Question

Given the following two complex numbers: Z1 = - j/2πfC

Z2 = j2πfL a) Solve for f in terms of L and C such that |Z1| = |Z2|. Assume that both L and C are positive real numbers.

Answer 1

To solve for f in terms of L and C such that |Z1| = |Z2|, we need to equate the magnitudes of Z1 and Z2.

The magnitude of a complex number Z = a + bj is given by |Z| = sqrt(a^2 + b^2).

For Z1 = -j/2πfC, the magnitude is:

|Z1| = sqrt((0)^2 + (-1/(2πfC))^2)

= sqrt(1/(4π^2f^2C^2))

= 1/(2πfC)

For Z2 = j2πfL, the magnitude is:

|Z2| = sqrt((0)^2 + (2πfL)^2)

= sqrt(4π^2f^2L^2)

= 2πfL

Now, we can equate the magnitudes of Z1 and Z2:

|Z1| = |Z2|

1/(2πfC) = 2πfL

To solve for f, we can rearrange the equation:

1/(2πC) = 2πfL

Now, isolate f:

f = 1/(4π^2CL)

Therefore, f in terms of L and C such that |Z1| = |Z2| is f = 1/(4π^2CL).

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