Frequency Response of the Op-Amp

For the ideal op-amp, the gain is infinite and it has infinite bandwidth. But the actual op-amp has finite bandwidth and finite gain. And the gain versus frequency curve is shown in figure 1. The Y-axis on the curve is the voltage gain of the op-amp in dB, while the X-axis is the frequency in the logarithmic scale.

Fig. 1 Frequency Response of the operational amplifier

As shown in Fig.1, The gain of the op-amp is constant up to a certain frequency and beyond that frequency, the gain of the op-amp reduces at a constant rate of -20 dB/dec.

Cut-off Frequency of the op-amp: The frequency at which the gain of the op-amp reduces by 3dB from the maximum value is known as the cut-off frequency of the op-amp. As seen from the above frequency response curve of the op-amp, the cut-off frequency is very low. Typically for the op-amp, it used to be in the range of 10 to 100 Hz. And up to cut-off frequency, the op-amp provides very high gain.

Although we typically say that, the gain of the op-amp is very high. But actually, the op-amp provides a very high gain up to cut-off frequency. The reason is, all the op-amps are internally compensated. That means all the op-amps have an internal compensation capacitor. And this internal compensation capacitor ensures that the op-amp has a stable response at the high frequencies.

Because of this internal compensation capacitor, the op-amp has a single break frequency up to the gain of the op-amp reaches to unity. Without the internal compensation capacitor, the op-amp can have a multiple break frequency till the gain of the op-amp becomes unity. (As shown in Fig.2)

Fig.2 Multiple break-frequencies in the op-amp frequency response without internal compensation capacitor

The multiple break frequencies can occur because of the stray capacitances and load capacitance. And because of the multiple break frequencies, the op-amp becomes unstable at high frequency. And that’s why all op-amps are internally compensated. And because of this internal compensation, in the open-loop condition, the cut-off frequency of the op-amp is very low.

Unity Gain Frequency : The frequency where the gain of the op-amp is unity is called unity gain frequency.

As, shown in Fig.3, because of the internal compensation, it is easy to understand the behavior of the op-amp with frequency (particularly in the closed-loop configuration). And the product of gain and frequency remains constant till the unity gain frequency for the op-amp, which is known as the gain-bandwidth product of the op-amp.

Fig. 3 Gain-Bandwidth Product of Op-Amp

For example, as shown in Fig.3, at 1 kHz frequency, the gain of the op-amp is 60 dB = 10³. Therefore, the gain-bandwidth product (GBP) is 1000Hz x 10³ = 10⁶

On the other end, at 1MHz, the gain of the op-amp is 1. Therefore, the GBP is 10⁶.

The Gain Bandwidth Product

Using the gain-bandwidth product, it is easy to identify the cut-off frequency of the op-amp, in the closed-loop configuration. Let’s say in the closed-loop configuration, the gain of the op-amp is 40 dB (100). In that case, the frequency response of the op-amp is shown in Fig.4.

Fig.4 The Frequency Response of the op-amp in the closed loop configuration

As shown in Fig.4, the gain of the op-amp is flat up to a certain frequency. And then it starts reducing at 20 dB/dec. The frequency from where the gain starts reducing is known as the cut-off frequency in the closed-loop configuration. And it can be found using the Gain Bandwidth Product.

For example, the gain of the op-amp is 100. (40 dB) and the gain-bandwidth product is 10⁶. Therefore, the cut-off frequency in the closed-loop configuration is 10⁶ / 100 = 10 kHz. As, seen from the above calculation, when the op-amp is used in the closed-loop configuration, then the cut-off frequency of the op-amp increases. Or in other words, using the op-amp in the closed-loop configuration, the gain up to which we get a constant gain (Flat gain response) can be increased. Also, here the product of closed-loop gain, and the cut-off frequency is equal to Gain Bandwidth Product (GBP).

GBP = A_CL x f_CL

Using this gain-bandwidth product, at a particular closed-loop gain, we can find the frequency up to which the gain of the op-amp will remain constant. For example, in the above case, when the Gain Bandwidth Product of the op-amp is 10⁶ and closed-loop gain 100, then up to 10 kHz, the gain of the op-amp will remain constant. Beyond that, it will start reducing.

That means to use the op-amp at high frequency with high gain, the gain bandwidth product of the op-amp should be sufficient to provide the constant gain at the operating frequency.

For example, with a required gain of 40dB (100), if the op-amp needs to be operated at 100 kHz, then the Gain Bandwidth Product of the op-amp should be at least 100 x 10⁵ = 10⁷ = 10 MHz. And sometimes, it becomes a very costly solution to use the op-amp of a very high gain-bandwidth product.

Instead, by cascading multiple stages, the bandwidth (the usable frequency up to which the gain of the op-amp is almost constant) of the overall cascaded system can be increased.

Cascading of op-amps to increase the overall bandwidth

As shown in Fig.5, let’s say for one op-amp, the Gain Bandwidth Product is 10⁶. And the gain of the op-amp in the closed-loop configuration is set to 100. In that case, the cut-off frequency of the op-amp is 10⁶ / 100 = 10⁴ = 10 kHz.

Fig. 5 Closed loop cut-off frequency of the Non-inverting op-amp

That means, in this configuration, the op-amp can provide a fixed gain only upto 10 kHz frequency. If we want to use the op-amp at a higher frequency with the same gain, then we need to choose an op-amp of high gain-bandwidth product. But the same can be achieved using the cascade connection of two op-amps. As shown in Fig.6, the two op-amps are cascaded.

Fig. 6 Cascade connection of the two op-amps to increase the overall bandwidth

As shown in figure.6, the gain-bandwidth product of each op-amp is 10⁶. And the gain of each op-amp is set to 10. That means the combined gain of the two op-amps is approximately equal to 100.

But now the cut-off frequency of the overall cascaded system is approximately equal to 64 kHz. If we have used a single op-amp with gain of 100. Then the cut-off frequency of that op-amp would have been 10 kHz. But with the cascaded connection, now the usable frequency range has been increased.

If fc is the cut-off frequency of the single stage, then for n- cascaded stages, the cut-off frequency is

where, n- number of stages and fc is the cut-off frequency of the single stage

In the above expression, when we put fc = 100 kHz and n = 2, we get f’_cl = 64 kHz

That means instead of using a single stage op-amp, by using a multiple stages, the overall bandwidth of the op-amp can be increased for the same gain.

For more information about Gain Bandwidth Product, check this video tutorial.

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