Which came first, the chicken or the egg? I was faced with just such a quandary when I set down to create the original edition of this book. The way that I found people got the most out of the topics was to get some basic ideas and concepts down first; however, those ideas were built on a presumption of a certain amount of knowledge. On the other hand, I realized that the knowledge that was to be presented would make more sense if you first understood these concepts—thus my chicken-vs.-egg dilemma.
Suffice it to say that I jumped ahead to explaining the chicken (the chicken being all about using electricity to our benefit). I was essentially assuming that the reader knew what an egg was (the “egg” being a grasp on what electricity is). Truth be told, it was a bit of a cheat on my part,1 and on top of that I never expected the book to be such a runaway success. Turns out there are lots of people out there who want to know more about the magic of this ever-growing electronic world around us. So, for this new and improved edition of the book, I will digress and do my best to explain the “ egg.” Skip ahead if you have an idea of what it’s all about, or maybe stick around to see if this is an enlightening look at what electricity really is.
The electron—what is it? We haven’t ever seen one, but we have found ways to measure a bunch of them. Meters, oscilloscopes, and all sorts of detectors tell us how electrons move and what they do. We have also found ways to make them turn motors, light up light bulbs, and power cell phones, computers, and thousands of other really cool things.
What is electricity though? Actually, that is a very good question. If you dig deep enough you can find Really Smart Persons all over the world who debate this very topic. I have no desire to that join that debate (having not attained RSP status yet). So I will tell you the way I see it and think about it so that it makes sense in my head. Since I am just a hick from a small town, I hope that my explanation will make it easier for you to understand as well.
We need to begin by learning about a very small particle that is referred to as an atom. A simple representation of oneFigure 0.1.
Atoms are made up of three types of particles: protons, neutrons, and electrons. Only two of these particles have a feature that we call charge. The proton carries a positive charge and the electron carries a negative charge, whereas the neutron carries no charge at all. The individual protons and neutrons are much more massive than the wee little electron. Although they aren’t the same size, the proton and the electron do carry equal amounts of opposite charge.
Now, don’t let the simple circles of my diagram lead you to believe that this is the path that electrons move in. They actually scoot around in a more energetic 3D motion that physicists refer to as a shell. There are many types and shapes of shells, but the specifics are beyond the scope of this text. You do need to understand that when you dump enough energy into an atom, you can get an electron to pop off and move fancy free. When this happens the rest of the atom has a net positive charge and the electron a net negative charge. Actually they have these charges when they are part of the atom. They simply cancel each other out so that when you look at the atom as a whole the net charge is zero.
Now, atoms don’t like having electrons missing from their shells, so as soon as another one comes along it will slip into the open slot in that atom’s shell. The amount of energy or work it takes to pop one of these electrons loose depends on the type of atom we are dealing with. When the atom is a good insulator, such as rubber, these electrons are stuck hard in their shells. They aren’t moving for anything. Take a look at tFigure 0.2.
In an insulator, these electron charges are “stuck” in place, orbiting the nucleus of the atom—kinda like water frozen in a pipe. Do take note that there are just as many positive charges as there are negative charges.
With a good conductor like copper, the electrons in the outer shells of theatoms will pop off at the slightest touch; in metal elements these electrons bounce around from atom to atom so easily that we refer to them as an electron sea, or you might hear them referred to as free electrons. More visuals of this idea Figure 0.3.
You should note that there are still just as many positive charges as there are negative charges. The difference now is not the number of charges; it is the fact that they can move easily. This time they are like water in the pipe that isn’t frozen but liquid—albeit a pipe that is already full of water, so to speak. Getting the electrons to move just requires a little push and away they go. One effect of all these loose electrons is the silvery-shiny appearance that metals have. No wonder that the element that we call silver is one of the best conductors there is
One more thing: A very fundamental property of charge is that like charges repel and opposite charges attract. If you bring a free electron next to another free electron, it will tend to push the other electron away from it. Getting the positively charged atoms to move is much more difficult; they are stuck in place in virtually all solid materials, but the same thing applies to positive charges as well.
