Lecture 12 Test 1

Problem 1

Given: V_a, V_b, V_x, R₁, R₂, R₃.

Find: V_c

Solution:

V_a - V_c = I₁ R₁

V_b - V_c = I₂ R₂

V_x - V_c = I₃ R₃

I₁ + I₂ + I₃ = 0

(V_a-V_c) / R₁ + (V_b-V_c) / R₂ + (V_x-V_c) / R₃ = 0

V_a / R₁ - V_c / R₁ + V_b / R₂ -V_c / R₂ + V_x / R₃ -V_c / R₃ = 0

V_a / R₁ + V_b / R₂ + V_x / R₃ = V_c / R₁ +V_c / R₂ +V_c / R₃

V_a / R₁ + V_b / R₂ + V_x / R₃ = V_c (1 / R₁ +1 / R₂ +1 / R₃)

V_c = (V_a / R₁ + V_b / R₂ + V_x / R₃) / (1 / R₁ +1 / R₂ +1 / R₃)

V_a = V
V_b = V
V_x = V
R₁ = Ω
R₂ = Ω
R₃ = Ω

V_c = V
V_c ≈ V

Problem 2

Given: R₁, R₂, ε₁, ε₂. 

 Find: I₁ 

 Solution: 

 I₁ R₁ - I₂ R₂ = ε₁ + ε₂ 

 I₁ + I₂ = 0 

 I₁ R₁ + I₁ R₂ = ε₁ + ε₂ 

 I₁ (R₁ + R₂) = ε₁ + ε₂ 

 I₁ = (ε₁ + ε₂) / (R₁ + R₂)

Problem 3

Solution Strategy:

Circuit Kirchhoff's Equations

 I₁ R₁ - I₂ R₂ = ε₁ - ε₂ 

 I₁ + I₂ = 0 

Solve the equations for the desired parameter