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The closed-loop gain of an op-amp with negative feedback can be calculated easily using the concept of a virtual short-circuit (also known as a virtual short or virtual ground^{*}).

The concept of a virtual short is that the V_{IN (+)} and V_{IN (-)} terminals of an op-amp with negative feedback have almost the same potential regardless of the input signal when it has a large open-loop gain.

Consider as follows to intuitively understand a virtual short.

An op-amp amplifies a difference in voltage between V_{IN(+)} and V_{IN(-)} by a factor of 100,000 or more (called the open-loop gain). However, a real op-amp has a finite output. Therefore, when a distortion-free output is obtained with an amplifier using an op-amp, the difference in voltage between the V_{IN(+)} and V_{IN(-)} inputs should be negligible.

In the case of the negative-feedback amplifier (inverting amplifier) shown in Figure 2-16, the output is connected to the input in such a manner that an increase in output causes a decrease in input. As a result, the output signal fits between the power supply and ground. (Suppose, for example, that an inverting amplifier has an input voltage of 1 V_{pp} and a gain of 3 (R_{2} = 3 × R_{1}). Then, the output voltage becomes 3 V_{pp}.)

At this time, the op-amp is operating with an open-loop gain of 100,000. Since the output voltage is 3 V_{pp}, the input voltage is 3 V_{pp}/100,000 = 30 μV_{pp}. Hence, V_{IN(-)} ≈ V_{IN(+)}.

Next, let’s use simple calculations to understand this. Figure 2-16 shows a negative-feedback amplifier (inverting amplifier) using an op-amp.

Suppose that the op-amp is the ideal one. Then, the following are true:

- Infinite open-loop gain (AV)
- Infinite input impedance
- Zero output impedance

Since the input impedance is infinite, all of the current flowing through R_{1} (i_{1}) flows through R_{2}.

i_{1} ＝ (V_{i} – V_{IN(−)}) / R_{1} = (V_{IN(−)} - V_{o}) / R_{2} (1)

The output voltage of the op-amp is given by the equation: V_{o} = A_{V} × (V_{IN(+)} – V_{IN(−)}) (2)

From Equations 1 and 2, V_{IN(+)} is calculated as follows:

_{IN(＋)} = V_{IN(−)} from Equation 3.

Hence, the voltage at the V_{IN(−)} input is equal to that of the V_{ IN(＋)} input connected to GND.

In this case, the condition of the V_{IN(−)} input is called a virtual short.

_{IN(-)} is called a virtual ground since it is virtually equal to GND.

Next, let’s calculate the closed-loop gain (A_{V}) of the noninverting amplifier shown in Figure 2-17 using a virtual short and the ideal op-amp. Let’s express the output voltage (V_{o}) as a function of V_{i}. From the concept of a virtual short, V_{IN(-)} = V_{IN(+)} = V_{i}.

Therefore, the current flowing through R_{1} (i_{1}) is calculated as follows:

I_{1} = V_{IN(-)} / R_{1} = V_{i} / R_{1}

No current flows to the op-amp input since it has infinite impedance. Letting the current flowing through R_{2} be I_{2}, I_{1} = I_{2}. Hence, the voltage across R_{2} (V_{R2}) is:

V_{R2} = R_{2} × I_{2} = R_{2} × V_{i} / R_{1}

Hence, V_{o} is calculated as:

V_{o} ＝ V_{R1} + V_{R2
}= V_{i} + R_{2} × V_{i} /R_{1} = V_{i} × (R_{1} + R_{2}) / R_{1}

A_{V} = V_{o} / V_{i} = (R_{1} + R_{2}) / R_{1}

You can easily find the closed-loop gain equation.

The closed-loop gain (A_{V}) of the inverting amplifier shown in Figure 2-18 can also be calculated in the same manner.

V_{IN(-)} = V_{IN(+)} = 0 V (GND)

I_{1} = V_{1} / R_{1} = I_{2
}V_{o} = V_{R2} = R_{2} × I_{2} = R_{2} × V_{1} / R_{1}

Hence, the closed-loop gain is:

A_{V} = V_{o} / V_{i} = R_{2} / R_{1}

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2. Using an op-amp

2.1. Feedback (positive and negative feedback)

2.2. Open-loop and closed-loop gains (Increasing the bandwidth of an amplifier)

2.3. Oscillation

2.4. Basic op-amp applications

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