So now we have an idea of what insulators and conductors are and how they relate to electrons and atoms. What is this information good for, and why do we care? Let’s focus on these charges and see what happens when we get them to move around.
First, let’s get these charges to move to a place and stay there. To do this we’ll take advantage of the cool effect that these charges have on each other, which we discussed earlier. Remember, opposite charges attract, whereas the same charges repel. There is a cool, mysterious, magical field around these charges. We call it the electrostatic field. This is the very same field that creates everything from static cling to lightning bolts. Have you ever rubbed a balloon on your head and stuck it on the wall? If so you have seen a demonstration of an electrostatic field. If you took that a little further and waved the balloon closely over the hair on your arm, you might notice how the hairs would track the movement of the balloon. The action of rubbing the balloon caused your head to end up with a net total charge on it and the opposite charge on the balloon. The act of rubbing these materials together caused some electrons to move from one surface to the other, charging both your head and the balloon.
This electrostatic field can exert a force on other things with charges. Think about it for a momefigure out a way to put some charges on one end of our conductor, that would push the like charges away and in so doing cause those charges to move.
Figure 0.4 shows a hypothetical device that separates these charges. I will call it an electron pump and hook it up to our copper conductor we mentioned previously.
In our electron pump, when you turn the crank, one side gets a surplus of elec trons, or a negative charge, and on the other side the atoms are missing said electrons, resulting in a positive charge.
If you want to carry forward the water analogy, think of this as a pump hooked up to a pipe full of water and sealed at both ends. As you turn the pump, you build up pressure in the pipe—positive pressure on one side of the pump and negative pressure on the other. In the same way, as you turn the crank you build up charges on either side of the pump, and then these charges push out into the wire and sit there because they have no place to go. If you hook up a meter to either end you would measure a potential (think difference in charge) between the two wires. That potential is what we call voltage.
It’s important to realize that it is by the nature of the location of these charges that you measure a voltage. Note that I said location, not movement. Movement of these charges is what we call current. (More on that later.) For now what you need to take away from this discussion is that it is an accumulation of charges that we refer to as voltage. The more like charges you get in one location, the stronger the electrostatic field you create.
Okay, it’s later now. We find that another very cool thing happens when we move these charges. Let’s go back to our pump and stick a light bulb on the ends of our wires,Figure 0.5.
Remember that opposite charges attract? When you hook up the bulb, on one side you have positive charges, on the other negative. These charges push through the light bulb, and as they do they heat up the filament and make it light up. If you stop turning the electron pump, this potential across the light bulb disappears and the charges stop moving. Start turning the pump and they start moving again. The movement of these charges is called current. The really cool thing that happens is that we get another invisible field that is created when these charges move; it is called the electromagnetic field. If you have ever played with a magnet and some iron filings, you have seen the effects of this field.
So, to recap, if we have a bunch of charges hanging out, we call it voltage, and when we keep these charges in motion we call that current. Some typical water analogies look at voltage as pressure and current as flow. These are helpful to grasp the concept, but keep in mind that a key thing with these charges and their movements is the seemingly magical fields they produce. Voltage generates an electrostatic field (it is this field repelling or attracting other charges that creates the voltage “pressure” in the conductor). Current or flow or move- ment of the charges generates a magnetic field around the conductor. It is very important to grasp these concepts to enhance your understanding of what is going on. When you get down to it, it is these fields that actually move the work or energy from one end of a circuit to another.
Let’s go back to our pump and light bulb for a minute,Figure 0.6.
Turn the pump and the bulb lights up. Stop turning and it goes out. Start turning and it immediately lights up again. This happens even if the wires are long! We see the effect immediately. Think of the circuit as a pair of pulleys and a belt. The charges are moving around the circuit, transferring power from one location toFigure 0.7.
Fundamentally, we can think of the concept as shown in thFigure 0.8.
