Math

[mac]

Simply speaking, mathematics is a language developed to simplify ideas that would take a lot of words to say or write. That is, instead of writing "the rate of change in time of a particle's momentum is proportional to the vectorial sum of all the external forces acting upon it", which is Sir Isaac Newton 2nd Law, we simply write dp/dt = Fext.
A beautiful language but not very easy to learn. English was easy for me...  but it took me 5 years at college to learn maths...

Maths, or "mate"  in spanish, have been among us from the very begining. I mean albebra, geometry, etc. Geometry for example helped Erathostenes to measure the circumference of the earth with an error less than 4% two centuries before Christ.
But calculus is relatively new. Newton shaped the work of Galileo and Copernicus, and laid the foundations of calculus on which all physics is based. In a way, he was forced to invent calculus. We could say that the lack of calculus delayed the development of physics, and some historians say that the church also played a key role.

Now, the equations we normally use to find the bias of a fuzz face are simple and mostly linear
( http://geocities.com/guitarfxs/english/ffmath_eng.html ).
This is because we are solving the stationary state, when all temporal derivatives are constant, ie dq/dt = i = const. But when we want to know what will happen to an incoming 1khz sine wave hiting a a pair of nkt275 we need to solve differential equations. It's up to you the method to solve them.

So, the beginner need basic math & electronic knowledge to build their pedals. There is a lot of info in the web and in this forum to start with. But if you are serious about making your own projects you will eventually need to analyze things in real time. Maybe spice is all you need, but some calculus will 'drive' you far far away, where no spice has gone before.

Is calculus that hard? Well, it depends on what are you dealing with. Fluid equations are non linear and harder to solve than Maxwell eqs. Pedals circuits are somehow simple, being the tricky part trying to model a transistor or opamp. Normally you analyze how a sine voltage is modified by the circuit. You write down the eqs and after a litre of coffee and several pages you get a result. Can someone with no formal education do it? Mmmmmhhhhh... In my opinion it seems unlikely but yet possibly.

mac

[KMS]

Actually, all the formulas are the result of calculus, well not all of them.  Ohms Law is not. But calculus is the axis for all of Engineering and you won't ever degree without it.   The answer to your questions is....Yes.

...speaking of Maxwell's equations, actually, Ohms law is derived using calculus.  Since thinkers such as Maxwell have done the groundwork, we don't need to even think twice about the calculus done that gives us the confidence in the integrity of the simple algebraic and arithmetic operations we do in circuit design.

So when it comes to the more mathematically involved operations such as frequency response and phase functions, we still don't actually need to do much calculus.  I can't think of any application in analog guitar pedal design that requires chugging through convolution, Laplace transforms and inverse Laplace transforms, or the likes. 

Here's an example:

The current/voltage relationship in a capacitor is expressed by the following in the time domain:

I(t) = C*d V(t)/dt

Impedance is generally expressed as Z= V(t)/I(t)
This is a hard one to express in the time domain for a capacitor, but it is fortunate that we have Fourier transformations.  The Fourier transform of this equation (presented without going through the mathematical process to get there) is this:

I = jwC*V

==> V/I = 1/jwC

Where w=2*pi*f, which is the radian frequency, or 2*pi*frequency
j = square root of negative one (imaginary number which mathematicians and physicists reference as "i")

Now this is easy.  We can just treat the capacitor in an equation like a resistor for circuit analysis just by saying that the impedance of the capacitor is Zc=1/jwC

Ok that's kind of a big jump there...."we can just treat the capacitor in and equation like a resistor".

Yes we can do that but proof that with a Maxwell equation.

I agree with most of that reply and I not gonna tell you that you can't derive Ohm"s Law from Maxwell's wonderful equations (I've never seen it done before and why would anyone need to do that) but I would like to point out a little history. 

Ohm's law was not derived or discovered from calculus nor to my knowledge has it ever been since George Ohm discovered his directly proportional relationship.  George Ohm conducted experiments and found many materials to be what we now call ohmic and  discovered a formula I=V/R from empirical data where not all materials obey his law for which such materials are know as nonohmic.

