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Old 28th Nov 2017, 6:54 pm   #1
Skywave
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Default Attenuators - theory and the design of.

Just a couple of questions that have me really puzzled: I'd appreciate some guidance, please.

1. Many classic sources on attenuator designs involve equations that produce the required value of the discrete resistors in that attenuator, based on the required attenuation and the 'impedances' of the load and source. It's the use of that word 'impedance' that bothers me. An impedance, Z, will contain resistance and reactance. Yet the equations always give the resistor values in terms of impedances and the attenuation factor, k. That implies functional relationships of the form 'Rx = f(Za, k) = f(Ra, Xa, k)', so Xa must be zero, yes?
(Please note: here I am not referring to the application of attenuators in transmission lines.)

2. As a follow-on to that, is it theoretically possible to design a non frequency-sensitive attenuator of known attenuation whose source and load impedances are known, say Za and Zb?

In anticipation, thank you for your thoughts.

Al.
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Old 28th Nov 2017, 8:24 pm   #2
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Default Re: Attenuators - theory and the design of.

I will leave the theory to more knowledgable members but for an attenuator with different input and output impeadance there is the pi circuit.

Ready reckoned here.
http://www.chemandy.com/calculators/...calculator.htm
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Old 28th Nov 2017, 8:46 pm   #3
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Default Re: Attenuators - theory and the design of.

Or its dual, the Tee.

http://www.chemandy.com/calculators/...calculator.htm

If the source and load impedances aren't purely resistive, it all gets more complex
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Old 28th Nov 2017, 8:51 pm   #4
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Default Re: Attenuators - theory and the design of.

Purely-resistive attenuators, matched to the ssource and sink impedances [assuming these are frequency-independent] are really rather simple.

If the source/sink impedances are inconsistent, these can be 'tamed' by making the attenuation something like 10dB or so - downstream of that you'll really only be seeing the characteristic-impedance of the attenuator, not the source.

Slug the incoming side with a load-resistor that can easily dissipate whatever the source can throw at it.

In my experience the biggest problem with attenuators has been dissipating tens/hundreds of Watts of peak RF energy with 50,000:1 duty-cycles.

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Old 28th Nov 2017, 8:52 pm   #5
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Default Re: Attenuators - theory and the design of.

The term 'impedance' covers pure resistance too. In the case of attenuators it will almost always be the case that X=0.

I'm not clear what your question 2 is asking. Source and load different? Or source and load not pure resistance? If the former, yes. If the latter, it depends on how the impedances vary with frequency but for some cases it should be possible.
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Old 28th Nov 2017, 11:18 pm   #6
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Arrow Re: Attenuators - theory and the design of.

Quote:
Originally Posted by G8HQP Dave View Post
In the case of attenuators it will almost always be the case that X=0.
Quite: it is the case when X is not zero that I was referring to.
But I'll try to be more explicit . . . .

Source impedance = Za = Ra + jXa; Load impedance = Zb = Rb + jXb
where X can be +ve or -ve.
Source voltage = Va; load voltage = Vb. Required attenuation = k = Vb/Va.

So our 'knowns' are k, Za and Zb.
Between source and load we have a 'black box' which produces the required attenuation of k. That attenuation is not variable with frequency.

What are the equations that determine the configuration of the components and their values inside that 'black box' in terms of the 'knowns'?

Al.
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Last edited by Skywave; 28th Nov 2017 at 11:41 pm.
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Old 28th Nov 2017, 11:35 pm   #7
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Arrow Re: Attenuators - theory and the design of.

The Chemandy calculators are clearly useful. But they are not applicable to my question. My Q. clearly stated that it was not in relation to transmission lines, which the Chemandy article does exclusively.

I can readily understand and appreciate that the original use of 'attenuators' (and their appropriate design equations) were originally for use with transmission lines, which have a characteristic 'impedance' which appears as a resistance, but is not a real, physical resistance. Hence, the use of 'Z' in the equations. But my post refers to attenuators that are not used in connection with TX lines, but actual impedances of the form Z = R jX, where X is not zero.

