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

The open-loop gain (GV) of an op-amp has the same frequency characteristics as a first-order RC lowpass filter as shown in Figure 2-3. At frequencies higher than the corner frequency (fC) at which the open-loop gain is 3 dB lower than the DC gain, the open-loop gain decreases at a rate of 6 dB per octave (20 dB per decade). In this frequency range, the decibel open-loop gain of the op-amp (GV) decreases by 6 dB (i.e., the linear open-loop gain (AV) halves) when the frequency doubles. Hence:

fc × AV = constant

The frequency at which the gain is equal to 1 (0 dB) is called the unity gain cross frequency (fT). Therefore, the above equation can be restated as follows. This is called the gain-bandwidth product (designated as GBWP, GBW, GBP, or GB).

fc × AV = fT

Note that this equation is true in the frequency range in which the open-loop gain decreases at a rate of 6 dB per octave.

Now, let’s consider what occurs when an input signal with a frequency of 2±1 kHz is applied to an op-amp that has frequency characteristics as shown in Figure 2-3. In the case of an op-amp under this condition, the gain at 3 kHz is roughly 10 dB lower than the gain at 1 kHz. Normally, the op-amp cannot be used under this condition. Negative feedback solves this issue.

The input (Vin) and the output (Vout) have the following relationship. This relationship is called a closed-loop gain (represented as GCL in dB scale and ACL in linear scale). The 20 log rule is used to convert a linear voltage gain into a decibel voltage gain: G = 20 × log A.

Vout / Vin = ACL = AV / (1 + AV × B)
= 1 / {B (1 + 1 / AV × B)}

where AV is the open-loop gain of an amplifier and B is the feedback factor. (AV × B) is called the loop gain. The denominator, (1 + AV × B), is called the amount of feedback. In the case of negative feedback, AV × B < 0. An op-amp has a very high AV. Hence, | AV x B | >> 1. Therefore, the amount of feedback is calculated as (1 + AV × B) ≈ AV × B (loop gain). Hence, the above equation can be simplified as follows:

Vout / Vin = ACL = 1/B

Figure 2-5 shows this relationship. The op-amp has a bandwidth of fC. With negative feedback, its closed-loop bandwidth expands to fCL. fCL is calculated as follows from the gain-bandwidth product equation:

fCL = fT / ACL

When the closed-loop gain (GCL) or the bandwidth (fCL) is insufficient, it is necessary to select an op-amp with high fT.

## Chapter2 Using an op-amp

2. Using an op-amp
2.1. Feedback (positive and negative feedback)
2.3. Oscillation
2.4. Basic op-amp applications
2.5. Virtual short (virtual ground)

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