Maxwell's equations are all about classical electromagnetism which can be reduced and manipulated to describe much more that just electromagnetism, but as far as I know you cannot directly reduce or derive the result I=V/R starting with only a Maxwell equation without making a jump somewhere and inserting Ohm's relationship or setting Ohm's relationship equal to some other relationship.

Since Ohms discovery many equations have been derived using calculus from Ohm's Law by inserting I=V/R or setting I=V/R to be equal to another relationship to become one of the variables that go into the design of formulas that calculate time vs. current, current vs. resistively, Voltage vs. current, change in time vs. current, change in resistance, change in current, so on and so forth but Ohm's Law in itself has no roots in calculus.

The "empirical" relationship is quoted as such in my physics text book on page 844 by Serway and Beichner, Physics for Scientists and Engineers, Fifth Edition, Saunders College Publishing.

There is also an interesting write-up at this link http://inventors.about.com/library/inventors/blohm.htm

The link also states the relationship is empirical.

Just thought everyone should know that calculus does not find all answers.

[Transmogrifox]

I agree.  Most of our "simple" relationships were first derived empirically.  However, to a very clever person many such relationships could be predicted with no testing based on the knowledge we have now.  I have seen "V=IR" derived from Maxwell's Equations in my universtiy Electromagnetic Theory class--and it is done without leaning on the empirical relationship it's deriving.  We could also use similar analysis tools to derive the rate of charge on a capacitor, though this relationship was also discovered empirically--essentially "letting the physical world do the math for us and spit out the result" approach.  That's part of what makes science so interesting: you can explore a physical phenomenon to any level of complexity, yet a deep understanding of all its workings is not necessary to comprehend it at a tangible level.  You can just "watch and learn".

This is why a breadboard is so great.  You can try things and tweak things and just observe what happens like many of the early scientists who contributed to the development of higher level theory and the language of mathematics to explain it.

So in the end, relationships derived from Maxwell's Equations can be used to explain why George Ohm observed that V=IR...

it took the experimentation process for our feeble minds to understand the physical world to the point where we can start understanding many things we can't measure directly...and our understanding of science barely scratches the surface.

So let's just take V=IR for granted and build some stompboxes 

[KMS]

I'm with you on that.....nothing quite like playing around with a circuit to get it just the way you want it.

While we are on the subject.....electromagnetic phenomena.......I have noticed some minor but noticeable differences in stompboxs with different enclosures.

Steel vs Aluminum is primarily what I referencing.

The steel box seems to have a little more feedback potential with a given circuit than the aluminum box, but the steel box seems to do a better job keeping RF from interfering with the circuit than the aluminum box.

Now I'm sure this relasionship could be explained with a Maxwell equation......but what do we do to reduce the feedback potential of a steel box just slightly (not too much reduction).....I kind of like feedback.....the kind I can predict that is.


I know.....wrap the steel box in aluminum foil.....right?   maybe?   Any ideas?

[mac]

Yes we can do that but proof that with a Maxwell equation.

...

Just thought everyone should know that calculus does not find all answers.

Let's see... this will be technical and boring... you may skip this and turn on your amp

R, C and L can be treated in the same way, since they are variables that depend on the geometry.
But while C and L can be calculated directly using Maxwell eqs. or some other eqs. that are derived from Maxwell's, the calculation of R, in terms of the E field and the properties of the conductor, can not because you need to mix Maxwell and Newton. Note that the resistance of  C and L depend on the freq which mean that they are 'conservative' quantities, while R does not. This is because electrons are flowing and colliding with other electrons dissipating energy.

Using some transport considerations ( see Boltzmann transport eq. and ideas ) one can calculate the value of the resistivity

r = n.n.e.L/(m.v)

where r : resistivity, n: number of free electrons, e: electron charge, L; mean free path length, m: mass of the electron, v: mean root sqr velocity of electrons.

In a copper wire R = r.length of wire/area of wire

resistivity, thermal conductivity, viscocity are all calculated using transport theory. Since changes of momentum of a particle are involved, calculus need to be used. Ok, minimal calculus... but calculus.
Ohm's law can be justified in the same way as the linear thermal conductivity in many materials. One can think they are the two  sides of the same coin.

