Real Analog: Circuits1.12b

Course: "Real Analog", Circuits 1
Lecture 12b - Derivation of maximum power transfer
Related educational materials: Chapter 5.1 - 5.4

For the free "Real Analog" textbook and other information, please visit us at:
www.digilentinc.com/classroom/realanalog

>> Now, I want to spend some time talking about operational amplifiers.

Operational amplifiers are a device that gets used a lot in electrical circuits.

You will see them over and over again this semester.

They get used a lot in instrumentation systems.

They get used a lot in control systems, etc, etc.

They're quite often the basis for electrical circuits,

which perform mathematical operations.

That's why they're called operational amplifiers.

Now, there's one point that has to be made very clear up front.

These are not a passive device.

So far, with the exception of our power sources,

all of our circuit elements have been passive than resistors, essentially.

That means that the energy delivered by the circuit to the element is non-negative.

This element does not create power out of somewhere else and provide it to the circuit.

Okay, it has to get any energy that it has from the circuit.

Operational amplifiers are in active device.

Okay. They will deliver power to your circuit.

The way they deliver power to your circuit is because they have an external power supply.

There's some other magical device somewhere that is feeding

these guys power, which these guys can then provide to your circuit.

I will tend to abbreviate operational amplifiers as op-amps.

That's very common, primarily, just to save syllables.

Quick overview of operational amplifiers.

We're going to think of operational amplifiers as a device.

It's something that performs some task.

So, we're going to think of them as a black box.

There is a bunch of internal circuitry in these guys.

We won't be analyzing these on that level.

Okay. They're going to be a black box that has

essentially some input output characteristic.

That's all we care about.

One of the drawbacks of dealing with things this way is that it may

appear as if KCL and KVL don't apply to these guys.

That's not true.

If you model the internal circuitry, these guys do satisfy KVL and KCL.

It's just that there's something very complicated going on inside there.

Number one, they've got an external power supply that's

feeding them current or voltage or whatever,

that we are generally not going to worry too much about when we're

looking at the op-amp as part of an overall circuit.

So, what we're going to end up with are several rules

for how the op-amp is going to behave.

Okay. Those are based on an analysis of the internal circuitry,

but we aren't going to worry about that translation.

We're just going to have a few rules that we're going to say

this is the way this device behaves,

we're going to forget about it until a 400-level class later on.

So, we're going to use op-amps to perform operations,

but we don't need to actually design and build the operational amplifiers

themselves or analyze them on a detailed level at this stage in our career.

Here's a schematic of a very common 741 operational amplifier.

You can see that it's pretty complex.

It has a whole bunch of bipolar junction transistors in it.

It has a bunch of resistors. It has a couple of inputs.

It has an output. It has a couple of external power supplies and

it's got some stuff here that we don't even need to worry about yet.

But we aren't going to deal with this internal view of

the operational amplifier, we're going to treat it as a black box.

Okay, our high level view of an operational amplifier is going to be to

represent it just as this rightward pointing triangle.

This device has three terminals. There are two input terminals.

They have a positive and a negative sign associated with

them and one output terminal here.

V_n is the voltage applied at the inverting or negative input terminal.

V_p is the voltage applied at the non-inverting or positive input terminal.

The out comes at the output of the operational amplifier.

Now, there are a number of parameters that

this operational amplifiers operation is going to be characterized relative to.

They're not necessarily these individual values, they're something else.

The first of these is the difference in

voltage between the inverting and non-inverting terminals.

The change in voltage between V_p and V_n.

So, Delta V_in is V_p minus V sub n. That's the voltage difference.

Keep in mind that generally, according to our operational amplifier behavior,

we don't care what these individual voltages are,

we just care what the difference is between them.

The other thing that you use to characterize

operational amplifier behavior are the currents into the input terminals.

We'll have some current into the positive or non-inverting terminal and

some other current into the negative or inverting terminal.

Okay, these parameters are what we're going to base

our rules of operational amplifier behavior on.

Now, I want to provide the rules by which we will

characterize the operational amplifiers behavior.

