3.1 Basic Amplifier Configurations

Through the strategic use of external feedback components, Op-Amps can be configured to create a variety of fundamental linear amplifier circuits. A key principle for analyzing these circuits is the concept of the “virtual short,” which states that for an Op-Amp in a stable feedback configuration, the voltage difference between the inverting and non-inverting terminals is effectively zero. This is a direct consequence of the amplifier’s extremely high open-loop gain.

Inverting Amplifier

An inverting amplifier receives an input signal through a resistor connected to its inverting terminal. It produces an output that is both amplified and inverted, meaning it is phase-shifted by 180° relative to the input. With the non-inverting terminal connected to ground, the virtual short concept dictates that the inverting terminal is also at zero volts. The nodal equation at this terminal is:

\frac{0-V_i}{R_1}+ \frac{0-V_0}{R_f}=0

Solving for the voltage gain (\frac{V_0}{V_i}) yields:

\frac{V_0}{V_i}= \frac{-R_f}{R_1}

The negative sign in the equation signifies the 180° phase inversion between the input and output signals.

Non-Inverting Amplifier

A non-inverting amplifier receives an input signal directly at its non-inverting terminal. As its name implies, it amplifies the signal without inverting its phase. The input voltage V_i is applied to the non-inverting terminal, and due to the virtual short, the voltage at the inverting terminal is also V_i. Using the voltage division principle for the feedback network, we can write:

V_{i} = V_{0}\left(\frac{R_1}{R_1+R_f}\right)

Rearranging this to find the voltage gain gives:

\frac{V_0}{V_i}=1+\frac{R_f}{R_1}

The positive sign indicates that the output signal is in phase with the input signal.

Voltage Follower

The voltage follower is a special case of the non-inverting amplifier where the feedback resistor (R_f) is zero and the input resistor (R_1) is infinite. This is achieved by directly connecting the output to the inverting input. In this configuration, the output voltage perfectly follows the input voltage. Since the output is tied to the inverting terminal, V_0 appears at this terminal. Due to the virtual short, this voltage must equal the input voltage at the non-inverting terminal:

V_{0} = V_{i}

This results in a voltage gain of exactly one. Voltage followers are commonly used as buffers to isolate stages in a circuit.

Having established these basic amplification roles, we can now explore how Op-Amps are used to perform arithmetic operations.

3.2 Arithmetic Circuits

Arithmetic circuits are those designed to perform mathematical operations such as addition and subtraction. By leveraging specific resistor configurations, Op-Amps can be used to create simple yet powerful analog computers.

Adder (Summing Amplifier)

An adder, or summing amplifier, is an electronic circuit that produces an output voltage equal to the sum of the input voltages applied to its inverting terminal. Since the non-inverting terminal is grounded, the inverting terminal is at a virtual ground (0V). The nodal equation at this node is the sum of the currents through the input resistors and the feedback resistor:

\frac{0-V_1}{R_1}+\frac{0-V_2}{R_2}+\frac{0-V_0}{R_f}=0

Solving for the output voltage, V_0, gives the general equation:

V_{0}=-R_{f}\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}\right)

In the special case where all resistors are equal (R_{f}=R_{1}=R_{2}=R), the equation simplifies to:

V_{0}=-(V_{1}+V_{2})

The negative sign indicates that the output is an inverted sum of the inputs.

Subtractor (Difference Amplifier)

A subtractor, or difference amplifier, produces an output proportional to the difference between the two input voltages. Its analysis is best approached using the superposition theorem.

• Step 1: First, consider only the input V_{1} and ground the input V_{2}. The circuit behaves as a non-inverting amplifier. The output voltage contribution from V_{1}, denoted as V_{01}, is: V_{01}=V_{1}\left(\frac{R_3}{R_2+R_3}\right)\left(1+\frac{R_f}{R_1}\right)
• Step 2: Next, consider only the input V_{2} and ground the input V_{1}. The circuit now behaves as a simple inverting amplifier. The output voltage contribution from V_{2}, denoted as V_{02}, is: V_{02}=\left(-\frac{R_f}{R_1}\right)V_{2}
• Step 3: The total output voltage V_0 is the sum of the individual contributions: V_{0}=V_{01}+V_{02} = V_{1}\left(\frac{R_3}{R_2+R_3}\right)\left(1+\frac{R_f}{R_1}\right)-\left(\frac{R_f}{R_1}\right)V_{2}

In the common case where all resistors are of equal value (R_{f}=R_{1}=R_{2}=R_{3}=R), the expression simplifies dramatically to:

V_{0}=V_{1}-V_{2}

This demonstrates the circuit’s ability to directly calculate the difference between two signals. From arithmetic, we can move to circuits that perform calculus-based operations.

3.3 Differentiator and Integrator Circuits

Op-Amps can be configured to perform the fundamental mathematical operations of differentiation and integration, which are critical in analog signal processing, control systems, and waveform generation.

Differentiator

A differentiator is a circuit that produces an output signal proportional to the first derivative (the rate of change) of its input signal. This is achieved by placing a capacitor in the input path to the inverting terminal. With the inverting terminal at virtual ground, the nodal equation is:

C\frac{\text{d}(0-V_{i})}{\text{d}t}+\frac{0-V_0}{R}=0

Solving for the output voltage, V_0, yields the equation for differentiation:

V_{0}=-RC\frac{\text{d}V_{i}}{\text{d}t}

If the product of resistance and capacitance (RC) is equal to 1 second, the circuit provides unity gain. The negative sign indicates a 180° phase inversion.

Integrator

An integrator circuit produces an output signal that is the mathematical integral of its input signal over time. This is accomplished by using a capacitor in the feedback path and a resistor in the input path. The nodal equation at the virtual ground of the inverting terminal is:

\frac{0-V_i}{R}+C\frac{\text{d}(0-V_{0})}{\text{d}t}=0

Rearranging and integrating both sides gives the output voltage equation:

V_{0}=-\frac{1}{RC}\int V_{i}{\text{d}t}

As with the differentiator, a time constant of RC=1 \text{ sec} results in unity gain, and the negative sign signifies phase inversion.

Beyond these mathematical operations, Op-Amps are also essential for converting between fundamental electrical quantities.

3.4 Converters of Electrical Quantities

In many circuit design scenarios, it is necessary to convert a voltage signal into a proportional current or vice versa. Op-Amps facilitate the creation of circuits that perform these conversions, which are defined by their transconductance and transresistance characteristics.

Voltage to Current Converter

A Voltage to Current (V-to-I) converter produces an output current that is directly proportional to its input voltage. In this configuration, the input voltage V_i is applied to the non-inverting terminal. Due to the virtual short, the same voltage V_i appears at the inverting terminal. The nodal equation at this terminal is:

\frac{V_i}{R_1}-I_{0}=0

This yields the relationship for the output current:

I_{0}=\frac{V_i}{R_1}

The gain of this circuit is known as Transconductance, defined as the ratio of output current to input voltage, which is equal to \frac{1}{R_1}.

Current to Voltage Converter

A Current to Voltage (I-to-V) converter produces an output voltage that is directly proportional to its input current. Here, the input current I_i is fed into the inverting terminal, which is held at virtual ground. The nodal equation is:

-I_{i}+\frac{0-V_0}{R_f}=0

Solving for the output voltage gives:

V_{0}=-R_{f}I_{i}

The negative sign indicates a 180° phase difference between the input current and the output voltage. The gain of this circuit is called Transresistance, defined as the ratio of output voltage to input current, which is equal to -R_f.

These linear applications showcase the Op-Amp’s flexibility, which extends into the realm of non-linear circuit functions.