This discussion is similar to the thermodynamics expressions. One can say that all thermodynamics is empirical and no calculus is needed to find the pressure of a litre of oxygen at 25C. Just use the following eq. and a calculator, and bingo!

P = N.K.T/V       where N: number of particles, K: Boltzmann constant, T: temp, V; volume.

Very easy. But, are you sure that no calculus was involved? Think quantum statistical mechanics.
To justify this simple equation you have to:
1. calculate the eigenvalues of a single particle in a box. You need to solve Schrodinger eq., a differential eq. Calculus, aagghhh.
    Ei = const(nx.nx + ny.ny + nz.nz)          nx, ny, nz: from 0 to infinite.
2. add millions and millions of particles in the box and write down the total energy as a function of the eigenvalues of every particle.
    E = sumEi  i:from 1 to N
    The first particle can have this values (1, 300, 55), the second (44, 23423, 4) and so on. The next instant all values change and you have a different energy. Can you imagine all the possible combinations?
3. find the most probable energy, ie, calculate the partition function. You need to sum close values so you have an integral equation. auchhh.
4. after all this calculations you get a surprinsingly simple expression for the mean energy of an ideal gas:
    E = 3.N.K.T/2.

A chemical or industrial eng. use this expression all the time. He may tell you that calculus is not needed to find the pressure in a tank, and he possibly could live without knowing a thing about calculus... but the real truth is that the very foundation of his simple macroscopic formulas were justified from the microscopic level with the aid of quantum statistical mechanics and calculus.

Calculus is to physics what a tube amp to a guitar.
Calculus does not find the answers, physics & money do

mac

[Transmogrifox]

Calculus is to physics what a tube amp to a guitar.
Calculus does not find the answers, physics & money do

[KMS]

Nice try mac....good effort.  And, no I don't think it is boring but I'm done playing with my amp for the night.

A proof has no substitutions that cannot be proofed mathematically (with another proof) and should be included for full proof. I still don't see the complete mathematical path from Maxwell or Newton to........ I=V/R.   Anytime you substitute one thing in place of another with no mathematical proof that such things are in FACT equal then your making a jump and the mathematical relationship is not complete.  The proof would go from Maxwell\Newton if you like then to Ohm's Law with no comments inserted into the equation like

Ohm's law can be justified in the same way as the linear thermal conductivity in many materials. One can think they are the two  sides of the same coin.

I agree with your statement mac, I just don't have any mathematical proof that linear thermal conductivity in many materials=I=V/R.

Ok...another puzzle............Magnetism, Gravity, and Electronic charge attraction all have very similar properties but there is no mathematical proof to tie them together, at least none that I have seen.

Work on that one,  the rewards will be much better than trying proof Ohm's Law from a Maxwell equation.

Tubes?.......what about transistors.........just junk right?

[mac]

Here I go again... (I wish this forum were in spanish to explain myself better...)

1. As I said before, proof of Ohm's law, and thermal conductivity & viscosity, are derived using transport techniques, not using Maxwell's eqs alone. Ohm's law is proved only when the drift velocity of the electrons due to an external field is much less than the electrons thermal energy. Being very, very, ..., very crude, ( you may or CAN object) for each electron,
m.dv/dt = sum( r* ) k.e.e / | r - r* |^2  + e.Eext
If the external field does not alter the mean thermal velocity much, then r-r* is the mean free path between collisions, a constant.
Then, the sum over r* is N times a mean internal electric field.
m.dv/dt = N.k.e.e / d^2 + e.Eext = e.Eint + e.Eext
dv = e.E.dt/m
(I guess that if the Eext is small then the Eint is not modified and can be thought almost zero in the average???)
v = e.E.dm/(m.vm) + const
Leaving the constant aside, multiplying by N.e,
N.e.v = j = [N.e.e/(m.vm)].E
j = g.E
Which is the form of Ohm's law in terms of the current density. Roughly.
After a couple of margaritas and being 4:50am I guess this is pretty.

2. About similarities with the thermal conductivity, the flux of heat is proportional to the gradient of temperature in the same way the current is proportional to the gradient of potential, ie, electric field.