In order to do that, I want to give a slightly more

complete symbol for the operational amplifier.

I said that the operational amplifier requires an external power supply to do its job.

The power supplies are provided here and here.

There are actually two additional terminals

that we need to worry about for five terminals in all.

Generally, when I'm analyzing a circuit, I'll leave these off

and just in the back of my mind, recognize that they're there,

but for right now, I need to put them back in.

Now, I said earlier, that the operation is going to be characterized by this difference in

voltage delta V_in and the currents into the non-inverting and inverting input terminals.

So, when we see one of these devices,

we're going to make the following assumptions.

These assumptions are relative to ideal operational amplifiers.

We will assume that the current into

the non-inverting and the inverting input terminals are both zero.

This device is not accepting any power into these terminals.

So, any power that comes into the output comes from the power

supplies that you've connected up here and here.

We're also going to assume that the difference in voltage

between the positive and negative terminals has to be zero.

Therefore, V_p is going to be equal to V_n.

One other thing has to be true for operational amplifiers, this output voltage.

The amount of output voltage you can get here is constrained by these two power supplies.

This voltage has to be greater than the negative power supply

and less than the positive power supply,

and those are strict inequalities.

Generally, most operational amplifiers you can only get to within

a volt or two of your power supply voltages.

Okay, a few points about operational amplifier behavior.

The output current is generally not known,

you cannot make any assumptions relative to the output current.

Right? It's provided by the external power supplies.

That's where people sometimes early on get themselves into trouble.

They say, "Okay, there's no current into the input terminals,

but I've got some current out of the output terminals.

KCL doesn't apply." Well, it does.

The current coming out of the operational amplifier is

coming from the external power supplies,

you know nothing about those.

You don't know anything about the output current unless you

analyze the circuit to determine what that is.

In general, when I'm analyzing an operational amplifier,

I will start out by applying KCL at the input nodes.

Okay. It doesn't always cure all of your problems,

but it's generally a good place to start.

The operation of the operational amplifiers is generally based on Delta V_in.

If I look at this amplifier as an input-output relationship,

what it looks like is that I have some Delta V_in,

I multiply that by some large number K.

That gives me V_out.

Now, for an ideal operational amplifier,

we assume that K goes to infinity.

Okay. For non-ideal or realistic operational amplifiers,

K is on the order of millions.

We'll assume it's infinite.

Also, I mentioned earlier the output voltage is limited by the external power supplies.

Okay, your output voltage must be lower than the positive power supply.

It must be higher than the negative power supply.

These two things in conjunction with one another lead

us to the conclusion that Delta V_in has to be equal to zero,

because if V_out is infinity times Delta V_in and V_out must be finite.

Right, we can only apply so much voltage at the power supply terminals.

In order to make this a finite number,

if this is infinite, this guy has to be zero.

Okay, let's do an example of analyzing an operational amplifier based circuit.

Here is my op-amp. It has two inputs and one output, okay?

It also has some other voltage supplies and some resistors hanging around here,

and what I haven't shown are the external power supplies.

I generally won't show those. Generally, when I start out analyzing one of these,

the first thing I'm going to do is employ my op-amp rules, okay?

There is no voltage difference between the inverting and non-inverting terminals.

So this delta V is zero. I've tied the non-inverting terminal to ground.

So, this voltage is zero volts. That means that since I

can't have a voltage difference between here and here,

this voltage is zero volts.

Likewise, the current here into the non-inverting terminal is zero,

the current here into the inverting terminal is also zero.

Now, I've labeled everything that I know about this operational amplifier.

I can go ahead and analyze it to determine V_out, okay?

As usual, I'll start out applying KCL at an input node.

KCL at an input node is kind of a good idea,

because you already know something's there.

You know there's no current into the op-amp itself.

So if I call this node A and do KCL at A,

the current through R_in is this voltage,

V_in minus this voltage which the op amp is constraining to be zero volts over RN.

So V_in minus zero over R_in,

this current into this node is equal to this current out of the node,

because there's no current flow through this leg, here.

This current is this voltage minus this voltage over R_f.

