Regulators are quite important to electronic circuits. Why? It's hard to stand up straight when the ground is shaking, isn't it? Same applies to electronics. There are two important regulators, constant current sources/sinks and constant voltage supplies, both of which work on roughly the same principles, thanks in part to Ohm's law and the nature of feedback circuits.

Math? Not much.

To make any useful evaluation of a regulator, we have to define some parameters first. We are interested in one of two things: constant voltage or constant current. Constant voltage means a small change ΔV for a large change in current ΔI, and constant current means the inverse. The ratio, ΔV / ΔI, is the source impedance. (If you don't already know it, a look at Thevenin and Norton source theory would be valuable here.) If we have a perfect Thevenin source, then it will have an open circuit voltage V at I = 0A, a voltage (V - I*R) at I > 0, V > 0, and a current I = V / R under short circuit conditions. Into a load resistance RL = R, VL = V / 2 and I = V / (2*R), for a maximal power dissipation in the load resistor (and in the supply, if you're counting!). By varying RL from infinity to zero, the voltage and current go all over the place. Not very interesting on a regulator page, right? Quite so. In fact, this shows why regulators only work over a narrow range! If we set current to, say, 1/10th of the short-circuit current or less, then voltage stays within 10% of the rated output voltage, which is suddenly a lot better. Note the caveat that power output must be a whole lot less than the power output maxima. But, efficiency is higher, because less power is dissipated in the supply. Fundamentally, this is what keeps your outlet at, say, 120V, as opposed to 60V when you turn on your toaster oven: the wiring resistance, fuses, transformers and generators between your oven and the -- whatever turns the generators at the power station -- is small enough that the voltage remains stable. And yes, this means that you can easily draw from the outlet peak currents more than 20 times the average rating! The power company knows this, and they therefore have regulations in place so your wiring is protected with fuses. In fact, those fuses are probably rated for something silly like "10kA" -- that's ten thousand amperes -- peak current, which I suppose is specified as not catching fire or turning into a dangerously large ball of plasma or something. Such current only lasts a few miliseconds, not enough time to heat up the wiring and cause a fire, but long enough that the lights (other lights in the house, your neighbor's, etc.) dim momentarily when you carelessly pass your Sawzall through a wall you didn't realize had live wires in it!

Current sources, likewise, have a very high source impedance. Interestingly, there aren't as many passive constant current devices; most depend on a Thevenin model, where a constant voltage source of absurdly high potential is reduced with a large, hot resistor, down to an approximately constant current. Some mechanisms have been devised, however. A design comes to mind for street lighting ballast transformers: a primary and secondary coil are placed together loosely, with the secondary mounted on a lever with a counterweight. Gravity holds the windings together and voltage output is high. When loaded, current goes from zero to something, and by Lenz' law, a force is developed between the windings, pushing them apart. The secondary comes to a new equilibrium location. The force against the secondary is proportional to the magnetic field, which is proportional to the current flow. All that's needed is a constant force against the secondary to set the current. Notably, the mass and inertia of such a swinging secondary allow ugly behavior such as a strong capacitive effect and the possibility of oscillations. When a large current is demanded, the secondary just sits there until the increased force gets it moving. Once moving, it will tend to keep moving, and swing too far. Then it falls back, and bounces off the primary, which repels more when the secondary swings in too fast. It acts like a spring, in fact. Clearly, a damper (electrical or mechanical) will be needed on such a system, and minimal mass of the secondary and its support are key to stability.

Reactive "Constant" Circuits

Capacitive CVS

Of interest to electronics, the simplest voltage source is a small resistor. Under some circumstances, this is quite sufficient. I've already described the resistive basics, so I needn't say more. I haven't said anything about reactive components, though. Where there are AC signals, reactances can come to the aid of many problems.

Here is a typical application for a capacitor, and as you might've guessed from the triode symbol, it is "ancient" in origin. Where you see a line with some signal on it that shouldn't be there, you may find a capacitor there, and such a capacitor is called a bypass. At low frequencies, the capacitor looks like an open circuit and it therefore has no effect on DC operation. At signal frequencies, reactance drops (XC = 1 / 2*pi*F*C) and signal current goes through the impedance R3 || XC (where "||" indicates paralleled resistance) = R3*XC / (R3 + XC), which drops as frequency and capacitance rise. Capacitor-filtered power supplies are better than anything else at high frequencies. Say there were a bypass capacitor from +280V to ground. At signal frequencies, the small reactance makes the power rail look like the ground rail (it is said to be AC-grounded), which is akin to shaking a weight back and forth with your body firmly anchored against a wall, as opposed to standing free. By the way, this is the reason that, in representative diagrams, you'll often see completely stupid things, like collectors and bases and emitters grounded through various things, with no obvious DC path: the diagram goes only for AC. Use your imagination to add DC bias and supply and AC bypasses to get an equivalent circuit that works.

