Impedance Matching when the Transmitter, Line and Load impedances are different

[The Tortoise-Man]

Assume we have a general rf system where source with impedance $Z_S$
transfers a signal to a load with impedance $Z_L$ through a transmission
line with impedance $Z_T$. We want to match this system
in order to maximize it's effiency:

In

https://en.wikipedia.org/wiki/Impedance_matching

there are two options how to do it if we deal with power transfer
from source to load through a transmission line:

There is one way to to maximize the power transfer; the
condition that should be fulfilled is $Z_S = Z^*_L$ and on the
other hand there is a way to minimize the reflexion of the signal waves.
To do this the relations between the source and transmission line
should satisfy $Z_S=Z_T$ and and between
load and transmission line $Z_L=Z_T$.

Obviously, it is not possible (if the system contains reactant components)
to fulfill both conditions, ie to maximize the power transfer AND
to minimize the signal wave reflexion.

My question is what to do in such situation? How to decide what to do
here in order to run the system with best effiency? How to decide
what is more important: to maximize the power transfer or
to minimize the signal wave reflexion? Is there a kind of guide
how to decide in such or similar cases?

 

[Baluncore]

Obviously, it is not possible (if the system contains reactant components)
to fulfill both conditions, ie to maximize the power transfer AND
to minimize the signal wave reflexion.

Since when is it not possible ?
The optimum solution may depend on line length and line losses, but you can have the best of both.

Match the transmitter to the line with a transmatch. Match the load or antenna to the line with an antenna matching unit. The length of the line and the wavelength are then not critical.

 

[DaveE]

You will have two matching problems to solve if you seek perfection, one at each interface.

 

[Baluncore]

You will have two matching problems to solve if you seek perfection, one at each interface.

I design the transmitter to be matched to the line.
I design the load or antenna to be matched to the line.
I use a low-loss transmission line.

Matching is not a problem. For power transfer it is an essential.
The half-baked matching of a 10 kW transmitter could be both expensive and embarrassing.

This thread presupposes incompetence and careless shortcuts.
If the job is worth doing, it is worth doing it properly.

 

[tech99]

It seems quite possible for the problem to happen in reality - its just a matter of practicality as with all engineering. For instance, a 75 Ohm antenna might be used with a 50 Ohm feeder and a transmitter of 20 Ohms. Transmitters are frequently not perfectly matched to their load, because there are other design calculations involved, such as linearity or efficiency. A straight forward mismatch does not necessarily incur very much loss. This can be seen by drawing the source and load as a potentiometer and calculating the power in each resistor. With a 50 Volt generator of 50 Ohm resistance working into a 50 Ohm load for instance, the power delivered is 12.5W, whereas with a 25 Ohm load it is 10.9W. However, if the feeder has some attenuation, it is then important to use the best match, because standing waves on the feeder will magnify its loss. So the rule is that mismatch at the receiving end of a lossy feeder should be avoided.

 

[DaveE]

It seems quite possible for the problem to happen in reality

Like that time in the lab when you have a 50Ω system but grab the 75Ω coax. It can take a surprisingly long time to figure out the mistake, LOL.

 

[The Tortoise-Man]

Since when is it not possible ?
The optimum solution may depend on line length and line losses, but you can have the best of both.

Match the transmitter to the line with a transmatch. Match the load or antenna to the line with an antenna matching unit. The length of the line and the wavelength are then not critical.

I think that this is not possible, because, let's try it:

1. We match the source to load via maximization
of the power transfer. So we get
$Z_S = Z^*_L$.

2. We match the source to the transmission line to
minimize the signal waves reflexion. So
$Z_S = Z_T$

3. We match the load to the transmission line again to
minimize the signal waves reflexion. So
$Z_L = Z_T$

If we try to obtain 1, 2, 3 simultaneously, we would obtain
$Z_S = Z^*_S$ and $Z_L = Z^*_L$

And in general as I said before that's pure mathematically wrong, if
our source and load have reactant components. That's
exactly my problem. How to resolve it or where is the
error in my reasonings above?

