I am constructing a magnetophotoluminescence system with a basic electromagnet, driven by a Keithley 2450 (fancy power supply). I would like to pulse it on and off quickly, to maximum current/field strength. The time response is limited by the inductance of the inductor, and I would like to know what handles and trade-offs we have. Good excuse to write an article!
Inductor basics
Inductance (ℒ) is the tendence of a conductor to oppose changes in the current flowing through it. The electric current produces a magnetic field, which if changing induces a back-emf on the circuit.
The inductance of a coil (ideal ferromagnetic core) is
ℒ = μ·N²·A / ℓ
where μ is the permeability of the core, N is the number of windings, A the cross-sectional area, and ℓ the length of the coil (parallel to screw axis).
The magnetic field from the electromagnet is
B = μ·N·I / ℓ
Note that this will saturate - think saturation in the B-H curve of ferromagnet. The gradient (B/I) will be far lower above the saturation of the core.
Critically the field may be increased without increasing the inductance by only adjusting the shape of the electromagnet. By reducing the cross-sectional area of the inductor for the same length, we win. However, this will cause a geometric tradeoff. The scaling of the solenoid's magnetic field strength and distribution depends on its shape: it decreases with solenoid length (1/ℓ) and spreads laterally, influenced by the ratio of cross-sectional area to length (A/ℓ). This trade-off is significant when it comes to concrete applications. Let's see if we can optimise the time-response for a given inductor geometry, then.
RLC circuit
An RL circuit has a fundamental rise/fall time constant
τ = L / R
due to the inductor resisting changes in the current by creating a back-emf. However, if there is any capacitance in the circuit (intended or parasitic, e.g. between windings in the electromagnet), we can get more complex time dependence in the step-response.
The RLC (effective) circuit equation is
ℒ d²i + R di + i/C = 0
where the d is a time derivative. This system has a natural, undamped (R=0) frequency
ω₀ = 1 / √(ℒC)
The damping ratio (ζ) determines how these oscillations decay over time for R≠0
ζ = (R/2) √(C/ℒ)
If ζ<1 the oscillations ring (underdamped), for ζ=1 we achieve optimal transition (critically damped, no ringing), and for ζ>1 there is a slow non-oscillatory transition (overdamped).
Optimising our RLC circuit
We want ζ=1, and we also want minimal τ = ℒ / R. Again, we assume we cannot reduce the inductance ℒ. We can, however, increase the resistance, which reduces τ.
If we are over-damped, there's not much we can do: we can't reduce the resistance or the capacitance. We might be able to add another inductor, but that would come with increased capacitance and resistance as well - this would be tricky to optimise.
If we start underdamped (small C and R compared with ℒ), then we can increase R, limited in how high we can go by the parasitic capacitance. This is kind of nice in the lab: keep increasing the resistance (if you have enough voltage to drive the electromagnet and cross the resistor) until you see minimal ringing and you know you have the 'best' time response.
Decade boxes
I have recently been doing a lot of electronics for the lab (see the lab-amp post). If I had planned this, I might have bought some decade boxes - these dramatically improve the prototyping experience. Basically, they are resistor/inductor/capacitors that vary across many decades. Very useful and ~AUD100 on ebay!
Outcome
Turns out my electromagnets were all sufficiently fast for the time-response of the systems I am measuring. If I change to measuring NV-diamond or hBN spins, I might need to optimise further. I must say I am happy I don't need to test a bunch of resistors
Outstanding Questions
We only have cheap electromagnets designed for simple applications, can I find some electromagnets with defined C, ℒ etc.? Are there electromagnets commercially available that are optimised for fast, strong switching?