Inductive CCS

Here is a typical example of a reactive constant current source. Often, a choke or transformer is used to couple signal from the plate. At DC, the inductor looks like a short, so we can immediately see plate voltage VP = B+ (realistically, minus IP / DCR, but I'll ignore that for now). At signal frequencies, however, the inductor has a reactance of XL = 2*pi*F*L, and for XL >> RP, the plate doesn't even know there's a load present and it can be called a constant current environment. Typical values for L range from nanohenries for UHF and microwave circuitry, to micro and mili- for RF chokes (RFCs) and SMPS filters and other medium to HF range filters. Narrow band (resonant) RF applications have the added bonus of a parallel capacitor, which resonates with the inductor, pushing reactance to infinity. For tubes, inductances up into whole henries are common. A low impedance audio power output stage might call for single digit H, while an interstage transformer or "reactance coupled" stage might call for hundreds. 10kohms reactance at 20Hz is a pretty large inductance indeed; fortunately, it need only handle a few miliamperes, so it's not physically as large as the number might imply.

And, incidentially, isn't there something odd about that? How I said the DC plate voltage is exactly B+, when talking about constant current? Ah, isn't that interesting then: a reactance acts as constant voltage or constant current, depending on what frequency you're looking at! In this case, at frequencies where XL ≤ RP || RL, it's roughly constant voltage. Likewise, the capacitor above acts as an open circuit (constant current) at low frequencies, and constant voltage at high frequencies. How much of each of course depends on your definition of "constantness", but the transition point is clear: when the reactance is on par with the rest of the circuit's impedance.

Active (DC) Regulators

What if we need an absolute voltage or current, one that doesn't depend on the supply? There are no absolutes in passives, only ratios, so resistors are out. Besides, the response has to be constant without regard to frequency. It just so happens that nonlinear not-quite-passive devices exist, and without too much trouble, can be used to make reasonable constants.

Constant Voltage

0G3 Voltage Stabilizer

A blast from the past, here is a plot of the 0G3 voltage stabilizer tube. A 0G3 (not "an OG3" -- it's a zero, not the letter "O") is a low voltage gas-discharge tube. The 0G3 is specified as 85V ±2V at a current of 6mA. Maximum ignition voltage is 125V. Whereas most gas regulators (0A3, etc.) are intended for direct use with a load, 0G3 is a precision reference and intended for operation at 6mA. Curiously, the plot above suggests 0G3 has positive dynamic resistance; most glow discharges have negative dynamic resistance. This is an important factor to remember when designing power supplies, because only a minimal amount of bypass capacitance can be used in parallel with gas tubes.

Zener Diode

One of the most common voltage regulators used today -- heck, used for the last four decades since they were introduced, zener diodes are pretty reasonable voltage sources. A typical I-V curve is shown above for the 1N4736A by Hitachi. Like most silicon devices, zeners typically have an exponential breakdown curve, which means it's pretty flat over a decade or so of current (10-100mA, say). Regulation seems to be better for the higher voltages, more than 5V or so. Zeners are quite well behaved; unlike gas regulators, they can be bypassed with considerable capacitance. True zeners (breakdown due to the zener effect) have a negative temperature coefficient, but the zener effect only occurs in the lower voltage "zeners" (under about 20V). At higher breakdowns, avalanche effect rules (which is essentially the semiconductor equivalent of a plasma discharge), which has a positive tempco. Inbetween, however, there is a proportion of both where the tempco is zero: this occurs right around 6.2V, making the 1N4735 and nearby values particularly stable and very useful as voltage references. (Incidentially, semiconductor avalanche can result in negative resistance under certain circumstances.)

