page 15
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R1 |
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Rn |
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Vn |
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First, sum the voltages about the loop, |
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∑ V = – V + V1 + V2 + … |
+ Vn = 0 |
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Next, use ohms law to replace voltages with the current, and then relate the values to the equivalent resistor Rs.
– V + IR1 + IR2 + … |
+ IRn = 0 |
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R1 |
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+ Rn = Rs |
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ASIDE: We can use a “single subscript notation” to indicate a voltage or current. To do this we need to add a direction arrow for current, or use ‘+’ and ‘-’ for voltages.
3.1.2 Node Voltage Methods
page 16
•If we consider that each conductor in a circuit has a voltage level, and that the components act as bridges between these, then we can try some calculations.
•This method basically involves setting variables, and then doing a lot of algebra.
•This is a very direct implementation of Kirchoff’s current law.
•First, let’s consider an application of The Node Voltage method for the circuit given below,
page 17
Find Vo |
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10Ω |
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20Ω |
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30V |
40Ω |
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5Ω |
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30Ω |
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First, we will add labels for nodes, currents and voltages,
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10Ω |
I1 |
node |
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20Ω |
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I3 |
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30V |
40Ω |
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V12 |
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5Ω |
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30Ω |
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node n2
Next, sum the currents at node 1,
∑ In1 = I1 – I2 – I3 = 0
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40Ω Vo
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40Ω Vo
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Then find equations for the three currents based on the the difference between voltage at nodes 1 and 2. For the center path,
V12 = I340Ω |
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V12 |
---------- |
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40Ω |
page 18
For the left branch, |
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V |
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= – 5I |
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+ 30 – 10I |
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V12 – 30 |
30 – V12 |
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I1 |
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–15 |
15 |
For the right branch, |
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V |
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= 20I |
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+ 40I |
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+ 30I |
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V12 |
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I2 |
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90 |
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Now, combine the equations, |
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0 = I1 – I2 |
– I3 = |
30 – V12 |
V12 |
V12 |
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-------------------15 |
– ------- |
– ---------- |
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40Ω |
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V12 = 19.5V
Finally, find the current in the right circuit branch, and then the voltage across R2,
I2 = |
V12 |
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19.5 |
= 0.217A |
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VR2 |
= I240 = ( 0.217) ( 40) = 8.7V |
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ASIDE: When using a voltage between nodes we are using the double subscript notation. When doing this we label nodes and then the voltage is listed as ‘Vab’ where ‘a’ is positive relative to ‘b’.
3.1.3 Current Mesh Methods
•If we consider Kirchoff’s Voltage law, we could look at any circuit as a collection of current loops. In some cases these current loops pass through the same components.
•We can define a loop (mesh) current for each clear loop in a circuit diagram. Each of these can be given a variable name, and equations can be written for each loop current.