Have a question on complex numbers.
Z1= 1+j2, Z2= 2+j5
Find total impedance in Cartesian and polar form of two impedances in parallel:
1/ZT = 1/Z1 + 1/Z2
I get to 39/58 + 83/58 j but then get a bit lost.
so, what's wrong with that as an answer? That's what I get, too.
http://www.wolframalpha.com/input/?i=1%2F%281%2F%281%2B2i%29+%2B+1%2F%282%2B5i%29%29
To find the total impedance in Cartesian and polar form for two impedances in parallel, follow these steps:
Step 1: Convert the given complex numbers to their polar form.
Z1 = 1 + j2 = √(1^2 + 2^2) * [cos(arctan(2/1)) + j*sin(arctan(2/1))] = √5 * [cos(63.43°) + j*sin(63.43°)]
Z2 = 2 + j5 = √(2^2 + 5^2) * [cos(arctan(5/2)) + j*sin(arctan(5/2))] = √29 * [cos(67.38°) + j*sin(67.38°)]
Step 2: Find the reciprocal of each impedance.
1/Z1 = 1 / (√5 * [cos(63.43°) + j*sin(63.43°)]) = 1 / √5 * [cos(-63.43°) + j*sin(-63.43°)]
1/Z2 = 1 / (√29 * [cos(67.38°) + j*sin(67.38°)]) = 1 / √29 * [cos(-67.38°) + j*sin(-67.38°)]
Step 3: Add the reciprocals of the impedances.
1/ZT = 1/Z1 + 1/Z2 = [1 / √5 * (cos(-63.43°) + j*sin(-63.43°))] + [1 / √29 * (cos(-67.38°) + j*sin(-67.38°))]
Step 4: Simplify the expression.
1/ZT = [√29 * (cos(-67.38°) + j*sin(-67.38°)) + √5 * (cos(-63.43°) + j*sin(-63.43°))]/(√5 * √29)
Step 5: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator (√5 * √29).
1/ZT = [√29 * (cos(-67.38°) + j*sin(-67.38°)) + √5 * (cos(-63.43°) + j*sin(-63.43°))]/(5 * √29)
Step 6: Combine the real and imaginary parts separately.
1/ZT = [√29 * cos(-67.38°) + √5 * cos(-63.43°)]/(5 * √29) + j[√29 * sin(-67.38°) + √5 * sin(-63.43°)]
Step 7: Convert the expression back to Cartesian form.
1/ZT ≈ [7.24 + 1.15j] + j[-1.38 - 1.04j]
Step 8: Simplify the expression.
1/ZT ≈ 7.24 - 1.38j + j*(-1.04 - 1.38) ≈ 7.24 - 2.42j
Therefore, the total impedance in Cartesian form is approximately 7.24 - 2.42j.
Step 9: Convert the Cartesian form back to polar form if desired.
To find the polar form of 7.24 - 2.42j, use the magnitude and angle formulas:
Magnitude (|ZT|) = √(7.24^2 + (-2.42)^2) ≈ √61.25 ≈ 7.82
Angle (θ) = tan^(-1)(-2.42/7.24) ≈ -18.70°
Therefore, the total impedance in polar form is approximately 7.82 ∠ -18.70°.