Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-In a parallel electrical circuit, the impedance $Z$ can be calculated using the equation
$$\frac { 1 } { Z } = \left( \frac { 1 } { Z _ { 1 } } + \frac { 1 } { Z _ { 2 } } \right)$$
where $Z _ { 1 }$ and $Z _ { 2 }$ are the impedances for the branches of the circuit. The phase angle measures the phase difference between the voltage and the current in an electrical circuit. If the impedance $\mathrm { Z }$ is expressed in the form $\mathrm { a } + \mathrm { bi }$, $\theta$ can be determined by the equation $\tan \theta = \mathrm { b } / \mathrm { a }$. Determine the phase angle $\theta$ (in degrees) for a parallel circuit in which $Z _ { 1 } = 10 + 20 \mathrm { i }$ and $Z _ { 2 } = 30 + 10 \mathrm { i }$.
A) $60 ^ { \circ }$
B) $30 ^ { \circ }$
C) $0 ^ { \circ }$
D) $45 ^ { \circ }$

Question

Quizplus · Accepted Answer

To find the total impedance $Z$, we first calculate the reciprocals of $Z_1$ and $Z_2$, then their sum, and finally take the reciprocal of that sum. For $Z_1 = 10 + 20i$, $\frac{1}{Z_1} = \frac{1}{10 + 20i}$. For $Z_2 = 30 + 10i$, $\frac{1}{Z_2} = \frac{1}{30 + 10i}$. The sum of these reciprocals gives the reciprocal of the total impedance $Z$. After calculating, we find $Z = a + bi$, where $a = b$. Thus, $\tan \theta = \frac{b}{a} = 1$, which means $\theta = 45^\circ$.

Perform the Indicated Operation ZZZ Can Be Calculated Using the Equation 1Z=(1Z1+1Z2)\frac { 1 } { Z } = \left( \frac { 1 } { Z _ { 1 } } + \frac { 1 } { Z _ { 2 } } \right)Z1​=(Z1​1​+Z2​1​)

Perform the Indicated Operation $Z$ Can Be Calculated Using the Equation $\frac { 1 } { Z } = \left( \frac { 1 } { Z _ { 1 } } + \frac { 1 } { Z _ { 2 } } \right)$