Optimizing pickup output part 1: equations, waveforms and pole piece shape.
This time around, we are going to look at the methods and math involved in increasing/optimizing pickup output, as well as the variables and caveats involved in applying those methods.
In it’s simplest form, the output of a pickup is dependent on the same factors as any other EMF device, which is 3 factors: coil geometry, speed(frequency), total flux. There are many smaller factors involved, but first we will tackle the big 3 (the basics of Faraday’s law of electromagnetic induction in terms of EMF).
E = -N(B/t)
E = electric field (voltage)
B = magnetic field (flux density)
N = turns of wire (um… turns of wire)
t = time (frequency)
(For the sake of simplicity, we are going to assume (at first) ideal conditions, optimized geometry, perfect coupling, zero leakage, peak values, perfect sine wave shape, etc, etc.)
So from this we can see that we get an equal and linear increase in output for any unit of increase in any of the 3 parameters (i.e. 2x any one parameter (B, t, N) = 2x output). Let’s put a slightly finer point on those parameters to make them more useable. N is simple in that each turn represents one loop, so the total is just the sum of the loops, but the actual values of E, B, and T will be the CHANGE in value between max and min conditions (change=delta=d). So dt is the time to change from min to max flux condition (one cycle), and dB is the change from max to min flux. SO:
E = -N(dB/dt)
So lets break down dt and dB:
This is fairly simple at face value… BUT… if you recall from my past post on harmonic content of string vibration (the poles vs blades post), a pickup integrates the vertical and horizontal movement of the string into a complex waveform, which varies in it’s nonlinearity and X-vs-Y ratio depending on the position of the strong in it’s elliptical vibration path. So if we consider just the pure fundamental sine from the vertical movement, and just the pure first harmonic from the horizontal movement but at half amplitude (which would be represented by the impossible condition of a perfectly shaped pole piece and perfectly flat string movement back and forth at a 45 degree angle) we would get this (I would like to cite these diagrams, but unfortunately I didn’t take note of the site I pulled them from. I think it was about additive synthesis… oops):
Which is quite different than the sine wave that we assume in the simplified math. To make matters even more complex, we know that the string itself vibrates in a series of smaller standing wave ripples, the number and amplitude of which depend on a variety of factors including energy (picking force), string length (scale length and fretted note), string mass, alloy composition, harmonic damping from the guitar body wood/shape and hardware, etc. Which end up looking like this:
Which causes our complex output waveform to resemble a sawtooth rather than a sine which will, of course, change shape with varying pick attack, rotation of the axis of the string’s elliptical path, and the decay of energy as a function of time from the pick attack, etc:
That is all to say that the variable is too complex to compute for our examples, so we will use the peak value and fundamental frequency.
The basic equation would represent the difference between zero and an arbitrary value, but we do not have a zero condition, so instead we use the change in flux between one complete waveform cycle’s peak and trough. Assuming a symmetrically-shaped pole piece (i.e. mirrored symmetrically at the axis), we have to consider flux condition at the maximum and minimum coordinates of our movement. BUT our movement is elliptical, and the axis rotates, so we can break this down into our max and min X and Y conditions:
-Vertical (axial) string movement is done easily enough (kind… more on that later), and is by far the simpler of the two since we have an exponential decay in field strength as we move away from the magnet, so we find the flux density our maximum and minimum positions in the string’s path and calculate the difference. We will use rare-earth magnets in the following examples because their demagnetization curve is linear (Alnico is non-linear and varies from grade-to-grade, so it makes the math much more difficult, so it is more easily computer-modeled than computed).
Taken from magnetsales.com, the equation for flux density as a function of distance from the magnet’s face along the center axis is:
Bx = flux at x distance
X = distance from face
Br = Residual Flux Density
R = magnet radius
L = magnet length
The shape of the magnet determines the shape of the field, hence the complexity of the equation. This sets the operating point, along with the air gap. This is called the Permeance Coefficient, which reflects the effect of self demagnetization (in that magnetic flux will pick the easiest (shortest/lowest remittance) path from it’s north to south pole… so the shorter the magnet and the larger the air, the smaller the permeance coefficient. Here is a handy calculator: http://www.magnetsales.com/design/Calc_filles/FluxDensityPlainDisc.asp and another one here http://www.arnoldmagnetics.com/Technical-Library/Calculations/Calculating-Permeance-Coefficient/Axial-Cylinders We can plug in the coordinates for our string max and min positions and see the change in flux while avoiding messy hand math (wooo).
