Parallel Circuits Analysis
Parllel circuits have the following important characteristics.
Here is a typical parallel circuit with two loads (resistors) across the voltage supply.
Circuit Legend:
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VT is the Total Voltage
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IT is the total Current
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V1, and V2 are the voltage drops across R1and R2
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I1, and I2 are the currents through R1and R2
Parallel Circuits Facts:
Voltage: The voltage is always the same -- VT = V1 = V2
Current: The sum of all the branch currents (the current through each load) adds up to the total (source) current -- IT = I1 + I2
Resistance: The Total resistance RT can be calculated as follows:
- 1 / RT = 1 / R1 + 1 / R2
- There is also a convenient way to calculate RT in the special case when you have only two resistors in parallel
- RT = VT / IT (applying Ohm's Law)
Power: The Total Power PT can be calculated in two possible ways:
- PT = P1 + P2
- PT = VT IT
Example:
In the circuit below, find all the unknown quantities:
GIVEN | FIND | FIND |
VTotal = VT = 40 V | V2 = ? | RT= ? |
ITotal = IT= 8 A | V1 = | P1 = ? |
R2 = 5.0 Ω | I1 = ? | P2 = ? |
I2= 3 A | R1 = ? | PT = ? |
Solution:
Use Ohm's law for each component. Start with the component where you have two variables out of the three (RT = VT / IT )
(i) Find I1 If I2= 3 A and IT = 8A, then I1 = IT - I2 = 8 A - 3 A = 5 A(ii) Find R1 R1 = V1 / I1 = 40 V / 5 A = 8.0 Ω(iii) Find V1 We know that VT = V1 = V2 (In parallel circuits the voltage is always constant)
Therefore V1= VT = 40V(iv) Now we can find R 2
R 2 = V 2 / I 2 = 40V / 3A = 13.3Ω(v) Find the Total Resistance RT
- Method #1 ----- 1/RT = 1/R1 +1/ R2 = 1/ 8 Ω + 1/13.3 Ω = 0.20 Ω
Careful Now ... if 1/RT = 0.20 Ω then, RT = 1/ 0.20 Ω = 5.0 Ω
- Method #2 ------ RT = VT / IT (Ohm's Law)
= 40 V / 8A = 5.0 Ω
You can see that both methods yield the same result(vi) Find P1 P1 = V1 x I1 = 40 V x 5 A = 200 W
This is the power used by R1(vii) Find P2 P2 = V2 x I2 = 40 V x 3 A = 120 W
This is the power used by R2(vii) Find PT PT = VT x IT = 40 V x 8 A = 320 W
This is the total power used by all three resistors together
Or, Alternatively, PT = P1 + P2 = 200 W + 120 W = 320 W
You can see that both methods yield the same results