More Than You Ever Wanted to Know About Electrical Engineering: A Simple Amplifier Model
We’re going to be looking more closely at amplifiers in the coming weeks. The specifics of how an amplifier is built can get pretty complex, so we’re going to want some kind of model that approximates the overall behavior of an amplifier and is simple enough for us to analyze relatively easily. Here’s a general one we can use:
This looks like a mess at first, but when you break it down, it’s not so bad. The left side is the input to the amplifier. v_s is the signal coming in. C_in and R_in represent the input impedance to the amp - that is, the impedance the signal sees on its way in. v_in is the signal the interior workings of the amp actually sees.
The right side is the stuff coming out of the amp. The triangle on the left is an imaginary voltage source that stands in for a lot of complicated amplifier stuff for us. All it does is produce a voltage equivalent to v_in multiplied by some internal gain, g. C_o and R_o here represent the output impedance of the amplifier.
We’ll be doing a lot more with this model and others like it.
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More Than You Ever Wanted to Know About Electrical Engineering, Part 13: Capacitance
Resistors are pretty easy to understand - you burn energy going through something that impedes the flow of charge through it. (If you’re talking about enough current, the burning may even be literal!) Now we’re going to start to look at circuit elements that can store energy, rather than just dissipating it. We’ll start with capacitors.
It’s really easy to make a capacitor. In fact, they’re so easy to make that you frequently make one by accident when building a circuit. (This can cause all kinds of headaches, depending on what you’re trying to do.) The only thing you need is two conductive surfaces separated by an insulating material (i.e. a dielectric material). That dielectric can be wire jacketing, or a special insulator, or even just air.
So. If you put this thing in a circuit, what happens?
For now, we’ll say that our voltage source is a battery. It provides a constant DC voltage of, say, 5 V. When you initially connect the battery to the circuit, current will start flowing, trying to make its way from the positive terminal of the battery to the negative terminal. However, it will find itself blocked by the dielectric of the capacitor. Charge will start to accumulate on the capacitor’s plates, and a voltage will develop across the capacitor, eventually equal to the 5 V of the battery in this case. Once you reach this point, there will be no difference in voltage between the top end of the capacitor and the positive terminal of the battery. No voltage difference means no current flow, so once the capacitor’s been charged up to this point, it will behave like an open circuit. You can think of this as storing energy in the electric field between the capacitor’s plates.
All of this happens fast, but it doesn’t happen instantaneously. So if you’re looking at the voltage across the capacitor while this is happening, it will look like this:
In math terms, it looks like this:
The C that appears here is the capacitance of the capacitor. It’s just a number describing how well the capacitor stores energy, and it depends on the size of the plates, the distance between them, and how good the dielectric is at preventing current flow.
Anyway. What this equation says is that current flowing through the circuit is equal to the rate of change of the voltage across the capacitor times the capacitance. If the voltage across the cap is steady (that is, it’s gotten to the same voltage as the battery), its time rate of change is zero and there’s no current. If the voltage across the cap is still increasing, the current that flows will be proportional to how fast it’s changing.
If you want to look at it in terms of voltage instead of current, you can rearrange that equation and integrate.
This is sometimes more useful than dealing with currents.
In any case, capacitors are one of the most useful and common circuit elements out there, and we’ll be looking at a number of ways to use them.
More Than You Ever Wanted to Know About Electrical Engineering, Part 4: DC Power
We've talked about how charges in a current lose some of their energy moving through an obstacle - that is, they drop some voltage moving through a resistance. In the illustration below, there's an electric potential energy difference of 9V between charges on one side of the resistor and charges on the other side.
The rate at which this energy is transferred is the power expended in going through the resistor. Power is defined as work per unit time, and is measured in watts - for electric charges moving through resistive loads, it is the voltage dropped across the load multiplied by the current flowing through the load.
P = VI
So for the circuit above, 0.081 W of power is consumed flowing through the resistor.
If you like, you can use Ohm's Law to rearrange the equation for power into other forms that may be more convenient. The (I^2)R form is particularly common, and you may hear references to "I^2 losses" sometimes.
So why do we care about power?
Well, in some cases, we're converting electrical energy to mechanical work and need to know what kind of current and voltages we'll see with a particular piece of machinery. That's pretty straightforward. But usually our concern is a little more subtle.
Where does all that energy go? Conservation of energy tells us it doesn't just vanish, but if there's no mechanical work being done, what happens to it?
It turns out that most of it is dissipated as heat. And heat is the great killer of electronics. Most devices come with a power rating, telling you how many watts they can safely dissipate, as well as a temperature rating. Exceeding those limits leads quickly to funky circuit behavior and (eventually) dead circuit boards. Or, in the case of power transmission and chassis wiring, a fire hazard. It's the designer's job to take these limitations into account and design for safe and reliable power management.
