If v 1 = 10 cos 2t mV and v 2 = 0.5t mV, find v o in the op amp circuit in
. Assume that the voltage across the capacitor is initially zero.

To find the output voltage \( v_o \) in the operational amplifier (op-amp) circuit given the input voltages \( v_1 \) and \( v_2 \), we first need to understand how the op-amp operates in this context. The op-amp can be configured in various ways, but let’s assume we are dealing with a basic inverting or non-inverting amplifier configuration. The key here is to analyze the inputs and how they affect the output.

Understanding the Inputs

We have two input voltages:

• v₁ = 10 cos(2t) mV: This is a cosine function that varies with time.
• v₂ = 0.5t mV: This is a linear function that increases with time.

Op-Amp Configuration

Assuming we are using a non-inverting op-amp configuration, the output voltage \( v_o \) can be expressed in terms of the input voltages. The general formula for a non-inverting amplifier is:

v_o = (1 + \frac{R_f}{R_i}) \cdot v_{in}

Where \( R_f \) is the feedback resistor and \( R_i \) is the input resistor. However, since we have two inputs, we need to consider how they interact. If we assume that \( v_1 \) is the primary input and \( v_2 \) is a secondary input affecting the output, we can express \( v_o \) as:

v_o = A(v_1 - v_2)

Here, \( A \) is the gain of the op-amp, which is typically very high in ideal conditions.

Calculating the Output Voltage

Now, let’s substitute the values of \( v_1 \) and \( v_2 \) into our equation:

v_o = A(10 \cos(2t) - 0.5t)

To find \( v_o \), we need to determine the gain \( A \). For simplicity, let’s assume \( A = 1 \) (unity gain) for this example, which is common in many applications:

v_o = 10 \cos(2t) - 0.5t

Initial Conditions

Given that the voltage across the capacitor is initially zero, we need to consider how this affects the output. If the circuit includes a capacitor in the feedback loop, the initial condition will influence the transient response. However, if we are looking for the steady-state output, we can focus on the above equation.

Final Expression

Thus, the output voltage \( v_o \) in the op-amp circuit, assuming unity gain and considering both inputs, is:

v_o(t) = 10 \cos(2t) - 0.5t

This expression indicates how the output voltage varies over time, influenced by both the oscillatory nature of \( v_1 \) and the linear growth of \( v_2 \). If you have specific values for the resistors or a different configuration, the output could change accordingly, but this gives you a solid foundation to understand the relationship between the inputs and the output in an op-amp circuit.