Parallel Circuits

From our studies of series circuits in the laboratory we found that they have several important characteristics.

 Circuit Legend:

  •     VT is the Total Voltage

  •     IT is the total Current

  •     V1, and V2 are the voltage drops across R1and R2

  •     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 in two possible ways:

  1. 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.
     
  2. RT  = VT / IT    (Ohm's Law)
Power

The Total Power PT  can be calculated in two possible ways:

  1. PT = P1 + P2
     
  2. PT  = VT IT   

 

Example:

In the circuit below, find all the unknown quantities:

GIVENFINDFIND
 VTotal  = VT = 40 VV2 = ?RT= ?
 ITotal  = IT= 8 AV1 =P1 = ?
R2 = 5.0 ΩI1 = ?P2 = ?
I2= 3 AR1 = ?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

  1. 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
     
  2. 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
 

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