|
|
Post by ms on Dec 13, 2018 13:18:36 GMT -5
Maybe using Thevenin's theorem is overkill in this case but it should give the right results, so why not? Thevenin's theorem is described here: en.wikipedia.org/wiki/Thévenin%27s_theorem. We need the version for ac impedances. Then we need to find the pickup impedance, either derive it as I did in the discussion started yesterday, or use Spice, for example, for actual numerical calculations as Aquinas did. The theorem does not say anything about magnetic coupling, but if we can show that this coupling results in an impedance, then we should be able to use the theorem. This is easy to do, and gives Aquinas's series representation. Application of the theorem has two steps, and it is trivial in this case. First, leave the terminals of the device open, compute the voltage at the terminals, and define a voltage source of this value. If we put a voltage Vi in series with the pickup coil, this voltage appears unmodified on the terminals when they are open, because no current flows. (Remember, we are neglecting the pickup coil capacitance. The capacitance can be added as an external device after we make the model.) Second, short the voltage source and compute the impedance looking into the terminals. This is just the pickup impedance, as derived in the attachments in the post of a discussion started yesterday, or you can use a Spice model. Now put the impedance in series with the voltage source, and you have a model for computing the output voltage as a function of the input voltage Vi with whatever external impedance you want to use. Vi could include eddy current shielding effects as Aquinas has. I will try this with the pickup impedance equation with a suitable value of C as an external device later and see how the results look. |
|
|
|
Post by aquin43 on Dec 13, 2018 14:51:02 GMT -5
Hello ms
You are describing my model exactly. Get the open circuit forward gain and the output impedance correct and you have a black box that is indistinguishable from the real pickup by any electrical test.
In my last but one post in the "new model" thread I showed how, using an algebraic equation describing the coupled coils and some resonant impedance measurements, it is possible to compute all of the necessary parameters for the impedance model. The fit can be quite good for some pickups, not so good for others.
The impedance can be calculated to the accuracy allowed by the model's fit but the forward gain can't. This is because, while the terminals give us good access to the heart of the pickup, the route from the strings or exciter does not. The simple model with magnetic coupling between the strings/coil and strings/parasitic-coil can add only a single low pass step to the forward gain. I soon found out that this is not enough to model a real pickup. At least one additional low pass section is necessary.
I have found that the core response of the pickup tends to have a higher Q than the one that shows in the forward gain. The response to a step of current in the output terminals will tend to ring much more than would be expected from the overall frequency response.
Do you do the algebra by hand or use a program? I have been using the free program Maxima, which prevents silly mistakes and can also output the solutions in a form that can be used directly in many programming languages.
Arthur
|
|
|
|
Post by JohnH on Dec 13, 2018 14:58:49 GMT -5
Yep Thevenin is a very powerful concept. For us, it can mean that there are various ways we can characterise a pickup, for use in circuit analysis. We just need to get open-cicuit voltage, impedance and phase correct (enough) vs frequency. This can be by developing various component-based models, with (or without) transformers, or it could be as a table of values of those three parameters, directly measured. The latter is a lot more data but fewer assumptions and simplifications needed.
|
|
|
|
Post by JohnH on Dec 13, 2018 15:30:42 GMT -5
I have found that the core response of the pickup tends to have a higher Q than the one that shows in the forward gain. The response to a step of current in the output terminals will tend to ring much more than would be expected from the overall frequency response.
My grasp of the maths is an order of magnitude more wooly than you guys, but I have an observation that may be related to, and support your point above. I put the Tucson model into my spreadsheet, to test it against my 6part LRC models. I wanted to see if the 6P model could track the Tucson model across a range of reactive loads. And in the case I tested, it did, within a db, demonstrating (but not proving) that although its empirically determined, it can still be a good black box. Now I had several goes at finding the optimum set of values for my model. And in each case, I settled on a component set that used only five of my six parts. The unused one was a simple resistive load. It was the last one I added to my model when I was developing it. I had hoped to find that it was in general, not needed. But when optimising models against real pickups, I found it to be essential if impedance and output vs frequency were to be adequately correct. My (speculative) point here is, this resistive load component, which was not needed to track the Tucson example, but which was needed to track real pickups, has the simple effect of reducing Q, which sounds very analogous to the correction you describe above. |
|
|
|
Post by aquin43 on Dec 13, 2018 17:36:42 GMT -5
My (speculative) point here is, this resistive load component, which was not needed to track the Tucson example, but which was needed to track real pickups, has the simple effect of reducing Q, which sounds very analogous to the correction you describe above. I think you may be correct. The pre-filter in the coupled coil model has the effect of preventing an actual sharp step from reaching the coil and so reduces the overshoot and ringing, while in your model you have to reduce the Q of the coil itself.
One thing that niggles about the coupled coils is the impedance anomaly at low frequencies. I have just measured an old Dimarzio humbucker and the fit, again deviates at 700Hz, just where the tone control works. I can fiddle it by more than doubling the dc resistance. But the dc resistance is the only parameter that can be directly measured an known to be correct. The model is obviously too simple.
Arthur
|
|
|
|
Post by ms on Dec 14, 2018 6:33:54 GMT -5
Hello ms
You are describing my model exactly. Get the open circuit forward gain and the output impedance correct and you have a black box that is indistinguishable from the real pickup by any electrical test.
In my last but one post in the "new model" thread I showed how, using an algebraic equation describing the coupled coils and some resonant impedance measurements, it is possible to compute all of the necessary parameters for the impedance model. The fit can be quite good for some pickups, not so good for others.
The impedance can be calculated to the accuracy allowed by the model's fit but the forward gain can't. This is because, while the terminals give us good access to the heart of the pickup, the route from the strings or exciter does not. The simple model with magnetic coupling between the strings/coil and strings/parasitic-coil can add only a single low pass step to the forward gain. I soon found out that this is not enough to model a real pickup. At least one additional low pass section is necessary.
I have found that the core response of the pickup tends to have a higher Q than the one that shows in the forward gain. The response to a step of current in the output terminals will tend to ring much more than would be expected from the overall frequency response.
Do you do the algebra by hand or use a program? I have been using the free program Maxima, which prevents silly mistakes and can also output the solutions in a form that can be used directly in many programming languages.
Arthur
I mostly do algebra by hand; since I am a Python user, I do use SymPy at times. This stuff for pickups is all by hand, never checked. Probably I should check it, but it appears to give all the right results in limiting cases. |
|
|
|
Post by aquin43 on Dec 14, 2018 8:48:01 GMT -5
I have found Maxima to be very liberating. For example, in the latest variant of my model, the resonant impedance equation to be fitted to the data ended up over 500 characters in length when written out programming style. There is no way I could have worked that out by hand even though the steps to derive it were quite simple algebra.
Arthur
|
|
