A system is anything with an input and an output. The idea is simple—take the input, shake it, squeeze it, do whatever, and then send it to the output. It can be represented with a block diagram, something like the one shown in Figure 2.47 .
All the magic happens inside the box. This magic is called thetransfer function . The transfer function is equal to the input over the output. It is the equation that you process the input through to get the output, so the following is true:
Output Magic*Input Eq. 2.17
A little algebra yields:
Output
Input Magic Eq. 2.18
Now you know how to fi nd what the magic inside the box is, and sometimes it is just that easy. Let’s try a simple example to see how. You put 12 miles into the input, wait … chugga, chugga, ding! … and out pops 19.32 km. As you might have guessed, the magic in this box is a metric converter, but what is the trans- fer function? According to the preceding equation, we simply divide the output by the input. That would be:
19 32 12 1 61 . . km miles km mile Eq. 2.19 The magic in our converter box looks like the one in Figure 2.48 .
Magic
Input Output
FIGURE 2.47
The magic box inside a system.
Keep It Under Control
691.61 Km/mile
Input Output
FIGURE 2.48
The magic revealed.
Note that the units made it into the box. This helps identify the type of units that will work at the input and what you will get at the output. Hopefully a lit- tle voice in your head is saying, “Isn’t this a rehash of the unit math chapter? ” It is, but this is a more formalized concept with some neat touches such as
R
Input Output
FIGURE 2.49
System diagram of a resistor.
the cool little boxes 37 you draw to help you understand the system. The next
question you should ask is, “How does this apply to electronics? ” Well, from the most basic to the most complex, you can represent any circuit with one of these magic (okay, some texts call them black ) boxes.
We’d better do another example. Take a resistor. A resistor can be thought of as a current-to-voltage converter. Put current into the input, apply magic, and get voltage at the output. What would be the transfer function of that? If you mum- bled a phrase with the words Ohm’s Law anywhere in it, you are probably right.
R V
I
Eq. 2.20
That would make the block diagram of the resistor look something what
Figure 2.49 shows .
In this transfer function,R is the value of the resistor in ohms, just in case you didn’t guess. (Note that 1 unit of ohms equals 1 unit of volts divided by 1 unit of amps, like good old Ohm’s Law says it does.)
Let’s step up the idea to something a little more complex, like a voltage divider. We already know the equation for this. It is:
Vo Vi Rg
Rg Ri
Eq. 2.21
Do you remember what Vo stands for? How about Vi? They are voltage output and voltage input. So let’s just use a little algebra to fi gure this out. The transfer function is equal to the output over the input, like this:
Vo Vi Rg Rg Ri Eq. 2.22
37 Drawing cool little boxes is fun! Why else do we represent what we think with schemat- ics and fl ow charts and the like? I suggest that way back when we were all apes around a campfi re, somehow drawing lines in the dirt gave us an evolutionary advantage. It must be important to the survival of the species, ‘cause I will say this: I loved to get out the crayons as a kid, and a good time sketching on the white board is still pretty darn satisfying!
The block diagram would look like Figure 2.50 .
This same concept can be used to describe all the circuits that we have seen so far. You might see block diagrams of this type where C orL has a littles by it. This is a mathematical trick known as a Laplace transform. It is used to sim- plify problem solving. If you transform all that time constant and frequency stuff using Laplace pairs, you can treat the transformed equations with simple algebra and then transform it all back when you are done. Laplace transforms are beyond the scope of this text, but do take note that the s rolls up all the frequency response of capacitors and inductors into a domain that can be han- dled easily when the equations get complex.
We can describe any system as a magic box with an input and an output. One way to determine what is in the magic box is to put a known signal into the input. Let’s take a look at what is probably the most important stimulus you can apply to the input of the magic box.