2.  Solving a circuit consists of finding unknown currents, the current direction, and the voltages in a circuit.  A multiloop circuit has more than one loop, and sometimes more than one battery. Topic 5.2 Extended B – Multi-loop circuits  A junction (or node) is a point in a circuit where three or more connecting wires are joined together.  A branch is a path connecting two junctions. 25  100  A B + - + - 50  15 V 10 V FYI: Each branch has its own current. I1I1 I2I2 I3I3 Snoop Dog Kirchhoff

3. Topic 5.2 Extended B – Multi-loop circuits  Start by assigning arbitrary directions to your branch currents: 25  100  A B + - + - 50  15 V 10 V FYI: This means that the sum of the currents going into a junction equals the sum of the currents leaving it. I1I1 I2I2 I3I3 I1I1 I1I1 I1I1 I1I1 I1I1 I2I2 I3I3 I3I3 I3I3 I2I2  Kirchhoff's first rule (the junction theorem) states that the algebraic sum of the currents at a junction is zero. FYI: The junction theorem is just a statement of the conservation of charge.  I = 0 Kirchhoff's Junction Theorem FYI: The convention is to assign a (+) to currents entering a junction, and a (-) to currents leaving the junction:  I = 0 I 3 - I 1 - I 2 = 0 (junction A) I 1 + I 2 - I 3 = 0 (junction B) FYI: At this point we have but 1 equation with three unknowns.

4. Topic 5.2 Extended B – Multi-loop circuits  Now we look at the voltages in any loop and invoke the second of Kirchhoff's rules: 25  100  A B + - + - 50  15 V 10 V I1I1 I1I1 I1I1 I1I1 I1I1 I2I2 I3I3 I3I3 I3I3 I2I2  Kirchhoff's second rule (the loop theorem) states that the algebraic sum of the voltage differences in any closed loop is zero.  V = 0 Kirchhoff's Loop Theorem FYI: The loop theorem is just a statement of the conservation of energy. FYI: A few conventions are used to determine whether the voltage difference is positive or negative. Here they are:  Traveling through a resistor in the direction of the current is a voltage DROP (-).  Traveling through a resistor opposite to the direction of the current is a voltage GAIN (+).  Traveling through a battery from (+) to (-) is a voltage DROP (-).  Traveling through a battery from (-) to (+) is a voltage GAIN (+).

5.  For loop 1: Topic 5.2 Extended B – Multi-loop circuits 25  100  A B + - + - 50  15 V 10 V I1I1 I1I1 I1I1 I1I1 I1I1 I2I2 I3I3 I3I3 I3I3 I2I2 15 - 50I 1 + 10 + 100I 2 - 50I 1 = 0 1 2 25 - 100I 1 + 100I 2 = 0 1 = 4I 1 - 4I 2  For loop 2: -10 - 25I 3 - 100I 2 = 0 -2 = 5I 3 + 20I 2 FYI: Counting the junction theorem equation, we now have three equations with three unknowns. I 3 - I 1 - I 2 = 0 (junction A)

6. Topic 5.2 Extended B – Multi-loop circuits 1 = 4I 1 - 4I 2 -2 = 5I 3 + 20I 2 I 3 - I 1 - I 2 = 0 I 3 = I 1 + I 2 -2 = 5(I 1 + I 2 ) + 20I 2 -2 = 5I 1 + 25I 2 5 = 20I 1 - 20I 2 8 = -20I 1 + -100I 2 13 = -120I 2 I 2 = -13/120 A FYI: The (-) in the I 2 means that we picked the wrong direction. 5 = 20I 1 - 20(-13/120 ) 5 = 20I 1 + 13/6 30 = 120I 1 + 13 17 = 120I 1 I 1 = 17/120 A FYI: The (+) in the I 1 means that we picked the right direction. I 3 = 17/120 + -13/120 I 3 = 4/120 I 3 = 4/120 A FYI: If you need to find any of the resistor voltages just use Ohm’s law: V = IR.