Detailed Circuit Analysis and Calculations

Circuit Analysis Problem

Problem Statement

Calculate the following values for the given circuit:

• Total resistance in the circuit
• Current in resistor 1
• Power dissipated by resistor 2
• Voltage across resistor 3

Circuit Parameters

• $V = 25.0 \text{ V}$
• $R_1 = 10 \Omega$
• $R_2 = 120 \Omega$
• $R_3 = 40 \Omega$
• $R_4 = 20 \Omega$

Solution

1. Calculate the equivalent resistance of the series combination of $R<em>3$ and $R</em>4$ ($R_{ST1}$):

• $R<em>{ST1} = R</em>3 + R_4$
• $R_{ST1} = 40 \Omega + 20 \Omega$
• $R_{ST1} = 60 \Omega$

2. Calculate the equivalent resistance of the parallel combination of $R<em>2$ and $R</em>{ST1}$ ($R_{ST2}$):

• $\frac{1}{R<em>{ST2}} = \frac{1}{R</em>2} + \frac{1}{R_{ST1}}$
• $\frac{1}{R_{ST2}} = \frac{1}{120 \Omega} + \frac{1}{60 \Omega}$
• $\frac{1}{R_{ST2}} = \frac{1}{120 \Omega} + \frac{2}{120 \Omega}$
• $\frac{1}{R_{ST2}} = \frac{3}{120 \Omega}$
• $R_{ST2} = \frac{120 \Omega}{3}$
• $R_{ST2} = 40 \Omega$

3. Calculate the total resistance ($R_{TOT}$):

• $R<em>{TOT} = R</em>1 + R_{ST2}$
• $R_{TOT} = 10 \Omega + 40 \Omega$
• $R_{TOT} = 50 \Omega$

4. Calculate the total current ($I_{TOT}$):

• Using Ohm's Law: $V<em>{TOT} = I</em>{TOT} \cdot R_{TOT}$
• $25 \text{ V} = I_{TOT} \cdot (50 \Omega)$
• $I_{TOT} = \frac{25 \text{ V}}{50 \Omega}$
• $I_{TOT} = 0.5 \text{ A}$

5. Calculate the voltage across resistor 1 ($V_1$):

• $V<em>1 = I</em>1 \cdot R_1$
• Since $R<em>1$ is in series with the rest of the circuit, $I</em>1 = I_{TOT} = 0.5 \text{ A}$
• $V_1 = (0.5 \text{ A}) \cdot (10 \Omega)$
• $V_1 = 5 \text{ V}$

6. Calculate the voltage across the parallel combination of $R<em>2$ and $R</em>{ST1}$ ($V_2$):

• Using Kirchhoff's Voltage Law (KVL): $V<em>{gain} = V</em>{drops}$
• $25 \text{ V} = V<em>1 + V</em>2$
• $25 \text{ V} = 5 \text{ V} + V_2$
• $V_2 = 20 \text{ V}$

7. Calculate the current through resistor 2 ($I_2$):

• $V<em>2 = I</em>2 \cdot R_2$
• $20 \text{ V} = I_2 \cdot (120 \Omega)$
• $I_2 = \frac{20 \text{ V}}{120 \Omega}$
• $I_2 = 0.167 \text{ A}$

8. Calculate the power dissipated by resistor 2 ($P_2$):

• $P = I \cdot V$
• $P<em>2 = I</em>2 \cdot V_2$
• $P_2 = (0.167 \text{ A}) \cdot (20 \text{ V})$
• $P_2 = 3.3 \text{ W}$

9. Calculate the current through the series combination of $R<em>3$ and $R</em>4$ ($I_{3-4}$):

• Using Kirchhoff's Current Law (KCL): $I<em>{in} = I</em>{out}$
• $I<em>{TOT} = I</em>2 + I_{3-4}$
• $0.5 \text{ A} = 0.167 \text{ A} + I_{3-4}$
• $I_{3-4} = 0.5 \text{ A} - 0.167 \text{ A}$
• $I_{3-4} = 0.33 \text{ A}$

10. Calculate the voltage across resistor 3 ($V_3$):

• $V<em>3 = I</em>{3-4} \cdot R_3$
• $V_3 = (0.33 \text{ A}) \cdot (40 \Omega)$
• $V_3 = 13.3 \text{ V}$