Even if the movement of the belt is slow, we see the effects on the pulley immediately, at the moment the crank is turned. It is the same way with the light bulb. However, the belt is replaced by the circuit, and it is actually the electromagnetic fields pushing charges around that transmit the work to the bulb. Without the effects of both these fields, we couldn’t move the energy input at the crank to be output at the light bulb. It just wouldn’t happen. Like the belt on the pulleys, the charges move around in a loop. But the work that is being done at the crank moves out to the light bulb, where it is used up making the light shine. Charges weren’t used up; current wasn’t used up. They all make the loop (just like the belt in the pulley example). It is energy that is used up. Energy is work; you turning the crank is work. The light bulb takes energy to shine. In the bulb energy is converted into heat on the filament that makes it glow so bright that you get light. But remember, it is energy that it takes to make this happen. You need both voltage and current (along with their associated fields) to transfer energy from one point to another in an electric circuit.
Do you remember your engineering introductory course? At most, I’ll venture that you are not sure you even had a 101 course. It’s likely that you did and, like the course I had, it really didn’t amount to much. In fact, I don’t remember anything except that it was supposed to be an “introduction to engineering.”
Much later in my senior year and shortly after I graduated, I learned some very useful general engineering methodologies. They are so beneficial that I sincerely wish they had taught these three things from the beginning of my coursework. In fact, it is my belief that this should be basic, basic knowledge that any aspiring engineer should know. I promise that by using these in your day-to-day challenges you will be more successful and, besides that, everyone you work with will think you are a genius. If you are a student reading this, you will be amazed at how many problems you can solve with these skills. They are the fundamental building blocks for what is to come.
This is a skill that one of my favorite teachers drilled into me during my senior year. Till I understood unit math, I forced myself to memorize hundreds of equations just to pass tests. After applying this skill I found that, with just a few equations and a little algebra, you could solve nearly any problem. This was definitely an “ Aha” moment for me. Suddenly the world made sense. Remember those dreaded story problems that you had to do in physics? Using unit math, those problems become a breeze; you can do them without even breaking a sweat.
With this process the units that the quantities are in become very important. You don’t just toss them aside because you can’t put them in your calculatofigure out the units you want in your answer and then work the probfigure out what you need to solve it. You do all this before you do anything with the numbers at all. This basic concept was taught way back in algebra class, but no one told you to do it with units. Let’s look at a very simple example.
When all the units that can be removed are gone, what you are left with is 60 mph, which is the correct answer. Now, you might be saying to yourself that that was easy. You are right! That is the point after all—we want to make it easier. If you follow this basic format, most of the “story problems” you encounter every day will bow effortlessly to your machinations.
Another excellent place to use this technique is for solution verification. If the answer doesn’t come out in the right units, most likely something was wrong in your calculation. I always put units on the numbers and equations I use in MathCad (a tool no engineer should be without). To see the correct units when all is said and done it confirms that the equations are set up properly. (The nice thing is that MathCad automatically handles the conversions that are often needed.) So, whenever you come upon a question that seems to have a whole pile of data and you have no idea where figure out which units you want the answer in. Then shape that pile of data till the units match the units needed for the answer.
My father had a saying: “‘Almost’ only counts in horseshoes and hand grenades!” He usually said this right after I “almost” put his tools away or I “almost” finished cleaning my room. Early in life I became somewhat of an expert in the field of “almost.” As my dad pointed out, there are many times when almost doesn’t count. However, as this bit of wisdom states, it probably is good enough to almost hit your target with a hand grenade. There are a few other times when almost is good enough, too. One of them is when you are trying to estimate a result. A skill that goes hand in hand with the idea of unit math is that of estimation.
The skill or art of estimation involves two main points. The first is rounding to an easy number and the second is understanding ratios and percentages. The rounding part comes easy. Let’s say you are adding two numbers, 97 and 97. These are both nearly 100, so say they are 100 for a minute; add them together and you get 200, or nearly so. Now, this is a very simplified explanation of this idea, and you might think, “Why didn’t you just type 97 into your calculator a couple of times and press the equals sign?” The reason is, as the problems become more and more complex, it becomes easier to make a mistake that can cause you to be far off in your analysis. Let’s apply this idea to our previous example. If your calculator says 487 after you add 97 to 97, and you compare that with the estimate of 200 that you did in your head, you quickly realize that you must have hit a wrong button.