Constant Current

One of the most mundane examples is a lightbulb. The filament is made of tungsten, whose resistivity increases with temperature. By running power through the filament, it heats up and resistance increases. So current decreases relative to an ohmic resistor. More voltage will cause more power and more heat and more resistance. Measurement of this is often a lab assigned in physics courses, and is a good exercise in busy-work and graphing for the intrepid student. I would include a plot here, but I don't have my lab book anymore, and all I can find online are course outlines describing exactly this assignment! Suffice it to say, the characteristic looks somewhat like the √x function, so as voltage increases, current increases slowly, making a lightbulb a rudimentary constant current source.

C1 Barreter

Advantage has been taken of this characteristic. Here is the V-I plot for a C1 barreter, taken from a Philips datasheet written about half a century ago. The basing diagram and symbol are in the lower right. If I recall correctly, these use an iron wire strung up in a hydrogen atmosphere, sealed in an envelope. Iron's resistivity increases dramatically with temperature, so it certainly should make an excellent constant current element. As we can see above, this device makes an excellent CCS for about 201mA. The curve crosses 200mA at 100V, suggesting it's designed to be operated at this point. This device was used in European radios and televisions, where the tube -- sorry, valve heaters were strung together. 100 and 150mA heaters are typical, but stacking all of them up to 240V makes for an impressive power-on surge. Inevitably, a small part of some heater heats up faster than the rest and blows out. This is bad, so what they often did is they knocked up heater current to 200mA, therefore requiring less voltage for the same heating power. The difference was taken up with a resistor, which limited the current to some extent, making things much easier, but still not great. Though the barreter is a heated filament itself, it heats up faster than the valve heaters and is able to protect them and supply a constant current, heating them up evenly. The barreter still has to heat up, so a resistor was still used, but not as much was otherwise needed.

2N3819 Drain Curves

Here's a more modern example. Field Effect Transistors make excellent current sources. Whereas the barreter depends on temperature, a slow effect, FETs operate almost instantaneously. Drain curves for the classic 2N3819 are shown above (from a Vishay data sheet). The curious advantage to FETs is, because they conduct a well-defined current at zero input (on the gate), you can short the input (tie gate to source) and you get a two-terminal current regulator. This particular transistor (keeping in mind 2N3819 is specified from 2-10mA Idss) evidently saturates at 6-7mA. They make these as standalone units: current regulating diodes (CRDs). Their hidden FET nature is evident from their distinctive temperature coefficient.

Amplified Regulators

Two-terminal regulators all have some physical feedback mechanism to explain their non-ohmic behavior (temperature feeds back to resistance in the barreter; ionization feeds back to voltage drop in the gas tube; width of a depletion region feeds back to resistance of the FET), so putting them aside from "amplified regulators" is a bit of a misnomer. To be more specific, here I will be talking about regulators designed with explicit error amplifiers. These are generally two- and three-terminal regulators (although many-terminal regulators exist, they are all combinations of three terminal regulators internally). Two terminal regulators are connected in parallel with the load, while three terminal regulators are (partly) in series with the load.

Active regulators come in two styles: pass and shunt. Pass regulators start with a constant voltage source and, using feedback, deliver constant voltage or current to the load. Shunt regulators start with a constant current source. Shunt regulators are less efficient because they waste current at idle; for this reason,

High Voltage Regulator

This circuit illustrates a typical discrete linear regulator. This is a three terminal, constant voltage, series connected regulator. Its purpose is to deliver a constant 80-120V at low current to a small load, from a low voltage supply, while tolerating a wide supply range. The left half is a power oscillator and voltage doubler which generates 150-300V, well above what is desired. The right half is a pass regulator which drops this to the required output.

The top-center 2N3439 is an emitter follower, biased up by a 330k resistor and pulled down by another 2N3439. Bottom center, a zener diode reference sets the voltage on the left side of the differential amplifier (a crude one built with resistors). If the output voltage were to rise, more voltage would appear on the right side, causing the left side's collector to rise, turning on the voltage amplifier and pulling the output back down. A current regulator is also provided to prevent short-circuit damage: the 2N3904 across the pass transistor will shunt current from the 330k through to the output, causing the emitter follower's output voltage to drop, when enough current flows through the 330 ohm resistor to forward-bias the transistor (about 0.7V, or 0.7V / 330Ω = 2.1mA). It's not a perfect current source, as the 330k resistor's current flows into the output as well, so the dynamic output impedance in current limiting will be around 300k.

Current Regulator

This is one of the simplest current regulators, and very useful.

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