Download Series and Parallel Resistances

Series resistors
1
R1
2
R2
3
R3
I
V1
0
Figure 1: Series circuit configuration.
What is the relationship between V1 and I?
𝑉1 = 𝐼(𝑅1 + 𝑅2 + 𝑅3 )
(1)
This is the same as if I had:
1
V1
Req
0
Figure 2: Equivalent circuit configuration.
Where:
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3 .
(2)
This is true for all series configurations.
1
R1
2
R2
3
R3
V1
0
Figure 3: Series circuit configuration – four resistors.
4
R4
I
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3 + 𝑅4 , 𝑉1 = 𝐼(𝑅1 + 𝑅2 + 𝑅3 + 𝑅4 )
(3)
How can I tell if I have a series configuration? The same current flows through all the resistors.
Parallel Resistors
1
I
R1
VS
R2
I2
I1
R3
I3
0
Figure 4: Parallel circuit configuration.
This time, the voltage across the resistors is the same. Note that for all of the resistors, one of their
terminals is connected to Node 1 and the other is connected to Node 0 (or ground).
In this case:
1
𝑅1
𝐼 = 𝑉𝑆 (
+
1
𝑅2
+
1
)
𝑅3
(4)
Note that this makes Req:
1
VS
Req
0
Figure 5: Equivalent circuit configuration.
𝑉𝑆
𝐼 = 𝑅 , so:
𝑒𝑞
1
𝑅𝑒𝑞
1
1
1
=𝑅 +𝑅 +𝑅
1
2
3
Note the difference between Equation (2) and Equation (5).
(5)
Analyzing Series Circuits
1
V1
R1
+ VR1 -
R2
2
R3
3
+ VR2 -
I
+ VR3 -
0
Figure 6: Calculating circuit variables in a series circuit.
Calculate all the currents and voltages in the circuit.
Step 1: Draw the equivalent circuit – calculate Req.
1
V1
I
Req
0
Figure 7: Using the equivalent circuit in a series circuit.
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3
These circuits are equivalent because for the same voltage, they provide the same current.
Step 2: Solve for current I:
𝐼=
𝑉1
𝑅𝑒𝑞
Step 3: Go back to original circuit, and solve for VR1, VR2, VR3.
𝑉𝑅1 = 𝐼 ∙ 𝑅1 , 𝑉𝑅2 = 𝐼 ∙ 𝑅2 , 𝑉𝑅3 = 𝐼 ∙ 𝑅3
Analyzing Parallel Circuits
1
I
R1
VS
R2
I2
I1
R3
I3
0
Figure 8: Calculating circuit variables in a parallel circuit.
Step 1: For parallel circuits, the equivalent circuit looks the same:
1
VS
Req
0
Figure 9: Using the equivalent circuit from a parallel configuration.
1
1
1
1
=
+
+
𝑅𝑒𝑞 𝑅1 𝑅2 𝑅3
Step 2: Calculate I:
𝐼=
𝑉𝑆
𝑅𝑒𝑞
Step 3: Calculate I1, I2, I3:
𝐼1 =
𝑉𝑆
𝑅1
𝐼2 =
𝑉𝑆
𝑅2
𝐼3 =
𝑉𝑆
𝑅3
Note that you don’t need to go through Step 2 to get to Step 3. You could complete Step 3 without Step
2 and calculate I as 𝐼 = 𝐼1 + 𝐼2 + 𝐼3 .
More complex equivalent resistors:
V2=VS
2
I
R1
VS
1
R3
V1
R2
I1
I3
0
V0=0
Figure 10: A more complex parallel/series configuration – calculating the equivalent resistor.
Note that the same current I1 goes through R1 and R2.
R1 and R2 are in series: 𝑅𝑒𝑞1 = 𝑅1 + 𝑅2
I
2
V2=VS
Req1
VS
0
I1
R3
I30=0
V
Figure 11: Incremental calculation of equivalent resistance.
You’ve “lost” the information on V1, but you can retrieve it later. Note that this does not change the
value of I, I1 or I2.
Note that now, the resistors Req1 and R2 are in parallel. So:
1
1
1
=
+
𝑅𝑒𝑞 𝑅𝑒𝑞1 𝑅2
1
I
VS
Req
0
Figure 12: Final equivalent circuit.
Note that this time, you have lost the information on I1 and I2, but you will retrieve that later as well.
𝑉𝑆
𝐼=𝑅 .
Solve for I:
𝑒𝑞
Now, go backward:
𝑉𝑆
From Figure 11, 𝐼1 = 𝑅
𝑒𝑞1
𝑉𝑆
and 𝐼2 = 𝑅 .
2
From Figure 10, 𝑉𝑅1 = 𝐼1 𝑅1 and 𝑉𝑅2 = 𝐼1 𝑅2 .
From Figure 10 again, 𝐼 = 𝐼1 + 𝐼3
What is the equivalent resistance of the following:
R1
2
R4
R2
3
R5
1
R3
V1
4