Ratios and percentages help you get an idea of how much one thing affects another. Say you have two systems that add their outputs together. In your design, one system outputs 100 times more than the other. The ratio of one to the other is 100:1. If the output of this product is way off, which of these two systems do you think is most likely at fault? It becomes obvious that one sys tem has a bigger effect when you estimate the ratio of one to the other.
Developing the skill of estimation will help you eliminate hunting dead ends and chasing your tail when it comes to engineering analysis and troubleshooting. It will also keep you from making dumb mistakes on those pesky finals in school! Learn to estimate in your head as much as possible. It is okay to use calculators and other tools—just keep a running estimation in your head to check your work.
When you are estimating, you are trying to simplify the process of getting to the answer by allowing a margin of error to creep in. The estimated answer you get will be “almost” right, and close enofigure out where else you may have screwed up.
In the game of horseshoes you get a few points for “almost” getting a ringer, but I doubt your boss will be happy with a circuit that “almost” works. However, if your estimates are “almost” right, they can help you design a circuit that even my dad would think is good enough.
Mechanical engineers have it easy. They can see what they are working on most of the time. As an EE, you do not usually have that luxury. You have to imagine how those pesky electrons are flittering around in your circuit. We are going to cover some basic comparisons that use things you are familiar with to create an intuitive understanding of a circuit. As a side benefit, you will be able to hold your own in a mechanical discussion as well. There are several reasons to do this:
Before we move on to the physical equivalents, let’s understand voltage, current, and power. Voltage is the potential of the charges in the circuit. Current is the amount of charge flowing2 in the circuit. Sometimes the best analogies are the old overused ones, and that is true in this case. Think of it in terms of water in a squirt gun. Voltage is the amount of pressure in the gun. Pressure determines how far the water squirts, but a little pea shooter with a 30-foot shot and a dinky little stream won’t get you soaked. Current is the size of the water stream from the gun, but a large stream that doesn’t shoot far is not much help in a water fight. What you need is a super-soaker 29 gazillion, with a half-inch water stream that shoots 30 feet. Now that would be a powerful water-drenching weapon. Voltage, current, and power in electrical terms are related the same way. It is in fact a simple relationship; here is the equation:
To get power, you need both voltage and current. If either one of these is zero, you get zero power output. Remember, power is a combination of these two items: current and voltage.
Now let’s discuss three basic components and look at how they relate to voltage and current.
Think about what happens when you drag a heavy box across the floor,Figure 1.1. A force called friction resists the movement of the box. This friction is related to the speed of the box. The faster you try to move the box, the more the friction resists the movement. It can be described by an equation:
Furthermore, the friction dissipates the energy loss in the system with heat. Let me rephrase that. Friction makes things get warm. Don’t believe me? Try rubbing your hands together right now. Did you feel the heat? That is caused by friction. The function of a resistor in an electrical circuit is equal to friction. The resistor resists the flow of electricity* just like friction resists the speed of the box. And, guess what? It heats up as it does so. An equation called Ohm’s Law describes this relationship:
Do you see the similarity to the friction equation? They are exactly the same. The only real difference is the units you are working in.
Let’s stay with the box example for now. First let’s eliminate friction, so as not to cloud our comprehension. TFigure 1.2 is on a smooth track with virtually frictionless wheels. You notice that it takes some work to get the box going, but once it’s moving, it coasts along nicely. In fact, it takes work to get it to stop again. How much work, depends on how heavy the box is. This is known as the law of inertia. Newton postulated this idea long before electricity was discovered, but it applies very well to inductance. Mass resists a change in speed. Correspondingly, inductance resists a change in current.
So what does a spring do? Take hold of a spring in your mind’s eye. Stretch it out and hold it, and then let it go. What happens? It snaps back into position,Figure 1.3 on the next page. A spring has the capacity to store energy. When a force is applied, it will hold that energy till it is released. Capacitance is similar to the elasticity of the spring. (One note: The spring constant that you might remember from physics texts is the inverse of the elasticity.) I always thought it was nice that the word capacitor is used to represent a component that has the capacity to store energy.
Take the basic tank or LC circuit. What does it do? It oscillates. A perfect circuit would go on forever at the resonant frequency. How should this appear in our mechanical circuit? Think about the equivalents: an inductor and a capacitor, a spring and mass. In a thought experiment, hook the spring up to the box from the previous drawing. Now give it a tug. What happens? It oscillates.
