Math Problem Statement

find Va, Vc, Vb

Find Va, Vc, Vb

Solution

To solve for vav_a, vbv_b, and vcv_c in this circuit, we'll need to analyze the voltage drops and apply Kirchhoff's Voltage Law (KVL) around the loops.

Step 1: Label the Points

Let’s label the nodes where the components connect:

  • Node at vav_a: Voltage across the component labeled vav_a.
  • Node at vbv_b: Voltage across the component labeled vbv_b.
  • Node at vcv_c: Voltage across the component labeled vcv_c.

Step 2: Analyze the Voltages

1. Loop Analysis (left loop including vav_a and the 5V source):

In the left loop, the 5V source is directly connected to vav_a. So: va=5 Vv_a = 5\text{ V}

2. Loop Analysis (middle loop including vav_a, the 15V source, and the 10V source):

Going clockwise: va+15 V+vb=10 Vv_a + 15\text{ V} + v_b = 10\text{ V}

Since va=5 Vv_a = 5\text{ V}: 5 V+15 V+vb=10 V5\text{ V} + 15\text{ V} + v_b = 10\text{ V} vb=10 V20 Vv_b = 10\text{ V} - 20\text{ V} vb=10 Vv_b = -10\text{ V}

3. Outer Loop Analysis (including vcv_c):

Considering the loop from the 5V source, through vcv_c, and back: va+vc=0v_a + v_c = 0

Given va=5 Vv_a = 5\text{ V}: 5 V+vc=05\text{ V} + v_c = 0 vc=5 Vv_c = -5\text{ V}

Final Answers:

  • va=5 Vv_a = 5 \text{ V}
  • vb=10 Vv_b = -10 \text{ V}
  • vc=5 Vv_c = -5 \text{ V}

Would you like any further details or explanations?

Here are some related questions you might find helpful:

  1. What are the basic principles of Kirchhoff's Voltage Law (KVL)?
  2. How do you apply KVL in more complex circuits?
  3. What is the significance of negative voltages in a circuit analysis?
  4. How can superposition theorem help in analyzing this circuit?
  5. What are the differences between series and parallel circuits?

Tip: When analyzing circuits, always be mindful of the sign conventions (positive/negative) for voltage drops and sources. This is crucial for correctly applying KVL and obtaining the right answers.

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Math Problem Analysis

Mathematical Concepts

Circuit Analysis
Kirchhoff's Voltage Law (KVL)
Voltage Drops

Formulas

KVL: ΣV = 0 for any closed loop
Ohm's Law: V = IR

Theorems

Kirchhoff's Voltage Law (KVL)

Suitable Grade Level

Undergraduate Electrical Engineering