## Introductory electronics 1: Basic passive elements

I’m going to start a series of posts about introductory electronics. I’ll essentially be dealing only with idealized devices since they are usually a good enough approximation especially when dealing with homebrew projects. Dealing with the non-idealities of actual devices would take us on a journey into solid state physics where as much time as we’d care to spend would be soaked up.

As a start, I’ll compare and contrast passive versus active devices. Which devices fall into which category is actually fairly open to interpretation and the definitions can vary. I’ll use the following definitions which I think are abundantly reasonable:

• A passive electronic device does not direct or amplify an electronic signal (some current). An easy way to think of them is sitting there not really doing anything interesting, passively. Not doing anything interesting in the sense that while the incident signal may get distorted, it still passes through in the same direction. Examples include resistors, capacitors, and inductors.
• An active electronic device can direct or amplify an electronic signal. Once you introduce an active device you can start doing really interesting things, like creating switches and amplifiers. Examples include diodes, transistors, and devices composed of transistors like op-amps.

Note that I include diodes under the heading of active devices since while normally we’d think of them as being passive, they do limit the affect the directionality of current and so under our definitions its inclusion is apt.

In this post I’ll deal with the primary passive circuit elements: The resistor, capacitor, and inductor. I write them in descending order at which you’re likely to encounter them, with inductors dead last since they are quite hard to miniaturize (though I did do a presentation on MEMS inductors last year). Resistors are capacitors are ubiquitous.

A resistor, as it name suggests, resists the flow of electric charge through a conductor which is just the current by introducing a voltage drop in the circuit. For an idealized resistor (the only one we’ll care about), this resistance is linear with a coefficient $\displaystyle R$ which we call the resistance of the resistor (this relation is called Ohm’s Law):

$\displaystyle V = R \frac{dq}{dt} = RI$

All conductors have some intrinsic resistance (except superconductors), but we’ll treat the interconnecting wires of a circuit as perfect conductors, so current passes through unimpeded. A collection of wires that connect with no intermediary components (passive or active) is called a node. We also often refer to wires as traces, especially in an integrated circuit or on a printed circuit board.

Symbol for a resistor

When electrons gather in a given region, they will repel electrons in a close (but, importantly, unconnected) region. These repelled electrons will leave behind a positive charge, and so a voltage difference will emerge between the negative electron-rich region and the positive electron-poor region. This system is called a capacitor. Note that the charge on each side of the capacitor will be equal to the other, $\displaystyle +q \mbox{ and } -q$. The voltage across the capacitor is proportional to this charge and we use the coefficient $\displaystyle C$ to represent the capacitance of the capacitor.

$\displaystyle V = \frac{1}{C}q$

(This equation is more often written as $\displaystyle q = CV$ but I wanted to have the voltage by itself on the left for continuity with the other two components). This equation seems like a bit of handful, since we essentially never deal with absolute charges when we design and use circuits. If we take the differential form of the equation, however, and divide both sides by $\displaystyle dt$ we’ll get

$\displaystyle \frac{dV}{dt} = \frac{1}{C} \frac{dq}{dt} = \frac{1}{C} I$

So we see that the current through a capacitor is related to how the voltage is changing, ie. a changing voltage will cause a current to pass through a capacitor. Happily we often deal with direct current (DC) systems where voltages are static, in which case no current would flow through a capacitor. Therefore, we have the following important result: A capacitor will act like an open circuit under a DC voltage.

Also note that the capacitor acts as an energy storage device, as the static charge buildup can be discharged to act as a source of energy. This is in contrast to a resistor which is only an energy sink (where energy is lost in the form of heat).

Symbol for a capacitor

Finally, we have the inductor which is a coil of wire that stores energy in the form of a magnetic field and acts to oppose changes in current. A changing current will induce a voltage linearly with a coefficient $\displaystyle L$ which we call the inductance of the inductor.

$\displaystyle V = L \frac{dI}{dt} = L \frac{d^2 q}{dt^2}$

Since the inductor will only have a voltage drop in presence of a changing current, we have the important result: An inductor will act like a short circuit under a DC voltage.

Symbol for an inductor

And thus we have equations for all three passive components in terms of their voltage and charge, so we can form differential equations. For example, if we have the three components in series we’ll have a time dependent voltage across the series as:

$\displaystyle V(t) = \frac{1}{C} q + R \frac{dq}{dt} + L \frac{d^2 q}{dt^2}$

RLC series circuit

Next time I’ll talk about how to combine these basic circuit elements when you have multiples of them in the same circuit (for example, if I have a $\displaystyle 10 \ \Omega$ and a $\displaystyle 100 \ \Omega$ resistor in series, can they be treated as a single resistor with some to-be-calculated resistance $\displaystyle R$? Yup).

EDIT: There’s been some argument about my definition of active vs. passive. To be sure, it makes the most sense to classify a diode as a passive element and restrict the term active to mean only components that consume power to amplify a signal with a positive gain. In the present day this essentially means transistors or things made from transistors (unless you’re one of the few, the proud, the people who still use vacuum tubes). I wanted to restrict discussion of passive elements to the idealized RLC components, and leave off diodes since they are almost like pseudo-transistors in that a transistor (specifically a MOSFET) can be thought of as being partly constructed by diodes. The other component that makes up a MOSFET, though, is a capacitor so it’s not really a good distinction. Therefore I added to the definition of an active device the idea of affecting the directionality of signal propagation.

This is essentially only an issue in that I wanted to have some way of structuring the introduction of circuit elements, and wanting to leave off diodes I classified them as something other than passive. Also, I didn’t want to focus on the idea of gain as the sole arbiter of activity, since in digital circuits we often have a whole collection of transistors that produce only a gain of unity (ie. no amplification). In practice this debate is totally meaningless, since worrying about whether a component or a sub-circuit is passive or active never occurs. If you never worried about passive versus active for the rest of your life after reading this, it would be a life well lived.