2.1 Fundamentals of the Operational Amplifier (Op-Amp)

The Operational Amplifier, or Op-Amp, is a versatile, high-gain, direct-coupled differential amplifier that serves as the fundamental building block for an extraordinary array of both linear and non-linear analog circuits. Its name derives from its original use in performing mathematical operations in analog computers. Today, its applications are far more widespread.

The fundamental purpose of an Op-Amp is to amplify the difference between two input signals. It can be configured to perform various linear operations (amplification, summing, integration), non-linear operations (comparison, logarithmic functions), and other mathematical tasks. It is capable of operating with both AC and DC signals.

An Op-Amp is constructed from several key internal stages: one or more differential amplifiers at the input, a level translator, and an output stage. The differential input stage gives the Op-Amp its two characteristic input terminals:

1. The inverting terminal (-): An input signal applied here will produce an output signal that is 180° out of phase with the input.
2. The non-inverting terminal (+): An input signal applied here will produce an output signal that is in phase with the input.

These terminals are named based on the phase relationship between their respective inputs and the final output of the amplifier. The performance of this powerful device is defined by a set of key parameters.

2.2 Key Characteristics of Op-Amps

The performance of an Op-Amp is quantified by several key characteristics or parameters. A thorough understanding of these parameters is essential for analyzing circuit behavior, predicting performance, and selecting the appropriate device for an application.

Open Loop Voltage Gain ()

The open loop voltage gain is the differential gain of the Op-Amp without any feedback path connected between the output and the inputs. It represents the maximum possible gain of the device.

• Formula: A_{v}= \frac{v_0}{v_1-v_2}
• Significance: This value is typically very high, often in the hundreds of thousands, which is a key attribute that enables many of the Op-Amp’s applications when used with a feedback network.

Output Offset Voltage

This is the voltage present at the output of the Op-Amp when the differential input voltage (the difference between the inverting and non-inverting inputs) is zero.

• Significance: Ideally, a zero-volt input difference should produce a zero-volt output. The output offset voltage is an imperfection resulting from mismatches in the internal components. For precision applications, this offset should be as low as possible.

Common Mode Rejection Ratio (CMRR)

The Common Mode Rejection Ratio (CMRR) is a measure of an Op-Amp’s ability to reject signals that are common to both input terminals. It is defined as the ratio of the closed-loop differential gain (A_d) to the common mode gain (A_c).

• Formula: CMRR=\frac{A_{d}}{A_{c}}
• Significance: A high CMRR is highly desirable. It indicates that the Op-Amp is very effective at amplifying the difference between its inputs while suppressing any noise or interference that appears simultaneously on both inputs (common-mode signals).

Slew Rate (SR)

The Slew Rate is defined as the maximum rate of change of the Op-Amp’s output voltage in response to a step input voltage.

• Formula: SR=Maximum\:of\:\frac{\text{d}V_{0}}{\text{d}t}
• Significance: Slew Rate is measured in volts per microsecond (V/\mu s). It determines the maximum frequency at which the Op-Amp can produce an undistorted output signal of a given amplitude. It is a critical parameter for high-frequency and high-speed applications.

These characteristics form the basis for comparing the theoretical “ideal” Op-Amp with its real-world “practical” counterpart.

2.3 Ideal vs. Practical Op-Amps

For the purpose of circuit analysis, it is often useful to employ a simplified model of the Op-Amp known as the ‘ideal’ Op-Amp. This theoretical model simplifies calculations, but it’s important to understand how its characteristics differ from those of a ‘practical’ Op-Amp found in the real world.

The Ideal Op-Amp

An ideal Op-Amp is a theoretical concept that does not exist in practice but provides a powerful tool for initial circuit design and analysis. Its characteristics are:

• Input impedance : This implies that the ideal Op-Amp draws no current from the input source.
• Output impedance : This implies that the Op-Amp can deliver any amount of current to the load without a voltage drop across its output.
• Open loop voltage gain : This infinite gain is the foundation for the “virtual short” concept used in analyzing feedback circuits.
• Zero output voltage for zero input voltage: An ideal device has no output offset voltage.
• Bandwidth is infinity: It can amplify signals of any frequency without attenuation.
• Common Mode Rejection Ratio (CMRR) is infinity: It perfectly rejects all common-mode signals.
• Slew Rate (SR) is infinity: Its output can change instantly in response to an input step.

The Practical Op-Amp

Practical Op-Amps deviate from these ideal characteristics due to imperfections in the manufacturing process. A real-world device exhibits the following:

• Input impedance, is in the order of Mega ohms. While not infinite, it is very high.
• Output impedance, is in the order of a few ohms. While not zero, it is very low.
• Open loop voltage gain, will be high, but finite (e.g., 10^5 to 10^6).

For optimal performance, a practical Op-Amp should have characteristics that approach the ideal:

• Input impedance should be as high as possible.
• Output impedance should be as low as possible.
• Open loop voltage gain should be as high as possible.
• Output offset voltage should be as low as possible.
• Operating bandwidth should be as high as possible.
• CMRR should be as high as possible.
• Slew rate should be as high as possible.

The IC 741 is the most popular and widely used example of a practical operational amplifier. In the following sections, we will apply these concepts to construct fundamental circuits using the Op-Amp as our core component.