Section 9.6: Kirchhoff's Laws
Kirchhoff's laws are used to analyze complex circuits: Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL).
\[ \sum I_\text{in} = \sum I_\text{out} \]
The total current entering a junction equals the total current leaving.
\[ \sum V_\text{around~loop} = 0 \]
The sum of all voltage drops and gains around any closed loop is zero.
Using KCL and KVL, you can set up simultaneous equations to solve for unknown currents or voltages in complex circuits.
Example 1: Simple Two-Loop Circuit
Consider a circuit with two loops and three resistors: \(R_1 = 2 \Omega\), \(R_2 = 3 \Omega\), \(R_3 = 4 \Omega\), and a battery of 12 V connected as shown. Find the currents \(I_1\) and \(I_2\) using Kirchhoff’s laws.
Assign loop currents \(I_1\) and \(I_2\). Using KVL for Loop 1:
\( 12 - 2 I_1 - 3(I_1 - I_2) = 0 \Rightarrow 5 I_1 - 3 I_2 = 12 \)
Loop 2: \( 3(I_2 - I_1) + 4 I_2 = 0 \Rightarrow -3 I_1 + 7 I_2 = 0 \)
Solve: \(I_1 = 4.2\,\text{A},~ I_2 = 1.8\,\text{A}\)
Practice Problems
- For a single junction with currents 2 A, 3 A entering, find the unknown current leaving.
- Two loops with resistors 5 Ω and 10 Ω in series and a 15 V battery. Apply KVL to find currents.
- A three-loop circuit has resistors 2 Ω, 3 Ω, 4 Ω and batteries 10 V and 5 V. Solve for all loop currents.
- Verify KCL at a junction where 1.5 A and 2.5 A enter and two unknown currents leave.
- Analyze a mixed series-parallel circuit with two loops using Kirchhoff's laws to find all branch currents.