Given the equivalent impedance of a circuit can be calculated by the expression \[ Z=\frac{Z_{1} Z_{2}}{Z_{1}+Z_{2}} \] If \( Z_{1}=-2-j \) and \( Z_{2}=-11-j \), calculate the impedance \( Z \) in both rectangular and polar forms. Rectangular form Polar form

We wish to compute   Z = (Z₁·Z₂)/(Z₁ + Z₂) with   Z₁ = –2 – j  and  Z₂ = –11 – j. Step 1. Compute the numerator: Z₁·Z₂   Z₁·Z₂ = (–2 – j)(–11 – j). Multiply out:   (–2)(–11) = 22,   (–2)(–j) = 2j,   (–j)(–11) = 11j,   (–j)(–j) = j² = –1. So,   Z₁·Z₂ = 22 + 2j + 11j – 1 = 21 + 13j. Step 2. Compute the denominator: Z₁ + Z₂   Z₁ + Z₂ = (–2 – j) + (–11 – j) = –13 – 2j. Thus,   Z = (21 + 13j)/(–13 – 2j). Step 3. Express Z in rectangular form. A common method is to multiply the numerator and denominator by the complex conjugate of the denominator. The conjugate of (–13 – 2j) is (–13 + 2j). So,   Z = [(21 + 13j)(–13 + 2j)] / [ (–13 – 2j)(–13 + 2j) ]. First, compute the denominator:   (–13 – 2j)(–13 + 2j) = (–13)² – (2j)² = 169 – 4(j²) = 169 – 4(–1) = 169 + 4 = 173. Now, expand the numerator:   (21 + 13j)(–13 + 2j)    = 21·(–13) + 21·(2j) + 13j·(–13) + 13j·(2j)    = –273 + 42j – 169j + 26j²    = –273 – 127j + 26(–1)    = –273 – 127j – 26    = –299 – 127j. Thus,   Z = (–299 – 127j) / 173, or, written in rectangular form,   Z = –299/173 – (127/173)j. If we approximate these values:   299/173 ≈ 1.728 and 127/173 ≈ 0.734, so   Z ≈ –1.73 – 0.73j  (in ohms). Step 4. Express Z in polar form. We need the magnitude (modulus) and phase angle. a) Magnitude:   |Z| = √[(–299/173)² + (–127/173)²]    = (1/173)·√(299² + 127²). Calculate:   299² = 89401  and  127² = 16129, so   √(299² + 127²) = √(89401 + 16129) = √105530 ≈ 324.93. Thus,   |Z| ≈ 324.93/173 ≈ 1.877 ohms. b) Angle: Since both the real and imaginary parts are negative, the number lies in the third quadrant. To begin, compute the acute angle:   θ₀ = arctan(|imaginary part|/|real part|) = arctan( (127/173)/(299/173) ) = arctan(127/299). Now, 127/299 ≈ 0.4247. Thus,   θ₀ ≈ arctan(0.4247) ≈ 22.97°. Being in the third quadrant, the actual angle is   θ = 180° + 22.97° ≈ 202.97°, or equivalently, one may express the phase as –157.03° (since 202.97° – 360° = –157.03°). Thus, the polar form is:   Z ≈ 1.88 ∠ –157.0°  (or equivalently 1.88 ∠ 203°). Summary of Answers: Rectangular form:   Z = –299/173 – (127/173)j ≈ –1.73 – 0.73j (ohms). Polar form:   Z ≈ 1.88 ∠ –157°  (ohms). Either expression for the angle (–157° or 203°) is acceptable as long as the quadrant is clear.