Application to Electrical Networks

Chapter 1 • Section 1-5

"Watt is love? Baby don't hertz me... no more! ⚡"

🔌 Important Symbols

Measured in Ohms ($\Omega$)

Measured in Volts ($V$)

Measured in Amps ($A$)

Which law relates Voltage ($V$), Current ($I$), and Resistance ($R$)?

Write an equation for each loop and solve for the currents $I_1, I_2, I_3$ in the following circuit.

Bottom Loop ($I_1$): $5I_1 + 3I_1 + 1(I_1 - I_2) + 2(I_1 - I_3) = -24$

Wait, let's check the slide... Ah, the slide says: $5I_1 + 3I_1 + I_1 - I_2 = -24$. (Simplification of shared branches)

Top Left Loop ($I_2$): $I_2 - I_1 + 4I_2 - 4I_3 + 2I_2 = 17$

Top Right Loop ($I_3$): $4I_3 - 4I_2 + 2I_3 + 2I_3 = -14$

Simplifying the equations gives us:

$$ \left[\begin{array}{rrr|r} 9 & -1 & 0 & -24 \\ -1 & 7 & -4 & 17 \\ 0 & -4 & 8 & -14 \end{array}\right] $$

Applying row operations...

$$ \left[\begin{array}{rrr|r} 1 & 0 & 0 & -5/2 \\ 0 & 1 & 0 & 3/2 \\ 0 & 0 & 1 & -1 \end{array}\right] $$

$$ I_1 = -2.5 A, \quad I_2 = 1.5 A, \quad I_3 = -1 A $$

Negative current means it flows in the opposite direction of the arrow! 🔄

If $I_3 = -1 A$, what is the actual direction of current flow compared to the loop arrow?