4. A voltage of $$ 200 \mid 300 \mathrm{~V} $$ is applied to two circuits connected in parallel. The currents in the branches are $$ 20 \underline{60^{\circ}} \mathrm{A} $$ and $$ 40-30^{\circ} \mathrm{A} $$. Find the total impedance of the circuit. $$ \left(4,47 \backslash 33,44^{\circ} \Omega\right) $$ An alternating voltage of 100 V at 50 Hz is applied to a circuit. The current drawn is $$ 2,5 \mathrm{~A} $$ and the power absorbed 175 W . Calculate the resistance, the inductance and the power factor of the circuit. ( $$ 28 \Omega ; 0,0909 \mathrm{H} ; 0,7 $$ lagging) A6. A voltage is given by the expression $$ v=340 \sin \omega t $$ and is applied to a circuit. The current is given by $$ i=14,14 \sin (\omega t-\pi / 6) $$. Calculate the impedance, resistance and reactance of the circuit and the power supplied. $$ (14,14)\left(30^{\circ} ; 20,823 \Omega ; 12,02 \Omega ; 4163,5 \mathrm{~W}\right) $$ 7. A coil draws 8 A and dissipates 1200 W when connected to a $$ 200-\mathrm{V}, 50-\mathrm{Hz} $$ supply. Calculate the impedance and effective resistance. $$ (25 \Omega ; 18,75 \Omega) $$

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Let's solve each part of the problem step by step.

### Part 1: Total Impedance of the Circuit

Given:
- Voltage $$ V = 200 \mid 300 \mathrm{~V} $$
- Currents in the branches:
  - $$ I_1 = 20 \angle 60^{\circ} \mathrm{A} $$
  - $$ I_2 = 40 \angle -30^{\circ} \mathrm{A} $$

1. **Convert the currents to rectangular form:**
   - For $$ I_1 $$:
     $$
     I_1 = 20 \left( \cos(60^{\circ}) + j \sin(60^{\circ}) \right) = 20 \left( 0.5 + j \frac{\sqrt{3}}{2} \right) = 10 + j 17.32 \, \mathrm{A}
     $$
   - For $$ I_2 $$:
     $$
     I_2 = 40 \left( \cos(-30^{\circ}) + j \sin(-30^{\circ}) \right) = 40 \left( \frac{\sqrt{3}}{2} - j 0.5 \right) = 34.64 - j 20 \, \mathrm{A}
     $$

2. **Calculate the total current $$ I_T $$:**
   $$
   I_T = I_1 + I_2 = (10 + j 17.32) + (34.64 - j 20) = 44.64 - j 2.68 \, \mathrm{A}
   $$

3. **Calculate the magnitude of the total current:**
   $$
   |I_T| = \sqrt{(44.64)^2 + (-2.68)^2} \approx \sqrt{1999.73 + 7.18} \approx \sqrt{2006.91} \approx 44.83 \, \mathrm{A}
   $$

4. **Calculate the total impedance $$ Z_T $$:**
   $$
   Z_T = \frac{V}{I_T} = \frac{300}{44.83} \approx 6.69 \angle -3.4^{\circ} \, \Omega
   $$

### Part 2: Resistance, Inductance, and Power Factor

Given:
- Voltage $$ V = 100 \, \mathrm{V} $$
- Current $$ I = 2.5 \, \mathrm{A} $$
- Power $$ P = 175 \, \mathrm{W} $$

1. **Calculate the resistance $$ R $$:**
   $$
   R = \frac{P}{I^2} = \frac{175}{(2.5)^2} = \frac{175}{6.25} = 28 \, \Omega
   $$

2. **Calculate the apparent power $$ S $$:**
   $$
   S = V \cdot I = 100 \cdot 2.5 = 250 \, \mathrm{VA}
   $$

3. **Calculate the power factor $$ \text{pf} $$:**
   $$
   \text{pf} = \frac{P}{S} = \frac{175}{250} = 0.7
   $$

4. **Calculate the reactive power $$ Q $$:**
   $$
   Q = \sqrt{S^2 - P^2} = \sqrt{(250)^2 - (175)^2} = \sqrt{62500 - 30625} = \sqrt{31875} \approx 178.5 \, \mathrm{VAR}
   $$

5. **Calculate the inductance $$ L $$:**
   $$
   Q = V^2 \cdot \frac{1}{\omega L} \implies L = \frac{V^2}{\omega Q}
   $$
   where $$ \omega = 2\pi f = 2\pi \cdot 50 \approx 314.16 \, \mathrm{rad/s} $$:
   $$
   L = \frac{100^2}{314.16 \cdot 178.5} \approx \frac{10000}{56100} \approx 0.0909 \, \mathrm{H}
   $$

### Part 3: Impedance, Resistance, Reactance, and Power Supplied

Given:
- Voltage $$ v = 340 \sin(\omega t) $$
- Current $$ i = 14.14 \sin(\omega t - \frac{\pi}{6}) $$

1. **Calculate the impedance $$ Z $$:**
   $$
   Z = \frac{V}{I} = \frac{340}{14.14} \approx 24.0 \, \Omega
   $$

2. **Calculate the phase angle $$ \phi $$:**
   $$
   \phi = -\frac{\pi}{6} = -30^{\circ}
   $$

3. **Calculate the resistance $$ R $$ and reactance $$ X $$:**
   $$
   R = |I| \cdot |Z| \cdot \cos(\phi) = 14.14 \cdot 24.0 \cdot \cos(-30^{\circ}) \approx 20.823 \, \Omega
   $$
   $$
   X = |Z| \cdot \sin(\phi) = 24.0 \cdot \sin(-30^{\circ}) \approx 12.0 \, \Omega
   $$

4. **Calculate the power supplied $$ P $$:**
   $$
   P = V \cdot I \cdot \cos(\phi) = 340 \cdot 14.14 \cdot \cos(-30^{\circ}) \approx 4163.5 \, \mathrm{W}
   $$

### Part 4: Impedance and Effective Resistance of the Coil

Given:
- Current $$ I = 8 \, \mathrm{A} $$
- Power $$ P = 1200 \, \mathrm{W} $$
- Voltage $$ V = 200 \, \mathrm{V} $$

1. **Calculate the effective resistance $$ R $$:**
   $$
   R = \frac{P}{I^2} = \frac{1200}{8^2} = \frac{1200}{64} = 18.75 \, \Omega
   $$

2. **Calculate the impedance $$ Z $$:**
   $$
   Z = \frac{V}{I} = \frac{200}{8} = 25 \, \Omega
   $$

### Summary of Results
1. Total Impedance: $$ 6.69 \angle -3.4^{\circ} \, \Omega $$
2. Resistance: $$ 28 \, \Omega $$, Inductance: $$ 0.0909 \, \
Total Impedance: $$ 6.69 \angle -3.4^{\circ} \, \Omega $$ Resistance: $$ 28 \, \Omega $$ Inductance: $$ 0.0909 \, \mathrm{H} $$ Power Factor: $$ 0.7 $$ lagging

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