Find the overall gain of the cascaded operational amplifier as shown below.
Understand the Problem
The question is asking to calculate the overall gain of a cascaded operational amplifier setup, which involves analyzing the relationship between the input and output voltages across multiple stages of amplification.
Answer
The overall gain of the cascaded operational amplifier is \( 20 \).
Answer for screen readers
The overall gain of the cascaded operational amplifier is ( A_{total} = 20 ).
Steps to Solve
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Calculate the gain of the first operational amplifier
The gain ($A_1$) of the first op-amp stage is calculated with the formula: $$ A_1 = 1 + \frac{R_{f1}}{R_1} $$
Given:- ( R_{f1} = 15k\Omega )
- ( R_1 = 5k\Omega )
Substitute the values: $$ A_1 = 1 + \frac{15k\Omega}{5k\Omega} = 1 + 3 = 4 $$
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Calculate the gain of the second operational amplifier
The gain ($A_2$) of the second op-amp stage is calculated similarly: $$ A_2 = 1 + \frac{R_{f2}}{R_2} $$ Given:
- ( R_{f2} = 8k\Omega )
- ( R_2 = 2k\Omega )
Substitute the values: $$ A_2 = 1 + \frac{8k\Omega}{2k\Omega} = 1 + 4 = 5 $$
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Calculate the overall gain of the cascaded amplifiers
The overall gain ($A_{total}$) of the cascaded op-amps is the product of the individual gains: $$ A_{total} = A_1 \times A_2 $$
Substitute the calculated gains: $$ A_{total} = 4 \times 5 = 20 $$
The overall gain of the cascaded operational amplifier is ( A_{total} = 20 ).
More Information
In a cascaded operational amplifier circuit, the overall gain can be found by multiplying the gains of each individual stage. This is a common approach in amplifier design to achieve the desired amplification level.
Tips
- Forgetting to add 1 in the gain formula for non-inverting amplifiers.
- Mixing up the resistors while substituting values leading to incorrect gain calculations.
- Not multiplying the gains correctly for cascaded setups.
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