3. If my memory dont cheat me, a group of physicists found that the em and the weak force were the same entity, at least in the lab. Not sure if a guy with a white afro look called Albert were working hard trying to unify the gravity, em, and the strong and weak force that tie neutrons and protons in an atom. A good excercise for the mind, very challenging, but not rewarding at all even if I were granted the nobel prize. I would make more money selling magic ge fuzz face$.

4. ... and the original post was??? 

mac

[Transmogrifox]

I think the making $ part from selling magic Ge Fuzz Faces was a good tie back to the post topic--all to say you don't need to be a math junkie to sell quality magic smoke.  If you don't let the magic smoke come out, the guitar tone will be good indefinitely

[mac]

I think the making $ part from selling magic Ge Fuzz Faces was a good tie back to the post topic--all to say you don't need to be a math junkie to sell quality magic smoke.  If you don't let the magic smoke come out, the guitar tone will be good indefinitely

Well, I made a mistake. I should have written "I WOULD make more money selling magic ge FF".
In fact, I don't sell pedals. 

And I agree with you, you don't need a lot of math to build a good fuzz face, for your own pleasure or to sell it. Being a regular reader of this forum maybe is all the diy'er needs.

mac

[mac]

I'm with you on that.....nothing quite like playing around with a circuit to get it just the way you want it.

While we are on the subject.....electromagnetic phenomena.......I have noticed some minor but noticeable differences in stompboxs with different enclosures.

Steel vs Aluminum is primarily what I referencing.

The steel box seems to have a little more feedback potential with a given circuit than the aluminum box, but the steel box seems to do a better job keeping RF from interfering with the circuit than the aluminum box.

Now I'm sure this relasionship could be explained with a Maxwell equation......but what do we do to reduce the feedback potential of a steel box just slightly (not too much reduction).....I kind of like feedback.....the kind I can predict that is.


I know.....wrap the steel box in aluminum foil.....right?   maybe?   Any ideas?

I heard about that.
A guy I met at guitar store told me something about "different potential energy inside the box when using aluminum instead of steel". That guy was so high I did not pay much attention to him.

mac

[KMS]

mac,


Interesting....hum....

Einstein died trying to tie magnetism, gravity, and electronic attraction/repulsion into one common mathematical model/formula.  He couldn't do it, so any field effects that you can completely explain mathematically as a direct link to collisions (collisions would be electronic attraction/repulsion) then at that point you will have done part of what Einstein could not do and you will be the most famous mathematical mind on earth when you do it. So keep trying. Einstein was sure it could be done....so it most likely can be done and if it can then a Maxwell equation could also be used.

Solving this would be of great significance in the world of science for many reasons.  One thing that would result from it would be a mathematical explanation of ohmic and nonohmic materials where the same formula/model would be used to calculate the resistivity of both ohmic and nonohmic elements and compounds. The model would not fit Ohm's law unless it was applied to the physical and chemical properties of ohmic materials and such a model would aslo hold true for nonohmic materials.  Field effects would also be calculted across nonohmic or non conductive materilas.  I think that such a model set up to define Ohm's Law would show that the nonconductor/nonohmic materilas would give results that would appear as a node on the graph or undefined or imaginary or negative/positive infinity, so on and so forth.


Tying Gravity into this would be all that is left after you finish up field effects and collisions.  I think your onto something but not quite done with it yet.

As far as the steel box vs the aluminum box...it is valid that there are small differences for circuits within such enclosures...I have just took a BOSS OD2 out of its original Aluminium box and installed into a steel enclosure and I get just a little more feedback.....at first I thought this was a problem but I think (emphasis on "think") that I am also hearing just a little more treble response from the BOSS OD2 in the steel enclosure.  I don't know what is causing it but by turning the tone to a mid-range setting on the BOSS OD2 instead of max treble like I usually have it set at when I get the feed back I have eliminated the problem.

Weird how the steel makes a difference.  I know that aluminum is a paramagnetic metal and steel is not and this difference in properties is most likely the root of the difference I hear in the sound.

[no one ever]

and whats the deal with mhos and darafs?

what component creates those?