Al.
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Old 29th Nov 2017, 12:16 am   #8
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Default Re: Attenuators - theory and the design of.

Quote:
Originally Posted by Skywave View Post
2. As a follow-on to that, is it theoretically possible to design a non frequency-sensitive attenuator of known attenuation whose source and load impedances are known, say Za and Zb?
Yes it is, but it's a sort of special case.

Consider where the source Z is a series resistor with a capacitance across it. Then make the load also a resistor with a capacitance across it. Make the R values and the C values have the same ratio and it's flat.

This is how a scope probe works.

There are also duals with series elements and duals of both with inductances.

David
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Old 29th Nov 2017, 3:32 am   #9
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Default Re: Attenuators - theory and the design of.

One practical problem is the impedance of the medium or system that the attenuator is working in.

For low impedance mediums such as 50 or 75R ,even for heavy levels of attenuation like 20dB the attenuator can simply be comprised of resistors and no reactive elements.

The typical attenuators you see for this that plug in series with coaxial cables with a male & female BNC are designed with resistors only and are a design called a "ladder attenuator" When you drop them in series with the coax or transmission line, they don't create an impedance bump because they are 50R in and 50R out as one example and they have a very wide bandwidth from DC into the UHF region often if they are well made.

However, all the rules for attenuators change when you go to a higher impedance medium, such as a 1meg input on a scope that also looks like 1M with a 15pF capacitance across it. For high levels of attentuation, while still maintaining a high input impedance the input capacitance has a severe LPF effect.

So for example with a x10 scope probe which essentially needs about a series 9 Meg resistor, the severe HF roll off has to be corrected by putting a small parallel capacitor across the 9M resistor to maintain the HF response through the attenuator.

So all of a sudden for any reasonable bandwidth your attenuator now requires a reactive element to work properly, but it didn't in the lower impedance medium where the effects of parallel and stray capacitance are minimized.
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Old 29th Nov 2017, 11:23 am   #10
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Default Re: Attenuators - theory and the design of.

Quote:
Originally Posted by Skywave
What are the equations that determine the configuration of the components and their values inside that 'black box' in terms of the 'knowns'?
Precisely the same equations as were used for pure resistances, except that you have to do complex arithmetic instead of real number arithmetic. You could even use an existing attenuator 'calculator' meant for resistance only and transform the solution to include the reactances, provided that the source and load had exactly the same frequency response. Just scale things.
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Old 29th Nov 2017, 2:48 pm   #11
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Question Re: Attenuators - theory and the design of.

Thank you, Dave - that certainly clarifies this matter for me.

So if in one of the traditional equations that starts 'Z = etc.', should I calculate the Z as √(R + X) and ignore the phase relationship between the R and the X?

Al.
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Old 29th Nov 2017, 4:13 pm   #12
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Default Re: Attenuators - theory and the design of.

If you're only concerned about the magnitude of the output, then yes, you can ignore the phase relationship. But since the requirement is for a non-frequency-dependent attenuator, the phase relationship is important. If we ignore the difference between real and imaginary parts, we've allowed the phase to vary, and it will vary with frequency, which isn't what we wanted. Therefore it's necessary to calculate using the real and imaginary parts of the input and output impedances.

Chris
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Old 29th Nov 2017, 4:23 pm   #13
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Default Re: Attenuators - theory and the design of.

Quote:
Originally Posted by Skywave
So if in one of the traditional equations that starts 'Z = etc.', should I calculate the Z as √(R + X) and ignore the phase relationship between the R and the X?
No, even if you are only concerned about magnitude. As I said, use a normal design method but then scale the result (or calculate it directly if you can do complex arithmetic).

Lets say you have a calculator for 50R impedance attenuators, but your load and source impedance is 100R+50pF (in series). Say one of the resistor values you calculated (for a 50R attenuator) is 35R. Then in the circuit in this position you put 70R+71.4pF. This maintains the same ratios.
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Old 29th Nov 2017, 4:35 pm   #14
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Question Re: Attenuators - theory and the design of.

