In the figure below, R = 27 ?? and \n
= -j39 ??. Find the magnitude of \nZ.
The Correct Answer and Explanation is:
To find the magnitude of the impedance ZZZ, we use the formula for the magnitude of a complex number:∣Z∣=R2+XC2|Z| = \sqrt{R^2 + X_C^2}∣Z∣=R2+XC2
Where:
- R=27 kΩR = 27\,\text{k}\OmegaR=27kΩ
- XC=−j39 kΩ⇒XC=−39 kΩX_C = -j39\,\text{k}\Omega \Rightarrow X_C = -39\,\text{k}\OmegaXC=−j39kΩ⇒XC=−39kΩ (since we take the imaginary part’s magnitude)
Now plug in the values:∣Z∣=(27)2+(−39)2|Z| = \sqrt{(27)^2 + (-39)^2}∣Z∣=(27)2+(−39)2∣Z∣=729+1521|Z| = \sqrt{729 + 1521}∣Z∣=729+1521∣Z∣=2250|Z| = \sqrt{2250}∣Z∣=2250∣Z∣≈47.43 kΩ|Z| \approx 47.43\,\text{k}\Omega∣Z∣≈47.43kΩ
So, the correct answer is:
47.4k ohms
Explanation:
In AC circuit analysis, impedance ZZZ is a complex quantity combining resistance RRR and reactance XXX. In this case, the reactance is purely capacitive, and its value is negative imaginary: XC=−j39 kΩX_C = -j39\,\text{k}\OmegaXC=−j39kΩ.
The impedance ZZZ is the vector sum of RRR (real part) and XCX_CXC (imaginary part). The phasor diagram forms a right triangle, with RRR as the base and XCX_CXC as the vertical side pointing downward (negative imaginary direction).
To find the magnitude of this vector ZZZ, we apply the Pythagorean theorem. This gives the length of the hypotenuse of the triangle, which corresponds to the magnitude of the total impedance.
The values are converted into squared components: R2=729R^2 = 729R2=729 and XC2=1521X_C^2 = 1521XC2=1521, summed to give 2250. Taking the square root gives approximately 47.43 kilo-ohms.
This calculation confirms the correct choice as 47.4k ohms.
