Determine the current in each branch of the circuit shown in...

Key Concepts

Ohm's Law

Ohm's Law is a fundamental relationship in electrical circuits that relates the current flowing through a resistor to the voltage across it and its resistance by the formula I = V/R. This law is critical to calculating the current through individual circuit elements once the voltage across them is known, thereby linking voltage, current, and resistance.

Circuit Analysis Techniques

Circuit analysis techniques, such as mesh (loop) analysis and node voltage analysis, provide systematic approaches to solving circuits with multiple components and branches. These methods combine Kirchhoff’s Laws and Ohm's Law to develop sets of simultaneous equations that, when solved, yield the currents in each branch. They are essential tools for decomposing complex circuits into manageable parts for analysis.

Kirchhoff's Circuit Laws

Kirchhoff's Circuit Laws, including Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL), are fundamental tools used in circuit analysis. KCL states that the sum of the currents entering a node must equal the sum of the currents leaving that node, while KVL dictates that the algebraic sum of all voltages around any closed loop in a circuit must equal zero. These principles are essential for setting up the equations needed to determine unknown currents in different branches of a circuit.

Transcript

00:01 In this question we're going to apply kirchhoff's rules for circuit analysis to determine the current in each branch of the circuit shown.

00:10 So our first thing to do is set up our two loops.

00:13 I'm going to start at the 12 volt battery and go counterclockwise around the outside loop.

00:20 I'm going to call this loop one with current one directed upward in that branch.

00:27 And then my second loop is going to start at the 4 volt battery and go counterclockwise around the left half of this circuit.

00:40 And we're going to call i2 directed upward in that branch.

00:44 And then of course at the junction at the top, above the 5 ohm resistor pictured and to the left of the 3 ohm resistor pictured is where we're going to have current i3 emerging when we had i2 entering from the bottom and i1 entering from the right.

01:11 So that gives our three rules to loop one junction.

01:16 So now we can go ahead and set up our first loop.

01:28 So remember kirchhoff's rules are basically, well the loop rules are conservation of energy.

01:32 So we're going to have a 12 volt potential rise and then we'll have i1 times 1 ohm as a potential drop.

01:40 And then we'll have i1 times 3 ohms as a potential drop.

01:45 And then we will have i3 times 8 ohms as a potential drop.

01:50 And then we're back to the battery so that's a total of zero.

01:52 So simplifying this we get 12 minus 4 i1 minus 8 i3 equals zero.

02:02 So rearranging this one so that the currents are on one side and the constant term for the battery is on the other.

02:13 4 i1 plus 8 i3 equals 12.

02:16 Now we can do the same thing for our second loop where we started with the 4 volt battery.

02:22 And then we will have i2 times 1 ohm as a potential drop.