 

[Baluncore]

How to resolve it or where is the error in my reasonings above?

You assume that step 1. matching of the source to load via maximization of the power transfer is needed. But that energy transfer efficiency comes by virtue of the fact that the device at each end of the line is matched to the line.

Maybe since you mention the presence of a reactive component, you are assuming that only the real R is to be matched, and that any reactive X will not be neutralised. But matching for energy transfer involves both R + X j , at both of the independent ends of the transmission line.

 

[The Tortoise-Man]

You assume that step 1. matching of the source to load via maximization of the power transfer is needed. But that energy transfer efficiency comes by virtue of the fact that the device at each end of the line is matched to the line.

So how my wrong step 1 approach should be replaced by correct one concretely? You suggest that the energy transfer efficiency comes by virtue of the fact that the device at each end of the line is matched to the line.
So you suggest to split the matching procedure in two steps: firstly to match the source with transmission line and next to match the transmission line with the load, right?

But if that's what you mean, then I`m facing the same problem as before: Let start firstly with source and transmission line matching separately. Should I perform here maximization the power transfer (ie $Z_S=Z^*_T$) or minimization of wave reflexion (ie $Z_S=Z_T$)? How to decide it?

 

[Baluncore]

How to decide it?

Rule number one. Do not accept standing waves on the transmission line, they are evil.

Match the source into the transmission line, with a perfectly matched termination or loss network at the far end, so there is no reflection possible. Then replace the perfect termination with the load, and match the load to the line.

So long as you use the line as part of the matching network, you must have standing waves. Standing waves indicate that energy is passing through the same line losses several times, which is not an intelligent way to match a high power line.

Low SWR may mean you have a perfect match. It may also mean you have a lossy line that can hide a mismatch. Where does the lost line energy go? It heats the line, melts the dielectric, then hopefully triggers a reflected energy automatic shutdown, before the fire starts.

 

[tech99]

Assume we have a general rf system where source with impedance $Z_S$
transfers a signal to a load with impedance $Z_L$ through a transmission
line with impedance $Z_T$. We want to match this system
in order to maximize it's effiency:

View attachment 292545

In

https://en.wikipedia.org/wiki/Impedance_matching

there are two options how to do it if we deal with power transfer
from source to load through a transmission line:

There is one way to to maximize the power transfer; the
condition that should be fulfilled is $Z_S = Z^*_L$ and on the
other hand there is a way to minimize the reflexion of the signal waves.
To do this the relations between the source and transmission line
should satisfy $Z_S=Z_T$ and and between
load and transmission line $Z_L=Z_T$.

Obviously, it is not possible (if the system contains reactant components)
to fulfill both conditions, ie to maximize the power transfer AND
to minimize the signal wave reflexion.

My question is what to do in such situation? How to decide what to do
here in order to run the system with best effiency? How to decide
what is more important: to maximize the power transfer or
to minimize the signal wave reflexion? Is there a kind of guide
how to decide in such or similar cases?

Not sure here what your constraints are. As people have mentioned, impedance transformers could be used at both ends of the line. I presume this is not possible?

 

[The Tortoise-Man]

Rule number one. Do not accept standing waves on the transmission line, they are evil.

Match the source into the transmission line, with a perfectly matched termination or loss network at the far end, so there is no reflection possible. Then replace the perfect termination with the load, and match the load to the line.

So long as you use the line as part of the matching network, you must have standing waves. Standing waves indicate that energy is passing through the same line losses several times, which is not an intelligent way to match a high power line.

Low SWR may mean you have a perfect match. It may also mean you have a lossy line that can hide a mismatch. Where does the lost line energy go? It heats the line, melts the dielectric, then hopefully triggers a reflected energy automatic shutdown, before the fire starts.