Let’s follow this reasoning for an LCR circuit. All we need to do is add a little resistance, or friction, to the mass-spring of the tank circuit. Let’s tighten the wheels on our box a little too much so that they rub. What will happen afteryou give the box a tug? It will bounce back and forth a bit till it comes to a stop. The friction in the wheels slows it down. This friction component is called a damper because it dampens the oscillation. What is it that a resistor does to an LC circuit? It dampens the oscillation.
There you have it—the world of electricity reduced to everyday items. Since these components are so similar, all the math tricks you might have learned apply as well to one system as they do to the other. Remember Fourier’s theorems? They were discovered for mechanical systems long before anyone realized that they work for electrical circuits as well. Remember all that higher math you used to know or are just now learning about—Laplace transforms, integrals, derivatives, etc.? It all works the same in both worlds. You can solve a mechanical system using Laplace methods just the same as an electrical circuit.
Back in the 1950s and 1960s, the government spent mounds of dough using electrical circuits to model physical systems as described before. Why? You can get into all sorts of integrals, derivatives, and other ugly math when modeling real-world systems. All that can get jumbled quickly after a couple of orders of complexity. Think about an artillery shell fired from a tank. How do you predict where it will land? You have the friction of the air, the mass of the shell, the spring of the recoil. Instead of trying to calculate all that math by hand, you can build a circuit with all the various electrical components representing the mechanical ones, hook up an oscilloscope, and fire away. If you want to test 1000 different weights of artillery at different altitudes, electrons are much cheaper than gunpowder.
I’m not sure if intuitive signal analysis is actually taught in school; this is my name for it. It is something I learned on my own in college and the workplace. I didn’t call it an actual discipline until I had been working for a while and had explained my methods to fellow engineers to help them solve their own dilemmas. I do think, however, that a lot of so-called bright people out there use this skill without really knowing it or putting a name to it. They seem to be able to point to something you have been working on for hours and say, “Your problem is there.” They just seem to intuitively know what should happen. I believe that this is a skill that can and should be taught.
There are three underlying principles needed to apply intuitive signal analysis. (Let’s just call it ISA. After all, if I have any hope of this catching on in the engineering world, it has to have an acronym!)
You should know immediately with something this basic that the answer is “smaller.” You should also know that how much smaller depends on the frequency of the signal and the time constant of the filter. What happens as you increase current into the base of a transistor? Current through the collector increases. What happens to voltage across a resistor as current decreases? These are simple effects of components, but you would be surprised at how many engineers don’t know the answers to these types of questions off the top of their heads.
Spending a lot of time in the lab will help immensely in developing this skill. If you look at the response of a lot of different circuits many, many times, you will learn how they should act. When this knowledge is integrated, a wonderful thing happens: Your head becomes a circuit simulator. You will be able to sum up the effects caused by the various components in the circuit and intuitively understand what is happening. Let me show you an example.
Now, at this time you might not have a clue as to what a transistor is, so you might need to file this example away until you get past the transistor chapter, but be sure to come back to it so that the “Aha!” light bulb clicks on over your head. The analysis idea is what I am trying to get across; you need it early on, but it creates a type of chicken-and-egg dilemma when it comes to an example. So, for now consider this example with the knowledge that the transistor is a device that moves current through the output that is proportional to the current through the base.
As voltage at the input increases, base current increases. This causes the pull-up current in the resistor to increase, resulting in a larger voltage drop across the pull-up resistor. This means the voltage at the output must go down as the voltage at the input goes up. That is an example of putting it all together to really understand how a circuit works.
One way to develop this intuitive understanding is by using computer simulators. It is easy to change a value and see what effect it has on the output, and you can try several different configurations in a short amount of time. However, you have to be careful with these tools. It is easy to fall into a common trap: trusting the simulator so much that you will think there is something wrong with the real world when it doesn’t work right in the lab. The real world is not at fault! It is the simulator that is missing something. I think it is best for the engineer to begin using simulators to model simple circuits. Don’t jump into a complex model until you grasp what the basic components do—for exam ple, modeling a step input into an RC circuit. With a simple model like this, change the values of R and C to see what happens. This is one way an engineer can develop the correct intuitive understanding of these two components. One word of warning, though: Don’t spend all your time on the simulator. Make sure you get some good bench time, too.
