phy 202 (blum)1 kirchhoff’s rules when series and parallel combinations aren’t enough
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PHY 202 (Blum) 1
Kirchhoff’s Rules
When series and parallel combinations aren’t enough
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Some circuits have resistors which are neither in series nor
parallel
They can still be analyzed, but one uses Kirchhoff’s rules.
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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.
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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.
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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!
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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
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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
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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
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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.
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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
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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
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Enter matrix in Excel, highlight a region the same size as the matrix.
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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
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Result of matrix inversion
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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
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Results of Matrix Multiplication