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Parallel Circuits

Benchmark Companies IncPO Box 473768Aurora CO 80047

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Total Voltage, Total Current, and Total Resistance in Parallel Circuits

Total Voltage in a Parallel Circuit

The voltage across any branch in parallel is equal to the voltage across any other branch and is also equal to the total voltage.

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Formula: VT = V1 = V2 = V3 = etc.

Total Voltage in a Parallel Circuit

Total Voltage, Total Current, and Total Resistance in Parallel Circuits

VT = VR1 = VR2 = VR3

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Total Current in a Parallel Circuit

The total current in a parallel circuit is equal to the sum of the currents in all the branches of the circuit.

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Formula: IT = I1 + I2 + I3 + etc.

Total Current in a Parallel Circuit

IT = IR1 + IR2 + IR3

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Total Resistance in a Parallel Circuit

The total resistance in a parallel circuit is found by applying Ohm’s law to the total values of the circuit.

VT = IT x RT

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A toaster, a waffle iron, and a hot plate are connected in parallel across a house line delivering 110 V. The current through the toaster is 2 A, through the waffle iron 6 A, and through the hot plate 3 A.

Example

Find:

(a) the total current drawn from the line

(b) the voltage across each device

(c) the total resistance of the circuitContinued

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a. Find the total current IT.1. Write the formula. IT = I1 + I2 + I3

2. Substitute numbers. IT = 2 + 6 + 3 = 11 A

Solution

b. