PHY 202 (Blum)1 Kirchhoff’s Rules When series and parallel combinations aren’t enough

2PHY 202 (Blum) Some circuits have resistors which are neither in series nor parallel They can still be analyzed, but one uses Kirchhoff’s rules.

3PHY 202 (Blum) Not in series The 1-k  resistor is not in series with the 2.2-k  since the some of the current that went through the 1-k  might go through the 3-k  instead of the 2.2-k  resistor.

4PHY 202 (Blum) Not in parallel The 1-k  resistor is not in parallel with the 1.5-k  since their bottoms are not connected simply by wire, instead that 3-k  lies in between.

5PHY 202 (Blum) Kirchhoff’s Node Rule A node is a point at which wires meet. “What goes in, must come out.” Recall currents have directions, some currents will point into the node, some away from it. The sum of the current(s) coming into a node must equal the sum of the current(s) leaving that node. I 1 + I 2 = I 3 I 1   I 2  I 3 The node rule is about currents!

6PHY 202 (Blum) Kirchhoff’s Loop Rule 1 “If you go around in a circle, you get back to where you started.” If you trace through a circuit keeping track of the voltage level, it must return to its original value when you complete the circuit Sum of voltage gains = Sum of voltage losses

7PHY 202 (Blum) Batteries (Gain or Loss) Whether a battery is a gain or a loss depends on the direction in which you are tracing through the circuit Gain Loss Loop direction

8PHY 202 (Blum) Resistors (Gain or Loss) Whether a resistor is a gain or a loss depends on whether the trace direction and the current direction coincide or not. II LossGain Loop direction Current direction

9PHY 202 (Blum) Neither Series Nor Parallel Draw loops such that each current element is included in at least one loop. Assign current variables to each loop. Current direction and lop direction are the same. JAJA JBJB JCJC

Currents in Resistors Note that there are two currents associated with the 2.2-kΩ resistor. Both J A and J C go through it. Moreover, they go through it in opposite directions. When in Loop A, the voltage drop across the 2.2-kΩ resistor is 2.2(J A -J C ) On the other hand, when in Loop C, the voltage drop across the 2.2-kΩ resistor is 2.2(J C -J A ) the opposite sign because we are going through the resistor in the opposite direction. PHY 202 (Blum)10

11PHY 202 (Blum) Loop equations 5 = 1  (J A - J B )  (J A - J C ) 0 = 1  (J B - J A )  J B + 3  (J B - J C ) 0 = 2.2  (J C - J A ) + 3  (J C - J B ) J C “Distribute” the parentheses 5 = 3.2J A – 1 J B - 2.2J C 0 = -1J A J B – 3J C 0 = -2.2J A – 3 J B + 6.9J C

12PHY 202 (Blum) Loop equations as matrix equation 5 = 3.2J A – 1 J B - 2.2J C 0 = -1J A J B – 3J C 0 = -2.2J A – 3 J B + 6.9J C

13PHY 202 (Blum) Enter matrix in Excel, highlight a region the same size as the matrix.

14PHY 202 (Blum) In the formula bar, enter =MINVERSE(range) where range is the set of cells corresponding to the matrix (e.g. B1:D3). Then hit Crtl+Shift+Enter

15PHY 202 (Blum) Result of matrix inversion

16PHY 202 (Blum) Prepare the “voltage vector”, then highlight a range the same size as the vector and enter =MMULT(range1,range2) where range1 is the inverse matrix and range2 is the voltage vector. Then Ctrl-Shift-Enter. Voltage vector

17PHY 202 (Blum) Results of Matrix Multiplication