A current i = 50sin 100πt mA flows in an electrical circuit. Determine, using integral calculus, its
a) Mean Value
b) RMS Value
each correct to 2 decimal places over the range t = 0 to t = 10 ms
I need to do this two way so I was thinking of using the trapezoidal rule and the Simpson's rule but I'm not sure how to.
Thank you for your help
Dylan
Tq for your help
To find the mean value and RMS value of the given current function, you can use integral calculus. Here's how you can do it using the trapezoidal rule and Simpson's rule:
1. Mean Value:
The mean value of a function over an interval can be found by taking the integral of the function over that interval and dividing it by the length of the interval.
Using the trapezoidal rule:
a) Divide the interval [0, 10 ms] into small subintervals. Let's say you divide it into 'n' subintervals.
b) The width of each subinterval, h, will be equal to (10 ms - 0 ms) / n.
c) Evaluate the function at the endpoints of each subinterval and sum them up.
For example, at the ith subinterval, evaluate the function at t = (i-1)h and t = ih, where i varies from 1 to n.
d) Multiply the sum by h/2 to get the approximate value of the integral.
So, the mean value, I_mean_trap, can be calculated as:
I_mean_trap = (h/2) * [f(0) + 2(f(h) + f(2h) + ... + f((n-1)h)) + f(10 ms)]
Using Simpson's rule:
a) Divide the interval [0, 10 ms] into subintervals. Let's say you divide it into 'n' subintervals.
b) The width of each subinterval, h, will be equal to (10 ms - 0 ms) / (2n).
c) Evaluate the function at the endpoints, midpoints, and endpoints of each subinterval and apply Simpson's rule formula.
For example, at the ith subinterval, evaluate the function at t = (2i-2)h, t = (2i-1)h, and t = 2ih, where i varies from 1 to n.
d) Sum up the values obtained in step c) and multiply the sum by h/3 to get the approximate value of the integral.
So, the mean value, I_mean_simp, can be calculated as:
I_mean_simp = (h/3) * [f(0) + 4(f(h) + f(3h) + ... + f((2n-1)h)) + 2(f(2h) + f(4h) + ... + f(2(n-1)h)) + f(10 ms)]
2. RMS Value:
The RMS value of a function over an interval can be found by taking the square root of the mean of the square of the function over that interval.
Using the trapezoidal rule:
a) Divide the interval [0, 10 ms] into small subintervals. The process is the same as for finding the mean value.
b) Evaluate the square of the function at each subinterval and sum them up. Multiply the sum by h/2 to get the approximate value of the integral.
c) Take the square root of the sum obtained in step b) to get the RMS value.
So, the RMS value, I_RMS_trap, can be calculated as:
I_RMS_trap = sqrt((h/2) * [f^2(0) + 2(f^2(h) + f^2(2h) + ... + f^2((n-1)h)) + f^2(10 ms)])
Using Simpson's rule:
a) Divide the interval [0, 10 ms] into subintervals. The process is the same as for finding the mean value.
b) Evaluate the square of the function at each subinterval and apply Simpson's rule formula.
c) Sum up the values obtained in step b) and multiply the sum by h/3 to get the approximate value of the integral.
d) Take the square root of the sum obtained in step c) to get the RMS value.
So, the RMS value, I_RMS_simp, can be calculated as:
I_RMS_simp = sqrt((h/3) * [f^2(0) + 4(f^2(h) + f^2(3h) + ... + f^2((2n-1)h)) + 2(f^2(2h) + f^2(4h) + ... + f^2(2(n-1)h)) + f^2(10 ms)])
Remember, the accuracy of the results will increase as the number of subintervals increases (i.e., as n increases).
These formulas will enable you to calculate the mean value and RMS value of the given current function using both the trapezoidal rule and Simpson's rule.