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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As its name suggests, the Operational Amplifier is one type of amplifier. The basic function of any amplifier is to amplify the input signal. But apart from amplifying the signal, it is also possible to perform different arithmetic operations using op-amp.

In the early days, when digital computers were not evolved, then op-amp was used to perform the different arithmetic operations like addition, subtraction, integration, and differentiation.

That’s the reason it is known as the Operational Amplifier. (An amplifier, which can perform different (arithmetic) operations)

The Circuit Symbol of Op-Amp

Fig. 1 The circuit symbol of an Operational Amplifier

As shown in the figure, the op-amp consists of two inputs, one output, and two power supplies. (positive and negative power supplies).

Some operational amplifiers work on the single power supply. (Such op-amps are known as single supply op-amp)

In the op-amp circuit symbol, the input terminal marked as positive is known as a non-inverting input terminal, and the terminal marked with a negative sign is known as the inverting input terminal. (as shown below in the figure)

V+ is the positive biasing voltage

V- is the negative biasing voltage

Fig.2  Inverting and non-inverting input terminals of the op-amp

Operation of Op-Amp in the open-loop configuration:

In the open-loop configuration, the op-amp is operated without any kind of feedback.

In the open-loop configuration, the op-amp amplifies the difference between the two input terminals (Between inverting and non-inverting inputs)

As shown in the fig.1, if V1 and V2 are the inputs at non-inverting and the inverting input terminals, then the output of the op-amp

Vo = Aol x (V1 – V2)

where Aol –  open-loop gain of the op-amp

As shown in Fig.3, if the input is applied at the non-inverting input terminal (V1) and another input  terminal is grounded then the output can be given as

Vo = Aol x V1

Fig. 3 Operation of the op-amp in open-loop configuration with the input applied at the non-inverting input terminal

Similarly, as shown in Fig. 4 when the input is applied only at the inverting input terminal and non-inverting input is grounded then the output can be given as

Vo = – Aol x V2

Fig. 4 Operation of the op-amp in open-loop configuration with the input applied at the inverting input terminal

If the differential input Vd is applied between the non-inverting and the inverting input terminal (as shown in fig. 5), then output in the open-loop configuration can be given as

Vo = Aol x Vd 

Fig. 5 Operation of the op-amp in the open-loop configuration with differential input

Typically, the open-loop gain of the op-amp is very high. (In the range of 10^5 to 10^6).

Even for very small differential input, the output of the op-amp will get saturated.

Example: 

if Vd = 1mV and Aol = 10^5, then Vo = 100V. (Theoretically)

But the output of the op-amp will be limited to positive and the negative saturation voltage (± Vsat).

For one op-amp, if the saturation voltages are ±12V, then for the above example, the output will be restricted to 12V,

And even for Vd = 5mV, Vo = 12V.

So, in such a case, it is said that the op-amp is operating in the saturation region. Typically the saturation voltage is less than the biasing voltages of the op-amp.

Voltage transfer curve of the op-amp in open loop configuration:

The same phenomenon explained above can also be explained with the help of the Voltage transfer curve of the op-amp.

Fig. 6 shows the voltage-transfer curve of the op-amp in the open-loop configuration.

Fig. 6 Voltage transfer curve of the op-amp in the open-loop configuration

As shown in the figure, the output voltage Vo is represented in the Y-axis and differential input to the op-amp is represented on the X-axis.

It is evident that, in the open-loop configuration,  for very small differential input only (typically in μV) the output of the op-amp will operate in the linear range. Beyond that, the op-amp will operate in the saturation region. (Because of the very high gain of the op-amp). In the linear region, the slope of the curve represents the open-loop voltage gain of the op-amp.

In the open-loop configuration, the op-amp can be used as a comparator. Apart from that, with the feedback, the op-amp can be used in various applications.

Here is the few important applications of the op-amp.

Applications of the Operational Amplifier:

1. Comparator
2. Active Filters
3. Oscillators and Multivibrators
4. Waveshaping and Waveform generating circuits
5. Analog to Digital Converter (ADC)
6. Digital to Analog Converter (DAC)
7. Linear Amplification
8. To perform the arithmetic operation on the signal (Addition, Subtraction, Multiplication, Integration, Differentiation etc.)

Equivalent Circuit of the Operational Amplifier:

Fig. 7 Equivalent circuit of the operational Amplifier

Fig. 7 shows the equivalent circuit of the op-amp.

Where,

V1, V2 – Non-inverting and inverting input of the op-amp

Vd = V1 – V2

Ri – Input resistance of the op-amp

Ro – Output Resistance of the op-amp

A- Open loop gain of the op-amp

Characteristics of Ideal Op-Amp: 

As, mentioned above, the op-amp is a very versatile IC and can be used in various applications. Because of its favorable characteristics, it is used in various applications.

Here is the list of characteristics of the ideal op-amp

1.  Infinite Input Impedance
2.  Zero Output Impedance
3.  Infinite Voltage Gain
4.  Infinite Bandwidth
5.  Infinite Slew Rate
6. Infinite Common Mode Rejection Ratio (CMRR )
7.  Zero input offset voltage (Output is Zero when input is Zero)

The characteristics of the actual op-amp will be different from the ideal op-amp. To get an idea, the characteristics of the very popular op-amp IC 741 is shown below:

Table 1: Characteristics of op-amp IC 741

To get more information about op-amp, check this video on Operational Amplifier .

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