Ok, so in case of the toy network from my opener post we should
match it as follows with only two steps:
1. M1: Match Source with Transmission Line via minimization
of reflected waves, so $Z_S= Z_T$

2. M2: Match Transmission Line with Load via minimization
of reflected waves, so $Z_T= Z_L$

And that's all? So my error in post #7 was has been
step 1. Should it considered as general strategy for
matching multistaged networkes by matching only
the immediate neighbours? And if the neighbour it the
transmission line, then the match is always realized
by minimization of signal reflexion?

If I have rephrased your idea correctly, can this idea be
generalized as follows:

Say we have a multistaged network between $n$ connected
subkomponents $S= S1, S2,..., Sn=L$:

and we want to match it. Can the general starategy be summarized as
follows:

1. The matches should be performed successively ONLY
between immediaty(!) neighboured components, right? ie there should be
always performed only $n-1$ matchings: between
$S1$ and $S2$, then between
$S2$ and $S3$, and ... and between
$S(n-1)$ and $Sn$. And no other matches especially between non neighbours are required?
In other words, eg it should be not tried to match
between not neghbours like eg $S2$ and $S5$?

So we obtain:

2. Next, how to match $Si$ to $S(i+1)$? In general as I
stated above there are two options:
maximizing power transfer (with $Z_{Si} = Z^*_{S(i+1)}$)
or minimizing reflected signal waves
(with $Z_{Si} = Z_{S(i+1)}$)

Having considered your explanations on the toy network from above, can it be stated/generalized as follows:

If $Si$ or $S(i+1)$ is
a transmission line, then we should
minimize reflected signal waves, so $Mi$ realizes
$Z_{Si} = Z_{S(i+1)}$. Otherwise, if neither
$Si$ nor $S(i+1)$ is tranmision line, we
maximize with $Mi$ power transfer with $Z_{Si} = Z^*_{S(i+1)}$.

Is it correct, or is the story here more complicated?

 

[The Tortoise-Man]

Not sure here what your constraints are. As people have mentioned, impedance transformers could be used at both ends of the line. I presume this is not possible?

If I understood Baluncore's point correctly, then my error was to try in the toy network from post #1 to match Source with Load directly, altough there is a transmission line, which by construction I regard as proper partial component of the network, which sits between source and load. So if I understand the philosophy on matching of multistaged networks correctly, then one should alway try ONLY to match the immediate neighbours, so in my toy example the source with TL and TL with load. or do I miss your or/and Baluncore's point?

See also my attempt to generalize it to $n$-staged network in last post. Does the approach make sense now?

 

[Averagesupernova]

The simplest answer: Match the load to the transmission line. So if you have a 50 ohm line, make the antenna matching network transform whatever impedance the antenna is so the transmission line 'sees' 50 ohms looking into the matching network. On the transmitter end make the matching network on that end so that the transmitter 'sees' what the transmitter is spec'd for. Ideally, the transmission line should be what the transmitter specs. Plug and play in that case.
-
Find some schematics for some HF ham transceivers that used tubes instead of output transistors. Many if not all used pi or T networks to match the output impedance of the tubes to the transmission line which was usually expected to be in the range of 50 to 75 ohms.

 

[Baluncore]

Is it correct, or is the story here more complicated?

That is correct, you now understand.
But by repeating the problem many times, you are making it seem more difficult than it really is.

You must consider what you class as a transmission line, is it long and lossy, or short and efficient? In all cases you might use a Smith chart to design and cut a line to perform the match, but that can be very costly for high power, or for long, or lossy lines.

It is easy to get carried away with using a Z₀ = √(Z₁*Z₂) transmission lines to match modules. When you are holding a hammer, everything looks like a nail.
Be very cautious before encouraging standing waves.

It is often the case that redesigning one module, to match its two adjacent modules, can eliminate two transmission lines and four matching networks.

 

[DaveE]

Yes, I think you've got it. There is a more general point here, which is for optimum performance, everything is matched to the adjacent things. Any "unmatched" interface causes reflections which may cause problems.

