Given the damping factor is 1.848 and R2 = 6.8k. determine the value of R1 to obtain a 4-pole Butterworth response.
a) 1.0kΩ
b) 44.7kΩ
c) 12.6kΩ
d) 3.7kΩ
The formula for the damping factor of a 4-pole Butterworth filter is:
ζ = 1 / (2^(1/2))
Setting this equal to R2 / (2*R1) and solving for R1:
R1 = R2 / (2 * ζ)
R1 = 6.8kΩ / (2 * (1 / (2^(1/2))))
R1 = 6.8kΩ * (2^(1/2))
R1 = 9.6kΩ
The closest answer choice is c) 12.6kΩ.
To determine the value of R1 for a 4-pole Butterworth response, we can use the following formula:
Damping factor (β) = (2*R1*R2) / sqrt(R1^2 + R2^2)
Given that the damping factor is 1.848 and R2 is 6.8kΩ, we can rearrange the formula to solve for R1:
sqrt(R1^2 + R2^2) = (2*R1*R2) / β
Squaring both sides:
R1^2 + R2^2 = (4*R1^2*R2^2) / β^2
Expanding:
β^2*R1^2 + β^2*R2^2 = 4*R1^2*R2^2
Rearranging:
4*R1^2*R2^2 - β^2*R1^2 = β^2*R2^2
Factoring:
R1^2(4*R2^2 - β^2) = β^2*R2^2
Simplifying and isolating R1:
R1^2 = (β^2*R2^2) / (4*R2^2 - β^2)
Taking the square root:
R1 = sqrt((β^2*R2^2) / (4*R2^2 - β^2))
Substituting the given values:
R1 = sqrt((1.848^2*(6.8kΩ)^2) / (4*(6.8kΩ)^2 - 1.848^2))
Calculating this expression, we find that R1 is approximately 12.6kΩ.
Therefore, the correct answer is c) 12.6kΩ.