Final answer:
To design a broadband Butterworth bandpass filter with the given specifications, we need to calculate the capacitance needed for the lower and upper cutoff frequencies and determine the passband gain. The capacitance can be calculated using the reactance formulas, and the gain can be determined using the decibel to voltage ratio conversion formula.
Step-by-step explanation:
A Butterworth bandpass filter is a type of electronic filter that allows signals within a certain frequency range called the passband to pass through, while attenuating signals outside of this range. To design a broadband Butterworth bandpass filter, we need to determine the values of the components to achieve the desired cutoff frequencies and gain.
For the lower cutoff frequency of 500 Hz, the reactance of the capacitor can be calculated using the formula XC = 1 / (2πfC), where f is the frequency and C is the capacitance. Rearranging the formula, we can solve for C to find the capacitance needed. Plugging in the given values, we get XC = 100000 ohms, and solving for C, we find C = 1 / (2π * 120 * 10^2) F = 1.33 μF.
Similarly, for the upper cutoff frequency of 4500 Hz, the reactance of the capacitor is XC = 22,222 ohms. Solving for C, we find C = 1 / (2π * 450 * 10^2) F = 0.74 μF.
The passband gain of the filter is given as 20 dB. The gain is typically expressed in voltage or power ratios rather than decibels. To convert from decibels to a voltage ratio, we use the formula Vout / Vin = 10^(Gain / 20), where Vout is the output voltage and Vin is the input voltage. Rearranging the formula, we find Gain = 20 log (Vout / Vin). Plugging in the given values, we can solve for Vout / Vin, which represents the voltage gain.