnetworkreflectionandtransmission.html. This post describes how to calculate the "Transmission Coefficient" through a lumpedelement network (and also its Reflection Coefficient) if it were inserted into a transmission line.
http://k6jca.blogspot.com/2021/09/doessourceimpedanceaffectswr.html. This post shows mathematically that source impedance does not affect a transmission line's SWR. This conclusion is then demonstrated with Simulink simulations.
Standard Caveat:
As always, I might have made a mistake in my equations, assumptions,
drawings, or interpretations. If you see anything you believe to
be in error or if anything is confusing, please feel free to contact
me or comment below.
And so I should add  this 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.