So, for the most trivial of examples, in a 1m long coax, the bit from 100cm to 101cm is matched to the bit from 101cm to 102cm, etc. i.e. match everything at every transition. IRL, this isn't always possible, so then you identify where the mismatch is and what effect it has on your circuit.

Also, as @Baluncore pointed out previously, you may have the flexibility to design your parts to match their source/load without the need for an additional matching network. This would be a good thing, avoiding adding extra stuff is good if you can do it.

 

[sophiecentaur]

I design the transmitter to be matched to the line.

I keep reading comments like this all over the place and I really have to question it as a general principle. If the Electricity Supply Companies used that approach then they would be dissipating many TW of power in their generators. Does a transmitter really have to dissipate internally, the same power that it feeds into the feeder line?
Obviously an aerial needs to be matched to the feeder to protect the line and transmitter from over volts and to reduce multiple signal paths and coloured frequency response (TV signals in particular) but that problem doesn't apply in the transmitter.
A Class C radio transmitter can have efficiency of up to 80% ('well known' rule of thumb) and that doesn't imply a conjugate impedance match.
A transistor audio amp doesn't dissipate half its total power; it has a low impedance voltage source output to feed a (say) 8Ohm speaker.

 

[Baluncore]

A Class C radio transmitter can have efficiency of up to 80% ('well known' rule of thumb) and that doesn't imply a conjugate impedance match.

Just like, but less efficiently than a switching power supply or inverter.

Does a transmitter really have to dissipate internally, the same power that it feeds into the feeder line?

No.
Impedance matching involves preventing the reflection of energy by keeping the ratio of voltage to current the same within the line. That can be done with lossless LC reactive components, it does not have to involve a resistor with heat generation.

 

[The Tortoise-Man]

You must consider what you class as a transmission line, is it long and lossy, or short and efficient? In all cases you might use a Smith chart to design and cut a line to perform the match, but that can be very costly for high power, or for long, or lossy lines.

Ok, so here seems the point to be should I regard a transmission line as a partial components of the whole network (ie as a $Si$ in the notation from comment #12 or not. And this depends on how so it is, with which characteristic frequencies I work and and and.

Ok, bit if we assume that I'm able to decide is the transmission line relevant for the network or not, then there are only two options how to proceed:

1. I recognize the transmission line as important for the matching procedure. Then, regarding it as network component $Si$, I should do two things, match it to $S(i-1)$ and $S(i+1)$ via minimization of reflection waves, ie match such that $Z_{S(i-1)}= Z_{Si}$ and $Z_{Si}= Z_{S(i+1)}$ and proceed to next stage $S(i+1)$

2. Recognize the transmission line as not relevant, eg id it is to short with respect the wave lengths I'm working with. Then ignore it just and proceed with the match between $S(i-1)$ and $S(i+1)$. If $S(i-1)$ and $S(i+1)$ are not transmission lines, do matching via maximization power transfer, ie $Z_{S(i-1)}= Z^*_{S(i+1)}$, if $S(i+1)$ is transmission line, decide again if it is relevant for the matching or nor and play the same game as before with it...

Is this "pseudo-algorithm" correct on dealing with transmission lines in multistaged network?

 

[Averagesupernova]

Here we go again...
-
https://www.physicsforums.com/threads/coaxial-cable-loss.731545/

 

[sophiecentaur]

Impedance matching involves preventing the reflection of energy

Yes. But what energy would there be, to be reflected at the transmitter, if the aerial is properly matched?
Where is a (Power) RF amplifier different from a switched mode Power supply or the National Grid? Why is matching a 'good thing'?

 

[Baluncore]

Is this "pseudo-algorithm" correct on dealing with transmission lines in multistaged network?

It will still be sub-optimal in many cases.

Do you need to delay or relocate a signal? If yes, consider using a transmission line.
Do you need to match two stages? If yes consider a short transmission line as one of the many possible solutions.