Thank you Chris: what you've written is also my understanding. And it marries closely with Q2. which I asked in my OP.

So . . . if I have an expression of the form R1 = f(Z1, Z2), where Z1 and Z2 are of the form R + jX, presumably I will need to replace R1 by (R1 + jX1); write down the resulting (and modified) corresponding equation; do the algebraic simplification with the aim of finally equating the real and imaginary (quadrature) components, thus arriving at the values for R1 and X1?

If that is correct, that I understand. It also would appear (to me) that the scaling method that Dave (G8HQP) mentions should give the same result; Dave's method looks a lot less cumbersome than the all-algebraic method mentioned in my above para.

Al.
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Old 29th Nov 2017, 6:23 pm   #15
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Default Re: Attenuators - theory and the design of.

Both methods, applied correctly, will give the same results for those situations where the load and source have the same phase angle. If the angle is different then you have to slog through the algebra.

What you can't do is ignore the phase angle - even if you only want to get the magnitude right. 50+j50 has a magnitude of 70 but you cannot substitute 70 for 50+j50 and get the same result from a circuit.
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Old 29th Nov 2017, 6:58 pm   #16
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Arrow Re: Attenuators - theory and the design of.

Thank you, Dave. What you have written above is exactly my understanding.

Al.
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Old 29th Nov 2017, 9:09 pm   #17
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Arrow Re: Attenuators - theory and the design of.

Quote:
Originally Posted by G8HQP Dave View Post
If the [phase] angle is different [between source and load], then you have to slog through the algebra.
Which I did do, using complex numbers. And, of course, 'slog' it was. I ended up with a number of cubic equations for the 'unknowns' with horrendous co-efficients consisting of the 'knowns'. The solution of similarly complicated quadratics I might contemplate, but cubics such as what I derived . . . . no chance! But it was an interesting exercise.

Al.
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Old 30th Nov 2017, 1:11 pm   #18
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Default Re: Attenuators - theory and the design of.

There is a formula for solving cubics but it is very complicated and probably rarely used.

You might be better doing complex algebra (which is no more difficult than real algebra) and get a complex formula for each impedance (unknown) in the network in terms of source and load (knowns). Then simply do complex arithmetic. If you use a common attenuator design (e.g. pi) then you may find the formula already written out. Doing this with separate R and X's just makes everything more complicated.
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Old 30th Nov 2017, 7:08 pm   #19
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Default Re: Attenuators - theory and the design of.

If you had a source of 50R in series with 50pF and a load that looked the same then one novel way to make a 'flat' attenuator would be to cancel each 50pF cap with a series -50pF cap. This would require an active circuit to create a -capacitor that looked like 'minus' 50pF in series.

Then use a conventional 50R resistive pad (10dB?) in the middle. In theory at least you would end up with a flat frequency response and flat phase and about 10dB attenuation. There would be limitations in terms of signal handling/linearity and bandwidth and it might be tricky to keep it all stable but if active circuits are allowed then this will work across a fairly wide RF bandwidth (in theory at least).
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Old 30th Nov 2017, 8:44 pm   #20
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Arrow Re: Attenuators - theory and the design of.

Quote:
Originally Posted by G8HQP Dave View Post
There is a formula for solving cubics but it is very complicated and probably rarely used.

You might be better doing complex algebra and get a complex formula for each impedance (unknown) in the network in terms of source and load (knowns). Then simply do complex arithmetic.
Which is exactly what I did. And the aforesaid cubics were the result.

As far as solving cubic equations, I do have the standard solution stored on this PC which runs under Excel. It's fine for 'simple' cubics with real number co-efficients, of course, but when those co-efficients are themselves algebraic expressions (especially cumbersome ones), i.e. the 'general case' for a given attenuator, obviously it isn't any use.

Upon reflection, though, these cubic equations might be amenable to an analysis to see if logical simplifications can be obtained on the basis that there might only be one real-valued solution (the other pair being a complex conjugate pair), as opposed to three!
But that's very much a 'maybe, one day' notion. Right now, I'm quite busy on a host of other tasks, not necessarily electrically-related.

Al.
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