# Investing amplifier formula derivation physics

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We know that v-out equals A times Now, it's usually v-plus minus v-minus. This is v-plus. This is v-minus. Usually the expression here is v-plus minus v-minus. And since v-plus is zero, we're just gonna put in minus v-minus.

This is equivalent to saying that v-minus equals minus v-out over A. So what else can we write for this circuit? OK, let's look at these resistors. Let's call this plus and minus vR1; and we'll call this one plus minus vR2. So there's a current flowing here, and that we'll call I. I equals vR1 over R1. Another way I can write that.

What's this voltage here? So I can write this in terms of v-minus, and that equals v-in minus v-minus over R1. That's the current going through this guy here. Now I'm gonna use something special. I'm gonna use something special that I know about this amplifier. What I know about an op-amp is that this current here is equal to zero. There's no current that flows into an ideal op-amp. So I could take advantage of that. What that means is that I flows in R2. So let me write and expression for I based on what I find over here, based on R2.

I can write I equals, let's do it, it's vR2 over R2. And I can write vR2 as: v-minus minus v-out over R2. All right, so I took advantage of the zero current flowing in here to write an expression for current going all the way through. So now we're gonna set these two equal to each other. Now we're gonna make these two equal to each other. Let me go over here and do that. V-in minus v-minus over R1.

That equals this term here, which is v-minus minus v-not, v-out rather, over R2. How many variables do we have here? We have v-out, we have v-in, and we have v-minus. And what I want is just v-out and v-in, so I'm gonna try to eliminate v-minus; and the way I'm gonna do that is this expression over here.

We're gonna take advantage of this statement right here to replace minus v-out over A. So I'll do that right here. So let me rewrite this. It's gonna be v-in minus minus v-out over A, so I get to make this a plus, and this becomes v-out over A all divided by R1. And that equals V-minus is minus v-out over A minus v-out over R2.

All right, let's roll down a little bit, get some room, and we'll keep going. What am I gonna do next? Next, I'm gonna multiply both sides by A, just to get A out of the bottom there. There's a lot of algebra here, but trust me, it's gonna simplify down here in just a minute.

All right, so now I'm gonna break this up into separate terms so I can handle them separately. Let's change colors so we don't get bored. Next, what I'm gonna do is start to gather the v-out terms on one side and the v-in terms on the other side. So that means that this v-out term here is gonna go to the other side. Av-in over R1 equals, let's do minus v-out over R2 minus Av-out over R2. And this term comes over as minus v-out over R1. Haven't made any sign errors yet.

Now let me clear the R1. We'll multiply both sides by R1. Yeah, the R1s cancel on that last term. Av-in equals. Out of here I can factor this term here. This small standing current in the output transistors establishes the output stage in class AB operation and reduces the crossover distortion of this stage. Small-signal differential mode[ edit ] A small differential input voltage signal gives rise, through multiple stages of current amplification, to a much larger voltage signal on output.

Input impedance[ edit ] The input stage with Q1 and Q3 is similar to an emitter-coupled pair long-tailed pair , with Q2 and Q4 adding some degenerating impedance. The input impedance is relatively high because of the small current through Q1-Q4. The common mode input impedance is even higher, as the input stage works at an essentially constant current. This differential base current causes a change in the differential collector current in each leg by iinhfe. This portion of the op amp cleverly changes a differential signal at the op amp inputs to a single-ended signal at the base of Q15, and in a way that avoids wastefully discarding the signal in either leg.

To see how, notice that a small negative change in voltage at the inverting input Q2 base drives it out of conduction, and this incremental decrease in current passes directly from Q4 collector to its emitter, resulting in a decrease in base drive for Q On the other hand, a small positive change in voltage at the non-inverting input Q1 base drives this transistor into conduction, reflected in an increase in current at the collector of Q3.

Thus, the increase in Q3 emitter current is mirrored in an increase in Q6 collector current; the increased collector currents shunts more from the collector node and results in a decrease in base drive current for Q Besides avoiding wasting 3 dB of gain here, this technique decreases common-mode gain and feedthrough of power supply noise.

Output amplifier[ edit ] Output transistors Q14 and Q20 are each configured as an emitter follower, so no voltage gain occurs there; instead, this stage provides current gain, equal to the hfe of Q14 resp. The output impedance is not zero, as it would be in an ideal op amp, but with negative feedback it approaches zero at low frequencies. Overall open-loop voltage gain[ edit ] The net open-loop small-signal voltage gain of the op amp involves the product of the current gain hfe of some 4 transistors.

Other linear characteristics[ edit ] Small-signal common mode gain[ edit ] The ideal op amp has infinite common-mode rejection ratio , or zero common-mode gain. In the typical op amp, the common-mode rejection ratio is 90 dB, implying an open-loop common-mode voltage gain of about 6.

The 30 pF capacitor stabilizes the amplifier via Miller compensation and functions in a manner similar to an op-amp integrator circuit. This internal compensation is provided to achieve unconditional stability of the amplifier in negative feedback configurations where the feedback network is non-reactive and the closed loop gain is unity or higher.

The potentiometer is adjusted such that the output is null midrange when the inputs are shorted together. The output range of the amplifier is about one volt less than the supply voltage, owing in part to VBE of the output transistors Q14 and Q Later versions of this amplifier schematic may show a somewhat different method of output current limiting.

Applicability considerations[ edit ] While the was historically used in audio and other sensitive equipment, such use is now rare because of the improved noise performance of more modern op amps. Apart from generating noticeable hiss, s and other older op amps may have poor common-mode rejection ratios and so will often introduce cable-borne mains hum and other common-mode interference, such as switch 'clicks', into sensitive equipment.

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The input signals are added with their own gain. The disadvantage is the inversion of the sum, which might not be desirable in some cases. Figure 1 Figure 1 shows the non-inverting summing amplifier with two inputs. Its transfer function is shown in equation 1. Each signal is added with its own gain created by the feedback resistor, Rf, and the corresponding resistor for that signal. But, why is that? Why is this transfer function a lot simpler than the non-inverting summing amplifier?

How can we derive this function? What is the transfer function of the inverting summing amplifier with 3, 4, or n inputs? This article answers all these questions. Figure 2 The key point here is to understand that the voltage level at the inverting input of the operational amplifier is zero volts. If this is clear to you, the rest is really simple. In the electronics jargon, it is said that the node is a virtual ground.

It is called like that because it presents zero volts, without being galvanically connected to ground. But why is the inverting input zero volts? Because the non-inverting input is connected to ground. The Op Amp will set the output level at a voltage that will bring its inverting input at the same level as the non-inverting input. The drift problem of Instrumentation Amplifiers can be improved if amplifiers A1 and A2 have offset voltages which tends to track the temperature.

The gain of an instrumentation amplifier can be varied by changing R1 alone. A high gain accuracy can be obtained by using precision metal film resistors for all the resistances. Because of the large negative feedback used, the amplifier has good linearity, typically about 0. The output impedance is also low, being in the range of milli ohms.

The input bias current of the Instrumentation Amplifier Circuit is determined by that of the amplifiers A1 and A2.

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