Ah, I see.. it's more elementary. Then...
dI/dt is the time rate of change in current. If everything stays constant over the time period dt, then you can change ds to deltas and the calculus is trivial:
V = L * deltaI / deltat
For a constant voltage and constant inductance, current rises linearly with time (or falls, if voltage is negative... more importantly, it's always in the same direction as the voltage is).
This is very handy, because pretty much all switching supplies have some part of the waveform where we hold the inductor voltage constant -- so we immediately know the current must be rising linearly during that time. Cool!
Notice the mean value of current does NOT depend directly on voltage, as it does in a resistor (V = I*R). The mean value can be anything -- the inductor only cares that the current changes. So, to get a certain amount of current (like the output current from a buck converter, to supply a load), it's the duty of the circuit to control the voltage, on and off, to keep current in the right range.
When working with magnetism, there are always two complementary aspects at work:
- Voltage and current change
- Flux and current (or more accurately, magnetomotive force, MMF -- amp-turns)
- Flux density and magnetic field strength
Flux is the time integral of voltage: although you measure voltage directly with a voltmeter or oscilloscope, you can also measure flux, too (of course, voltmeters aren't any good at measuring small flux... but scopes are great). Flux is in units of volt-seconds, the weber (Wb).
Volts and amps are the most important in circuit, because we have instruments to measure them. Both flux, amps and volts are all measured in circuit, between points.
So, flux is the time integral of voltage, and current is the time integral of, well... the change in current. Or voltage is the derivative of flux, and dI/dt is the derivative of current. Same thing, different direction. (If all this calculus is throwing you for a loop, do look up a basic explanation of calculus -- it's actually very simple and intuitive, concerning how variables change with respect to others. It's about curves and slopes and such. There are many good videos and websites out there that do a good job. Sadly, I don't have any to recommend offhand.)
Flux density (variable usually B, units tesla T == Wb/m^2) and magnetic field strength variable usually H, units A/m) are the fields-in-space, bulk equivalents to the others. Just as resistivity is the bulk equivalent of resistance (a circuit property, measured between two terminals), and so on. (Again, if fields are throwing you for a loop -- there are many great videos explaining what fields in space are, maybe even including vector fields, and some vector calculus -- yes, merely how directional variables change with respect to others! There's probably some good E&M tutorials too, hmm...)
Inductance is the conversion factor between flux and current, or between voltage and current rate-of-change. 1H == 1 Wb/A == 1 Vs/A.
Permeability is the conversion factor between flux density and magnetic field strength: mu_0 = 4pi x 10^-7 H/m.
So, permeability is the bulk equivalent of inductance.
By the way, any time you see a bulk property, it's usually that it's in units of "something-meters", or "something per meters" -- these usually use a times-two-divide-one sort of ratio to bring the "something" out. Like, resistance goes up with length and down with area, so resistivity is ohm-meters, and resistance (the circuit value) = resistivity (the bulk parameter) * length / cross-sectional-area. Other times, you see "something per cubic meters", which is a density. Ah, but sometimes you don't see either, like pressure, which is a force per area. Ah, but if you break down the units, you get 1 Pa == 1 N/m^2 == 1 J/m^3 -- pressure is actually energy density! Which is why it takes work (with a capital 'W') to pressurize something -- you're putting energy into some volume.
There are just a few bits missing before you can design magnetics from scratch:
mu = mu_r * mu_0
mu is the permeability of the core (H/m). mu_r is the relative permeabilty, typically 1 (vacuum, etc.), 10-100 (powdered irons, gapped ferrites), or up to ~10,000 for ferrites, or even more for special materials.
Since we have core materials of reasonably high relative permeability, we can make the approximation that, instead of flux flying around through all the damn space around a winding (needless to say, integrating over that space would be painful!), it's entirely confined to the core. This isn't true, but the value of an approximation is in its usability: as long as the permeability is more than 10, we can expect it's true to within an error of maybe 1/10th (10%). Or over 100, 1/100th (1%), and so on. So it's an approximation, but it's very reasonable for engineering purposes. And since your average powdered iron is around 75, or ferrite around 2000, it's probably a better approximation than the mechanical measurements of the cores themselves, so it's lost in the noise. Good approximation indeed!
Now that we have a core, which traps "all" the flux from the winding, we know where all that flux is going. Aha, now we can calculate things easier.
How long of a path does the flux travel?
How much is its cross sectional area?
You might guess from the units that flux density is flux per area, so we need to know the area. A_e is the effective area of the core. l_e is effective length. These, together with mu_r, are all we need to describe a core to reasonable accuracy. Now we can design inductors and transformers.
And now that we can completely simplify the geometry into a cheesy diagram, ALL the equations that are left are nothing more than dumb, easy ratios. Bitchin'.
Those are...
dB = V*dt / A_e (increase in flux density, in a core of cross section A_e, after applying a flux V*dt)
B = mu * H = mu_r * mu_0 * H (bulk inductivity)
H = N*I / l_e (magnetic field strength: N = number of turns around or through the core -- N*I is amp-turns or MMF)
A_L = mu * Ae / l_e (circuit inductivity -- the inductance you get from a single turn. Note the "times area per length" conversion from bulk inductivity mu into pure circuit inductance!)
L = A_L * N^2 (inductance for a winding of N turns)
How about some more complicated ones? Sadly, all our core materials saturate, so we don't want to force them to run at excessive flux density. Even then, we don't usually want to go very high, because core losses (heating) go up with B as well. You determine Bmax (the peak operating flux density) in this way, either based on saturation (less a safety factor) or how much heat you can allow in your core.
The peak flux of a square wave is V / (4*F). (Note that F in Hz == 1/s, so V/F is Vs = flux.)
