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Problem 4: Determining the value of $R_3$

The task is to add an additional resistor $R_3$ to limit the voltage gain to +2.0 dB.

1. Gain in dB to voltage ratio: The relationship between voltage gain $A_v$ and decibels (dB) is given by: $A_v = 10^{\frac{dB}{20}}$ Given that the gain is +2.0 dB: $A_v = 10^{\frac{2.0}{20}} \approx 1.26$

2. Resistor calculation: The voltage gain of an operational amplifier in non-inverting configuration is: $A_v = 1 + \frac{R_f}{R_1}$ where:

   • $R_f$ is the feedback resistor (which in this case would include $R_3$),
   • $R_1$ is the input resistor.

   Rearranging the equation to solve for $R_3$, using $A_v = 1.26$, and $R_1 = 10k\Omega$: $1.26 = 1 + \frac{R_3}{10k}$ Subtract 1 from both sides: $0.26 = \frac{R_3}{10k}$ Solving for $R_3$: $R_3 = 0.26 \times 10k = 2.6k\Omega$

Thus, $R_3 \approx 2.6k\Omega$.

Problem 5: Determining the Unity Gain Frequency

The unity gain frequency is the frequency at which the magnitude of the gain is 1. This can be determined by analyzing the impedance of the circuit involving the capacitor and resistors.

1. Impedance of the capacitor: The impedance of the capacitor $X_C$ is given by: $X_C = \frac{1}{2\pi f C}$ where $f$ is the frequency and $C = 0.01\mu F = 10^{-8} \text{F}$.

2. Impedance of the parallel combination of $R_3$ and $X_C$: The total impedance $Z_F$ of the feedback network is the parallel combination of $R_3$ and $X_C$: $Z_F = \frac{R_3 \cdot X_C}{\sqrt{R_3^2 + X_C^2}}$ For unity gain, the frequency is such that: $A_v = 1 = 1 + \frac{Z_F}{R_1}$

   Solving this requires inserting the values for $R_3$, $C$, and $R_1$ and finding the frequency $f$.

Would you like more details on this or have any other questions about the steps?

Follow-up questions:

1. How does the parallel impedance of a resistor and a capacitor affect the frequency response?
2. What are the common methods to stabilize voltage gain in operational amplifiers?
3. How would increasing the value of $C$ affect the unity gain frequency?
4. What happens to the circuit if $R_3$ is omitted entirely?
5. Can you explain the significance of dB gain in amplifiers and how it translates to voltage?

Tip: Always check component tolerances when designing circuits to ensure real-world accuracy!