Question

If admittance of a circuit is \( Y = 0.3 - j0.4\text{ s} \), then the magnitude of the impedance is:

• A. \( 1\ \Omega \)
• B. \( 2\ \Omega \)
• C. \( 5\ \Omega \)
• D. \( 10\ \Omega \)

Hint:

Recognize the common 3-4-5 Pythagorean triple scaled down by 10! The components \(0.3\) and \(0.4\) automatically form a vector magnitude of \(0.5\). Its reciprocal is simply \(\frac{1}{0.5} = 2\ \Omega\).

Answer 1

The Correct Option is B

Concept: Admittance (\(Y\)) represents the ease with which an alternating current flows through a circuit network, measured in Siemens (S) or mhos (\(\mho\)). Impedance (\(Z\)) represents the comprehensive opposition to that current flow, measured in Ohms (\(\Omega\)). These two complex parameters share a reciprocal mathematical relationship: \[ Z = \frac{1}{Y} \] When these values are expressed in complex rectangular forms:
• Admittance: \(Y = G + jB\) (where \(G\) is conductance and \(B\) is susceptance)
• Impedance: \(Z = R + jX\) (where \(R\) is resistance and \(X\) is reactance) The magnitudes of these complex numbers are related inversely: \[ |Z| = \frac{1}{|Y|} \]

Step 1: Calculating the magnitude of the admittance \(|Y|\).
We are given the complex admittance as \(Y = 0.3 - j0.4\text{ S}\). The modulus magnitude of any complex number \(A + jB\) is calculated using the Pythagorean theorem: \[ |Y| = \sqrt{G^2 + B^2} = \sqrt{(0.3)^2 + (-0.4)^2} \] Evaluating the squares of the decimal components: \[ (0.3)^2 = 0.09 \] \[ (-0.4)^2 = 0.16 \] Summing these squares inside the radical: \[ |Y| = \sqrt{0.09 + 0.16} = \sqrt{0.25} = 0.5\text{ S} \]

Step 2: Finding the magnitude of impedance \(|Z|\).
Using the inverse relationship between the magnitudes: \[ |Z| = \frac{1}{|Y|} = \frac{1}{0.5} = \frac{1}{\frac{1}{2}} = 2\ \Omega \] The magnitude of the circuit impedance is exactly \(2\ \Omega\), which corresponds to Option (B).