- The 4 appears because, if flux starts at zero, and the square wave starts in the middle of a flat top, then for the remaining half a flat, flux rises to peak, then the wave's flat bottom drops flux through zero to -peak, then the next flat top raises it back up through zero (where we started in the cycle) up to +peak, and so on.
- If you're working with a sine wave, instead of 4, you use 4.44 (~= sqrt(2)*pi), and Vrms. (For the square wave, Vpk = Vrms so I don't need to specify for that case.)
- If you're working with a single switch type converter (like a flyback, boost, buck, etc.), flux starts at zero and goes positive, then back to zero; it never goes negative. So you have to use 2 instead of 4. Or, just take t_on * VCC, and that's that.
Smushing together a few more, and you get...
N = V / (4*F*Bmax*Ae)
The number of turns you need for a transformer*, to support a square wave (balanced, not "single switch") voltage, at frequency F, peak flux density Bmax, and cross sectional area Ae.
Note that current and permeability don't appear in this equation. Ideally, transformers do not draw magnetizing current at all (permeability is infinite). So that's a good thing.
*Or inductor...
But, for an inductor, we WANT to draw current. How? By using a small permeability.
In the above equations, whenever you see mu, you can put in whatever the effective permeability is, as seen by the flux flowing through the loop. If you happen to put air gaps in that loop, you get a lower effective permeability.
So the last bit that's missing is, how lengths and gaps work together.
l_e is the effective length of the core. But the core has a very high permeability (ferrites and such). We can do two equivalent things: translate everything into terms of air gap alone, or in terms of average core alone.
If we think of the core as "air gap equivalent", this is simply l_c = l_e / mu_r. A core with A_e = 100 mm^2, l_e = 100 mm and mu_r = 2000 has the same inductivity as an imaginary wafer of air, with cross section 100 mm^2 and thickness 100/2000 = 0.05 mm (about the same as tissue paper). This isn't an imaginary quantity, though; it's very real. It would be fantastic to build a winding so it drives a really tiny volume of air, but this is impossible, so we use cores effectively to focus the magnetic field down to smaller volumes of air, so that we can get higher energy density!
(And guess what: as I said before, energy density is also pressure. In fact, the attractive or repulsive force you feel with magnets is precisely this pressure, which is the energy density of the fields around the magnets. The energy density is e = B^2 / (2*mu). If you run the numbers for mu_0 and 1.5T (typical of an NdFeB magnet at the face), the pressure's not bad at all -- which is why they can be so dangerous to handle!)
Note that you don't get much air to store energy in, if you just use an ungapped core. You can add more, by simply gapping it (shimming the core pieces apart, or grinding down the center peg -- note that, if you use shims, you must count them double! The gap length l_g is the total gap around the loop, not necessarily the physical gap length.)
Powdered iron cores already have fairly low permeabilities (10-100), and they're usually offered in formats (like solid toroids) that are inconvenient to gap. The explanation is that they're made up of powdered materials (which have high permeability, in and of themselves), which doesn't pack very well, so the air gap is distributed between particles, rather than concentrated at one gap.
Whatever the case, whether it's a distributed gap or an explicit one, you can find the air gap equivalent l_eff = l_c + l_g = (l_e / mu_r) + l_g. Since this length equivalent is also an air equivalent, use mu_r = 1 (or mu = mu_0) in the earlier equations.
The alternate view is to find the average permeability. Given the resistivity of copper, and typical core properties, it seems the best values are in the range of mu_r = 10 to 30. (I haven't yet evaluated why this is, but it's certainly the empirical case. The resistivity of the copper winding is important -- because, if we could just send infinite current through a teensy bit of wire, we wouldn't need cores at all!) Too low and you need so much wire that you incur more winding losses; too high and you end up needing way more core than you should (high mu --> low energy density!) and thus incur more cost and core losses.
In this view, we are calculating the average permeability by assuming l_eff = l_e. We already know l_eff (the mu_r = 1 equivalent, I'll call it l_eff* now) from above, so mu_eff = mu_r * l_e / l_eff* = mu_r * l_e / [(l_e / mu_r) + l_g] = mu_r / [(1 / mu_r) + (l_g / l_e)]. The last form is neat, because it's dimensionless: it's the reciprocal of the sum of reciprocals of mu_r (already a ratio of mu) and l_e/l_g (which is how many times longer the core path is than the air gap path).
Finally, selection.
Ferrite cores can be gapped to order (or ground or shimmed by hand), so it's best to design by flux density first. Once you know how many turns you need, you can calculate the required gap l_g by working backwards first from inductance, then from A_L, and then from l_eff or mu_eff.
Alternately, if you're working with gapped ferrites in a catalog, or powdered iron toroids that you can't gap, you can ignore the last, oh, ten paragraphs or so -- because, the Ae, l_e and mu_eff are what you get, end of story. You can design based upon required inductance (N = sqrt(L / A_L)) or flux density (the 'transformer method', if you will?), running the calculations for every part that's likely close enough for use, and finding the one the most suitable. Once you have some choices narrowed down, you can refine further: check the core losses, adjust turns for exact inductance (or as exact as possible), stuff like that.
Sadly, there is no formula to suggest an ideal core -- or perhaps fortunately, because physics would be really weird if that sort of thing were nailed down from so little information. You can make guesses based on typical geometries (an EE core is going to have certain ratios of l_e and Ae, etc.), you can include copper losses in the calculation (and how much area there is for winding the copper, the winding area Aw), and so on. Mostly, you're just cranking numbers over and over, because there isn't a comprehensive spreadsheet across manufacturers from which you can run all the calculations at once to pick the best parts. Alas, that's where engineers have to pick up, it seems.
Tim
Home
Help
Search
Login
Register