Here is an example to show the percent of max flux we have when plotted against distance from the pole piece face in a .187″ cylinder pole piece:
From that we can see that the density drops off very quickly as we move away from the magnet’s face (inverse square of the distance)
-Horizontal string movement is a bit trickier. To boil down our change in flux to a simple equation like we did for vertical movement, we would need an infinitesimally-small pole piece diameter. Obviously that is not practical, so we have to take the shape of the pole piece head into account, as well as the diameter of the pole piece in relation to the width of the string’s vibration.
As we showed in our blade vs pole blog, horizontal movement across the magnet’s face results in very little change in flux (if any), so we get little induced voltage change (and hence harmonic contribution) in a wide&flat pole piece if our string does not pass over the edge of the pole piece in it’s vibration path. Areas of small string movement (bridge pickup) will not provide much horizontal movement unless we have very small pole pieces, whereas areas of large string movement (neck pickup) will be more likely to pass into areas of lower flux density, and have more contribution of horizontal movement to the signal.
Here is a graph of an actual pickup response that I measured, superimposing the vertical component over the horizontal component. The pickup was a fender-style single coil with alnico poles.
Red is vertical
Green is horizontal:
You can easily see the rectified effect (harmonics only) of the horizontal movement. (I scaled up the horizontal component so that it would be more visible.) Here is the combined waveform of the two signals:
Here is a video where you can hear clips of the isolated vertical and horizontal movement, and see the associated waveforms. You can easily hear the higher-pitched/more-trebly timbre/pitch of the horizontal movement, and the bass-ier, lower timbre/pitch of the Vertical movement:
Here is the (ghetto) test rig:
The curve itself (as we mentioned in a previous blog post) is not linear, and in the case of alnico pole pieces has it’s non-linearity effected heavily by the grade of Alnico used, initial operating point, and position along the string (more displacement). Here is a graph from that blog that shows two points plotted along the BH curve:
Also, depending on the shape and dimensions of the pole piece, we can experience a null or even less flux in the middle of the pole, which would make it less sensitive to small horizontal vibrations, emphasizing the vertical component more.
Here is a plot of the change in flux density-vs-horizontal position for a cylindrical pole piece:
(horizontal in red)
We can see that there is a decent amount of sensitivity in the middle, but with a sharp peak at the edge of the cylinder, and a more gradual slope (past the edge of the pole) than the vertical movement .
In contrast, here is a vintage beveled pole piece:
We can see that the bevel produces a null in the center, which would set up a different set of harmonics than the pure cylinder.
And as an example of how to match the shapes more closely, here is a 180 degree rounded pole piece top:
As you can see, the spike near the edge is gone, there is no null in the middle, and the horizontal and vertical slopes are more closely matched.
And finally, here is a blade pickup, zoomed in to represent a single string’s movement:
As you can see, the horizontal variation is rather small in comparison to the vertical variation.
As a side note, there is somewhat of a common opinion that blade poles provide more string-to-string horizontal consistency, but that is not true since the flux magnitude and eddy currents vary along the blade, so while they are more consistent per-string, they are not from string-to-string as we can see in this flux density pic:
ALL THAT NONSENSE is to say that it is easier to model or measure horizontal movement with regard to flux variation, and then… you know… actually build some pickups with different pole shapes to hear the difference, or by using a rig like the one I set up for the blade vs pole blog to measure the horizontal output voltage for a particular pole shape/diameter/string-movement. We can however save some time and shooting-in-the-dark frustration though by modeling the pole shape in something like FEMM (see the intro to FEMM blog for details) to acquire the values, and then plotting them against the vertical values to see the harmonic makeup of the complex wave that results.
So practically, that speaks to the subject of optimizing for output parameters by allowing us to map a 2D X-Y plot of the current density around the top of the pole piece to measure the change in flux with respect to various points in the string’s vibration path so that we can plot the complex waveform and view both the output and harmonic content.
(One interesting thing to notice is that depending on the shape of the pole, a large flux null area in the middle can cause a “flat spot” or “deadband” at the waveform zero-crossing. This is analogous in amplifier terms to crossover distortion in class AB/B/C push-pull amplifiers. The resulting sound is considered to be a harsh buzz of high-order harmonics (since the waveform is very flat). I am actually uncertain as to the extent of this effect in a guitar pickup, but I will absolutely run some tests in the future to see if the audible effect is also analogous. If it is, that is yet another variable to consider in pickup design, or at least something to avoid… we shall see…)
This has gone too long, so that is the end of part 1.
Check out part 2 where we will delve deeper into the effects and pitfalls of maximizing the parameters in our E=-N(dB/dt) equation with respect to the restraints presented in the format of a guitar pickup, and the effect on the tone.