PHY 202 (Blum) 1

Kirchhoff’s Rules

When series and parallel combinations aren’t enough

2 PHY 202 (Blum)

Some circuits have resistors which are neither in series nor

parallel

They can still be analyzed, but one uses Kirchhoff’s rules.

3 PHY 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.

4 PHY 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.

5 PHY 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.

I1 + I2 = I3 I1 I2

I3The node rule is about currents!

6 PHY 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

7 PHY 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

Loo

p di

rect

ion

Loo

p di

rect

ion

8 PHY 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. I I

Loss Gain

Loo

p di

rect

ion

Loo

p di

rect

ion

Cur

rent

dir

ectio

n

Cur

rent

dir

ectio

n

9 PHY 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.

JA

JB

JC

Currents in Resistors

Note that there are two currents associated with the 2.2-kΩ resistor. Both JA and JC 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(JA-JC)

On the other hand, when in Loop C, the voltage drop across the 2.2-kΩ resistor is 2.2(JC-JA) the opposite sign because we are going through the resistor in the opposite direction.

PHY 202 (Blum) 10

11 PHY 202 (Blum)

Loop equations

5 = 1 (JA - JB) + 2.2 (JA - JC) 0 = 1 (JB - JA) + 1.5 JB + 3 (JB - JC) 0 = 2.2 (JC - JA) + 3 (JC - JB) + 1.7 JC

“Distribute” the parentheses

5 = 3.2 JA – 1 JB - 2.2 JC 0 = -1 JA + 5.5 JB – 3 JC 0 = -2.2JA – 3 JB + 6.9JC

12 PHY 202 (Blum)

Loop equations as matrix equation

5 = 3.2 JA – 1 JB - 2.2 JC 0 = -1 JA + 5.5 JB – 3 JC 0 = -2.2JA – 3 JB + 6.9JC

0

0

5

9.632.2

35.51

2.212.3

C

B

A

J

J

J

13 PHY 202 (Blum)

Enter matrix in Excel, highlight a region the same size as the matrix.

14 PHY 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

15 PHY 202 (Blum)

Result of matrix inversion

16 PHY 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

17 PHY 202 (Blum)

Results of Matrix Multiplication