3.3 Inductor Circuits
Inductors as discussed in Chapter 1, "Introduction to Electronics, Electrons, and Light Speed," simply store energy in a magnetic field. In embedded system design, you won’t actually use many inductors; they are more commonly found in audio and video circuits or when designing complex filters. So, we don’t need to cover them as in depth as we did with capacitors, so I just want to show you a couple circuits and some results.
3.3.1 Inductors Model
Figure 3.34 depicts both a more physical image of an inductor and a circuit diagram. To refresh your memory, the inductor works simply by passing a current through the inductor: As the current flows, a magnetic field is generated. This field strength is proportional to the number of turns of wire in the inductor, the geometry, etc.; these factors are lumped together and called inductance, L, and measured in henrys, named after the physicist Joseph Henry. In any case, as the current flows and the field is generated, this also develops a voltage across the inductor with the governing equation:
Figure 3.34 Physical inductor model.
Equation 3.2: Inductance.
- V = L*di/dt
That is, the rate change of current I in the circuit multiplied by the inductance is equal to the voltage, so 1 henry of inductance with 1 amp of current generates a voltage of 1V.
Inductors can be used just as capacitors to create low pass and high pass filters, but they would be huge in such cases. Therefore, as mentioned, inductors are only used in high frequency applications in most cases.
3.3.2 Inductors in Series and Parallel
Inductors sum just as resistors do in series and parallel combinations; thus given a collection of inductors in series, their equivalent inductance as shown in Figure 3.35 is
- Leq = L1 + L2 + ... + Ln
And in parallel
- 1/Leq = 1/L1 + 1/L2 + ... + 1/Ln
Figure 3.35 Inductors in series and parallel.
3.3.3 The Step Response of an Inductor
Referring to the governing equation of the inductor, we can surmise that the inductor seems like a short to a DC current; that is, the voltage across the inductor must be 0. Only when the current is changing is there a voltage developed. Let’s now take a look at what happens at this time. Assuming that we have a DC current of I0 as shown in Figure 3.36 at t=0- (notice at this point there is no voltage drop over the inductor since it’s a short to the current source), we switch the switch from position 1 to 2 and switch out the current source and switch in the load resistor R; we now have the circuit shown in Figure 3.36. Writing a loop equation, we get
- L*di/dt + R*I = 0
Figure 3.36 Inductor Analysis Circuit.
Which again is a first order differential equation with solution of the form A*ex; knowing that and knowing that the current at t=0 is I0, we arrive at this final solution as a function of time t
- I(t) = I0*e-(R/L)*t
Notice the (R/L) term? This is similar to the R*C term in the capacitor model, thus R/L is the time constant in an inductive circuit. These are all the results we need for now; we aren’t going to use inductors in our designs, so it’s not worth the time to pursue more analysis. Let’s take a look at some final results.
3.3.4 RLC Circuits
The final model of many circuit elements, including conductor paths and filters, is a combination of R, L, and C. There are a number of possible permutations of RLC, but let’s look at the two most obvious: series and parallel.
188.8.131.52 Series RLC
Figure 3.37 depicts the step response of an RLC circuit to a digital 5V pulse. This is very close to the real world and if you understand this you are 99% there to understanding what happens at the analog level of digital signals. In any case, referring to the figure, we send a digital signal through a transmission line—maybe it’s a data bus line, a control signal, etc.—in any case, due to the capacitance, inductance, and resistance of the connection, the line seems like a series RLC model, so what does the step function end up looking like? Figure 3.37 also shows this: We see that the signal rises exponentially as we suspected, but then it "rings" about the max; then when the step signal goes back to 0V, the RLC signals falls again to 0V exponentially, but rings again about 0V. So the inductance looks like its causing this ringing effect, and it is, thus, you must keep the inductance of your lines small, otherwise you will ring them so much that the data may get corrupted.
Figure 3.37 A series RLC circuit.
184.108.40.206 Parallel RLC
Okay, so that’s the first case, series RLC. Let’s take a look at a more interesting case: parallel RLC, as shown in Figure 3.38. When you have a resistor, capacitor, and inductor in parallel like that, you create a special kind of circuit called an RL-tank—basically the precursor to an oscillator—so when you excite the circuit with a step input, the circuit charges and then discharges slowly by oscillating the energy away. This is easy to understand if you think about the two energy storage elements (capacitor and inductor) feeding each other. So the bottom line is that if you place an RLC in parallel you might get oscillations! And if you didn’t intend that you would have serious trouble.
Figure 3.38 A parallel RLC circuit.