Question

Simplify and present results in polar and time domain forms.

[(1 + j5)(2-j) + 5∠45°] = ?

Answer 1

To simplify the expression [(1 + j5)(2-j) + 5∠45°] in polar and time domain forms, first multiply the terms, then add the result to 5∠45°. Convert the real part to polar form and combine real and imaginary parts to get the result in polar form. Finally, convert the polar form to rectangular form using trigonometric functions to obtain the result in time domain form.

Step-by-step explanation:

Simplifying and presenting results in polar and time domain forms:

To simplify the expression [(1 + j5)(2-j) + 5∠45°] in polar form:

1. First, perform the multiplication inside the brackets: (1 + j5)(2-j) = 1(2) + 1(j5) + j5(2) + j5(j5) = 2 + j5 - 2j - 5 = -3 + 3j
2. Next, add the result to 5∠45°: -3 + 3j + 5∠45° = (-3 + 5∠45°) + (3j)
3. Now, convert the real part to polar form: -3 = 3∠180°
4. Combine the real and imaginary parts to get the final result in polar form: (-3 + 5∠45°) + (3j) = (3∠180° + 5∠45°) + (3∠90°)

Therefore, the simplified expression in polar form is (3∠180° + 5∠45°) + (3∠90°).

To present the result in time domain form, convert the polar form to rectangular form by using the trigonometric functions: (3∠180° + 5∠45°) + (3∠90°) = (-3 + 5cos(45°) + j5sin(45°)) + (-3 + j3)

Simplifying further, we get: -3 + 5cos(45°) - 3 + j5sin(45°) + j3 = -6 + 5/√2 + j(5/√2 + 3) = -6 + 5/√2 + j8/√2.

Therefore, the result in time domain form is -6 + 5/√2 + j8/√2.