Question Three Two impedances, $$ Z_{1}=(3+j 6) \Omega $$ and $$ Z_{2}=(4-j 3) \Omega $$ are connected in series to a supply voltage of 120 V . Determine the magnitude of the current and phase angle relative the voltage.

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We are given a series circuit with impedances 
$$
Z_{1} = 3 + j6~\Omega \quad \text{and} \quad Z_{2} = 4 - j3~\Omega,
$$
and a supply voltage 
$$
V = 120~\text{V}.
$$

**Step 1. Calculate the Total Impedance**

Since the impedances are in series, the total impedance is the sum:
$$
Z_{\text{total}} = Z_{1} + Z_{2} = (3 + 4) + j(6 - 3) = 7 + j3~\Omega.
$$

**Step 2. Compute the Magnitude of the Total Impedance**

The magnitude of $$ Z_{\text{total}} $$ is given by:
$$
\left| Z_{\text{total}} \right| = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}~\Omega.
$$

**Step 3. Determine the Current**

Ohm's law for AC circuits relates the supply voltage to the current through:
$$
I = \frac{V}{Z_{\text{total}}}.
$$
The magnitude of the current is then:
$$
\left| I \right| = \frac{120}{\sqrt{58}}~\text{A}.
$$

**Step 4. Find the Phase Angle of the Total Impedance**

The phase angle of $$ Z_{\text{total}} $$ is:
$$
\theta_{Z} = \arctan\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) = \arctan\left(\frac{3}{7}\right).
$$

**Step 5. Determine the Phase Angle of the Current**

Because the current is given by $$ I = \frac{V}{Z_{\text{total}}} $$, the phase angle of the current is the negative of the phase angle of the impedance:
$$
\theta_{I} = -\arctan\left(\frac{3}{7}\right).
$$

**Final Answer**

- Magnitude of the current:
  $$
  \left| I \right| = \frac{120}{\sqrt{58}}~\text{A}
  $$
- Phase angle relative to the voltage:
  $$
  \theta_{I} = -\arctan\left(\frac{3}{7}\right) \quad \text{(in radians or convert to degrees as needed).}
  $$
The total impedance is $$ 7 + j3 \, \Omega $$. The magnitude of the current is $$ \frac{120}{\sqrt{58}} \, \text{A} $$ and the phase angle relative to the voltage is $$ -\arctan\left(\frac{3}{7}\right) $$.

Question Three Two impedances, \( Z_{1}=(3+j 6) \Omega \) and \( Z_{2}=(4-j 3) \Omega \) are connected in series to a supply voltage of 120 V . Determine the magnitude of the current and phase angle relative the voltage.

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