Mesh Analysis of Resistor Power

Mesh Analysis for Power Dissipation

• Mesh analysis is utilized to calculate the currents in a circuit with resistors.
• Variables defined:
  • ( I_3 = -I_1 ): indicating the relationship between currents in the circuit.
  • ( I_3 = 0.5I_0 - 0.5I_1 ): another expression relating current ( I_0 ).

Setting Up Equations

• The first mesh equation:
  • ( I_1 + 4(I_1 - I_3) + 56 = 0 ).
• Simplifying the equation:
  • ( I_1 + 4I_1 - 4I_3 + 56 = 0 ).
  • Upon substituting ( I_3 = -0.5I_1 ):
    • ( I_1 + 4I_1 + 2I_1 + 56 = 0 ) leading to ( I_1 = -8A ).

Calculating Other Currents

• From ( I_3 = -0.5I_1 ):
  • ( I_3 = 4A ).
• Using the second mesh equation:
  • ( -56 + 8(I_2 - I_3) + 2I_2 = 0 ) leads to:
  • ( -56 + 8(I_2 - 4) + 2I_2 = 0 ).
• Solving gives ( I_2 = 8.8A ).

Power Dissipation Calculations

• Resistor at 1Ω:

  • Power ( P_{1\Omega} = I_1^2 = 64W ).

• Resistor at 4Ω:

  • Power ( P_{4\Omega} = 4(I_1 - I_3)^2 = 576W ).

• Resistor at 8Ω:

  • Power ( P_{8\Omega} = 8(I_2 - I_3)^2 = 184.32W ).

• Resistor at 2Ω:

  • Power ( P_{2\Omega} = 2I_2^2 = 154.88W ).

Summary of Results

• Calculated currents: ( I_1 = -8A ), ( I_2 = 8.8A ), ( I_3 = 4A ).
• Power dissipated in various resistors spans from ( 64W ) to ( 576W ).