You will find this signal analysis skill very useful in diagnosing problems as well as in your design efforts. As your intuitive understanding increases, you will be able to leap to correct conclusions without all the necessary facts. You will know when you are modeling something incorrectly, because the result just won’t look right. Intuition is a skill no computer has, so make sure you take advantage of it!
Okay, so I came up with a fourth item.6 One of my engineering instructors (we’ll call him Chuck7) taught me a secret that I would like to pass on. Almost every discipline is easier to understand than you might think. The secret professors don’t want you to know is that there are usually about five or six basic principles or equations that lie at the bottom of the pile, so to speak. These fundamentals, once they are grasped, will allow you to derive the rest of the principles or equations in that field. They are like the old simple Legos®; you had five or six shapes to make everything. If you truly understand these few basic fundamentals in a given discipline, you will excel in that discipline. One other thing Chuck often said was that all the great discoveries were only one or two levels above these fundamentals. This means that if you really know the basics well, you will excel at the rest. One thing you can be sure of is thehuman tendency to forget. All the higher-level stuff is often left unused and will quickly be forgotten, but even an engineer-turned-manager like me uses the basics nearly every day.
Since this is a book on electrical engineering, let’s list the fundamental equations for electrical circuits as I see them:
We will get into these concepts in more detail later in the chapters, but let me touch on a couple of examples. You might say, “You didn’t even list series and parallel capacitors. Isn’t that a basic rule?” Well, you are right, it is fairly basic, but it really isn’t at the bottom of the pile. Series and parallel resistors are even more fundamental because all that really happens when you add in the caps is that the frequency of the signal is taken into account; other than that it is exactly the same equation! You would be better served to understand how a capacitor or inductor works and apply it to the basics than to try to memorize too many equations. “What about Norton’s theorem?” you might ask. Bottom line, it is just the flip side of Thevenin’s theorem, so why learn two when one will do? I prefer to think of it in terms of voltage, so I set this to memory. You could work in terms of current and use Norton’s theorem, but you would arrive at the same answer at the end of the day. So pick one and go with it.
You can always look up the more advanced stuff, but most of the time a solid application of the basics will force the problem at hand to submit to your engi- neering prowess. These six rules are things that you should memorize, under- stand, and be able to do approximations of in your head. These are the rules that will make the intuition you are developing a powerful tool. They will unleash the simulation capability that you have right in your own brain.
If you really take this advice to heart, years down the road when you’ve been given your “pointy hairs” and you have forgotten all the advanced stuff you used to know, you will still be able to solve engineering problems to the amazement of your engineers.
This can be generalized to all disciplines. Look at what you are tfigure out the few basic points being made, from which you can derive the rest, and you will have discovered the basic “Legos” for that subject. Those are the things you should know forward and backward to succeed in that field. Besides, Legos are fun, aren’t they?
Every discipline has fundamentals that are used to extrapolate all the other, more complex ideas. Basics are the most important thing you can know. It is knowledge of the basics that helps you apply all that stuff in your head correctly. It doesn’t matter if you can handle quadratic equations and calculus in your sleep. If you don’t grasp the basics, you will find yourself constantly chasing a problem in circles without resolution. If you get anything out of this text, make sure that you really understand the basics!
This, I believe, is one of the best-taught principles in school for the budding engi- neer or technician, and it should be. So why go over it? Well, two reasons come to mind: One, you can’t go over the basics too much, and two, though any engi- neer can quote Ohm’s Law by heart, I have often seen it ignored in application.
First, let’s state Ohm’s Law: Voltage equals current multiplied by resistance; itFigure 2.1 on the next page.
It is simple, but do you consider that resistance exists in every part of a circuit? It is easy to forget that, especially since many simulators do. I think the best way to drive this point home is to recount the way it was driven home to me.