When the system could be simplified by modification of a stage interface, avoid the complexity and cost of a kludge. https://en.wikipedia.org/wiki/Kludge

The quarter-wave transformers used in optics are implemented by coating the transmission surfaces. Quarter-wave transformers for RF can be bulky, and are often better replaced with a compact Pi or T, or for broad-band, with a transformer or a tapered LC ladder network.

A quarter wave transmission line on a PCB can be designed to have a tapered characteristic impedance by controlling the width. That is often better than having several quarter wave stages with geometrically related impedance steps.

 

[Baluncore]

Yes. But what energy would there be, to be reflected at the transmitter, if the aerial is properly matched?

If the load is properly matched then there is no further matching required.
If you did the job properly once, why in your right mind would you do it again?
You have made recursive, and over-generalised your straw man.

Where is a (Power) RF amplifier different from a switched mode Power supply or the National Grid? Why is matching a 'good thing'?

Why is there any need to match the impedance and maximise power transfer from the grid?

When the impedance of the source is low, the voltage is stable, and you take what current you require. The national grid adjusts the available current in real time to match the total load required by all consumers. A switching power supply does exactly the same thing.

You could play mind games by saying that you reflect back to the grid the immense current you do not require, so it can be used by others. The difference between the transmitted and reflected current is then what you use and what flows on your power line.

An RF power amplifier is not designed to absorb large amounts of reflected energy from the antenna. The RF energy was generated at great expense by the PA, to be radiated by the antenna.

 

[berkeman]

Thread closed temporarily for Moderation...

 

[berkeman]

Yes. But what energy would there be, to be reflected at the transmitter, if the aerial is properly matched?

If well matched, there will be very little reflected energy (SWR close to 1.0). But when matching to a short antenna (like many antennas used for 10m/20m/40m bands), the input impedance of the antenna is not real and a complex matching network is necessary.

Where is a (Power) RF amplifier different from a switched mode Power supply

That's a reasonable question. I haven't done the analysis, but the SMPS is using time-domain chopping of the input power and reactive output components to regulate the output DC voltage. So there probably is some analysis that shows how such a functionality basically matches the input power to the load. I'm not sure it would still be referred to as impedance matching, though.

Ok, so here seems the point to be should I regard a transmission line as a partial components of the whole network (ie as a $Si$ in the notation from comment #12 or not. And this depends on how so it is, with which characteristic frequencies I work and and and.

Ok, bit if we assume that I'm able to decide is the transmission line relevant for the network or not, then there are only two options how to proceed:

1. I recognize the transmission line as important for the matching procedure. Then, regarding it as network component $Si$, I should do two things, match it to $S(i-1)$ and $S(i+1)$ via minimization of reflection waves, ie match such that $Z_{S(i-1)}= Z_{Si}$ and $Z_{Si}= Z_{S(i+1)}$ and proceed to next stage $S(i+1)$

2. Recognize the transmission line as not relevant, eg id it is to short with respect the wave lengths I'm working with. Then ignore it just and proceed with the match between $S(i-1)$ and $S(i+1)$. If $S(i-1)$ and $S(i+1)$ are not transmission lines, do matching via maximization power transfer, ie $Z_{S(i-1)}= Z^*_{S(i+1)}$, if $S(i+1)$ is transmission line, decide again if it is relevant for the matching or nor and play the same game as before with it...

Is this "pseudo-algorithm" correct on dealing with transmission lines in multistaged network?

I think this is pretty close to correct. If the line segments are long enough to manifest TL effects (and thus generate reflections), it is best to use matching networks to minimize reflections and wasted energy. If the joining sections are short, the matching networks are probably not needed. Note though that even RF connectors are designed for a particular impedance generally, even though they are electrically short at RF frequencies.

Here we go again...
-
https://www.physicsforums.com/threads/coaxial-cable-loss.731545/

Yeah, I don't want a well-meaning argument between SAs to confuse the OP. I think we are all saying basically the same thing, with different approaches and experiences. I'm going to tie off this thread for now. If folks have further comments or questions, please contact me via PM. Thanks.