So that's equal to zero minus V_out over R_f.

V_in, as my input voltage,

I don't know what it is,

but I have to be told it before I can determine a number for V_out.

So V_out, let's multiply this by R_f,

is R_f over Rn times V_in taking this negative sign over here.

V_out is equal to minus R_f over R_in times V_in.

So whatever you give me for V_in,

I'm going to multiply that by a number,

take the negative of that,

and this op-amp will give you that as V_out.

This has a particular name.

This is an inverting voltage amplifier, okay?

It's amplifying voltage.

You give it a voltage in,

it gives you a voltage out.

It inverts that.

The voltage out you get as the negative of the voltage in.

It is also amplifying that,

according to whatever you choose for R_f and R_in.

If R_f is 10 ohms and R_in is one ohms,

then this is going to be 10 and the output voltage is

going to be negative 10 times whatever the input voltage is.

Let's analyze another operational amplifier circuit.

I want to find V_out with this operational amplifier based circuit.

Notice, again, that I have my three terminal device.

It has some non-inverting and inverting terminal in and output terminal.

I'm not showing my power supplies,

but if I wired this circuit up in the lab,

I would need to provide power to it, okay?

So let's find V_out as a function of V_in.

The first thing I want to do is apply my op-amp rules.

This voltage source is insisting that

the voltage at the non-inverting terminal is going to be set to be V_in.

The op-amp itself is insisting that

there is no voltage difference between these two terminals,

therefore I must have voltage V_in at this terminal.

Therefore, this voltage here is V_in.

Now, I have no current into these terminals.

Notice, very importantly, this voltage source is not providing any power, okay?

The current out of the voltage source is zero.

It's not providing any power in order to create this output voltage.

Any power in this output signal is coming from the external power supplies.

So these currents are zero.

I know something about that current.

Now I'm going to my old fallback standard position.

I'm going to do KCL at the input nodes.

Let me call this node A. I'll apply a KCL there.

Let me say that this current through the resistor Rf is I_f,

and this current through R_1 is I_1,

and those are going to be my positive directions.

So KCL, at A, tells me that the current going into node A

is equal to the current coming out of node A,

so I_f is equal to I_1.

This current is zero, right?

I don't need to list it in my KCL.

Now, I_f is this voltage minus this voltage, over R_f.

So, V_out minus V_in over R_f is equal to the current going through here,

which is just V_in minus,

I'm going to take this as my reference,

it's going to be zero volts.

V_in minus zero over R_1.

So V_out over R_f is equal

to V_in over R_f taking this term over to the other side,

plus V_in over R_1.

Grouping terms, V_out is equal to R_f

times one over R_f plus one over R_1 times V_in.

This becomes a one plus R_f over R_1.

So therefore, V_out is equal

to one plus R_f over R_1 times V_in.

This device takes a voltage V_in,

multiplies it by a number,

and actually a positive number.

In order to get V_out,

this is a non-inverting voltage amplifier.

Okay, you're still amplifying voltage by taking your input voltage,

multiplying it by a number to get the output voltage,

but you're not changing the sign, it not-inverting.

Now one quick thing I want to point out about both this example and the previous one.

Both of these had a resistor which was

feeding back from the output to one of the input terminals.

That is very typical of op-amp based circuits.

It's generally called the feedback loop.

Almost invariably you will feedback from the output to the inverting input terminal.

That's necessary in order to keep this entire device stable.

If I feedback to the positive terminal,

generally this device will do what is called going unstable,

the output will try to go to infinity.

So what you'll have happen is that your output voltage will either

go to the positive or the negative voltage rail, and stay there.

It can't get to infinity,

but it's going to go as high as it can go,

and it's not going to come back.

Okay, this concludes Lecture 12.

Next lecture, we'll do some more work with operational amplifiers.

We'll look at their operation in a little bit more depth,

we'll talk more about the voltage rails applied by the power supplies,

and we'll start looking at them as dependent sources.

Okay, later on in your schooling career you'll see a lot of dependent sources.

We'll just start kind of looking at things in that way in this class.