The basics are the most important; let me repeat that, the basics are important! Ohm’s Law is the most basic principle you will use as an electrical engineer. It is the foundation on which all other rules are based. The fundamental fact is that resistance impedes current flow. This impedance creates a voltage drop across the resistor that is proportional to the amount of current flowing through it. If it helps, you can think of a resistor as a current-to-voltage converter.
With that important point made, let’s consider two other types of impedance that can be found in a circuit. We will get into this in more detail later, but for now consider that inductors and capacitors both can act like resistors, depend- ing on the frequency of the signal. If you take this into account, Ohm’s Law still works when applied to these components as well. You could very well rewrite the equation to:
Think of the impedance Z as resistance at a given frequency.3 As we move on to the other basic equations, keep this in mind. Wherever you see resistance in an equation, you can simply replace it with impedance if you consider the frequency of the signal.
One final note: Every wire, trace, component, or material in your circuit has these three components in it: resistance, inductance, and capacitance. Everything has resistance, everything has capacitance, and everything has inductance. The most important question you must ask is, “Is it enough to make a difference?” The fact is, in my own experience, if the shunt resistor had been 100 times larger, that would have made the errors we were seeing 100 times less. They would have been insignifi cant in comparison to the measurement we were taking. The impedance equations for capacitors and inductors will help you in a similar way. Consider the frequencies you are operating at and ask yourself, “Is this compo- nent making a significant impact on what I am looking at?” By reviewing this sig- nificance, you will be able to pinpoint the part of the circuit you are looking for.
Next on our list of basic formulae is the voltage divider rule. Here is the Figure 2.5 shows a schematic of the circuit:
The most common way you will see this is in terms of R1 and R2. I have changed these to Rg (for R ground) and Ri (for R input) to remind myself which one of these goes to ground and which one is in series. If you get them back- ward, you get the amount of voltage lost across Ri, not the amount at the output (which is the voltage across Rg). If the gain of this circuit just doesn’t seem right, you might have the two values swapped.
You might also notice that the gain of this circuit is never greater than 1. It approaches 1 as Ri goes to 0, and it approaches 1 as Rg gets very large. (Note that as Rg gets larger, the value of Ri becomes less significant.) Since this is the case, it is easy to think of the voltage divider as a circuit that passes a percent- age of the voltage through to the output. When you look at this circuit, try to think of it in terms of percentage. For example, if Rg = Ri, only 50% of the voltage would be present on the output. If you want 10% of the signal, you will need a gain of 1/10. So put 1 K in for Rg, and 9 K in for Ri, and, voilà, you have a voltage divider that leaves 10% of the signal at the output.
Did you notice that the ratio of the resistors to each other was 1:9 for a gain of 1/10? This is because the denominator is the sum of the two resistor values. I’ll also bet you noticed that if you swap the two resistor values you will get a gain of 9/10, or 90%. This should make intuitive sense to you now if you recognize that, for the same amount of current, the voltage drop across a 9 K Ri will be nine times larger than the voltage drop across a 1 K Rg. In other words, 90% of the voltage is across Ri, whereas 10% of the voltage is across Rg, where your meter measuring Vo is hooked up. The voltage divider is really just an exten- sion ofigure), but it is so useful that I’ve included it as one of the basic equations that you should commit to memory.
Let’s consider for a moment what might happen to the previous voltage divider circuit if we replace Rg with a capacitor. It is still a voltage divider circuit, is it not? But what is the difference? At this point you should say, “Hey, a cap is just a resistor that’s value changes depending on the frequency; wouldn’t that make this a voltage divider that depends on frequency?” Well, it does, and this is commonly known as an RC circuit. Let’s draw one now,Figure 2.6.
Using your intuitive understanding of resistors and capacitors, let’s analyze what is going to happen in this circuit. We’ll do this by applying a step input. A step input is by definition a fast change in voltage. The resistor doesn’t care about the change in voltage, but the cap does. This fast change in voltage can be thought of as high frequencies,6 and how does the cap respond to high frequen- cies? That’s right, it has low impedance. So, now we apply the voltage divider rule. If the impedance of Rg is low (as compared to Ri), the voltage at Vo is low. As frequency drops, the impedance goes up; as the impedance goes up, based on the voltage divider, the output voltage goes up. Where does it all stop?
Think about it a moment. Based on what you know about a cap, it resists a change in voltage. A quick change in voltage is what happened initially. After that our step input remained at 5 V, not changing any more. Doesn’t it make sense that the cap will eventually charge to 5 V and stay there? This phenom- enon is known as the transient response of an RC circuit. The change in voltage on the output of this circuit has a characteristic curve. It is described by this equation:
The graph of this outpuFigure 2.7. The value of R times C in this equation is also known as tau, or the time constant, often referred to by the Greek letter τ.
For a step input, this curve is always the same for an RC circuit. The only thing that changes is the amount of time it takes to get to the final value. The shape of the curve is always the same, but the time it takes to happen depends on the value of the time constant tau. You can normalize this curve in terms of the time constant and the final value of the voltage. Let’s redraw the curve with multiples of τ along the time axis,Figure 2.8.
At 1 τ the voltage reaches 63.2%, at 2 τ it is at 86.6%, 3 τ is 95%, by 4 τ it is at 98%, and when you reach 5 τ you are close enough to 100% to consider it so.
This response curve describes a basic and fundamental principle in electronics. Some years ago I started asking potential job candidates to draw this curve after I gave them the RC circFigure 2.6. Over the years I have been dismayed at how many engineers, both fresh out of school and with years of experience, cannot draw this curve. Fewer than 50% of the applicants I have asked can do it. That fact is one of the main reasons I decided to write this book. (The other was that someone was actually willing to pay me to do it! I doubt it would have gotten far otherwise.) So, I implore you to put this to memory once and for all; by doing so I guarantee you will be a better engineer. Plus, if I ever interview you, you will have a 50% better chance of getting a job! If you understand this concept, you will understand inductors, as you will see in the next section.
Before we move on, I would like you to consider what happens to the current in this circuit. Remember Ohm’s Law? Apply it to this example to understand what the current does. We know that:
A little algebra turns this equation into:
A little common sense reveals that the voltage across R in this circuit is equal to voltage at the output minus voltage at the input. As an equation, you get:
We know the voltage at each point in time in terms of tau. At 0 τ, Vo is at 0. So the full 5 V is across the resistor and the maximum current is flowing. For all intents and purposes, the cap is shorting the output to ground at this point in time. At 1 τ, Vo is at 63.2% of Vi. That means Vr is at 36.8% of Vi. Repeat this process, connect the dots, and you get a curve that moves in the opposite direc- tion of the voltage curve, something like whaFigure 2.9.
Notice how current can change immediately when the step input changes. Also notice how the voltage just doesn’t change that fast. Capacitors impede a change in voltage, as the rule goes. What this also means is that changes in cur- rent8 will not be affected at all. Everything has its opposite, and capacitors are no exception, so let’s move on to inductors.
Now that we have thought through the RC circuit, let’s consider the RL circFigure 2.10. Remember that the inductor resists a change in current but not in voltage. Initially, with the same step input, the voltage at the output can jump right to 5 V. Current through the inductor is initially at 0, but now there is a voltage drop across it, so current has to start climbing. The current responds in the RL circuit exactly the same way voltage responds in the RC circuit.
Since you committed the RC response to memory, the RL response is easy. It is exactly the same from the viewpoint of current9; the current grapFigure 2.11 on the next page.
I hope you are saying to yourself, “What about the voltage response?” At this time, consider Ohm’s Law for a moment and try to graph what the voltage will do. What is the current at time 0? How about a little later? Remember Ohm’s Law—for the current to be low, resistance must be high. So initially the inductor acts like an open circuit. Voltage across the inductor will be at the same value as the input. As time goes on, the impedance of the inductor drops off, becoming a short, so voltage drFigure 2.12 shows the graph.
The inductor is the exact complement of the capacitor. What it does to current, the cap does to voltage, and vice versa.
There are two ways for compfigured in a circuit: series and parallel. Series components line up one after another; parallel components are hooked up next to each other. Let’s go over the formulas to simplify these com- ponent arrangements. Series resiFigure 2.13, are easy; you simply add them up, no mul- tiplication needed!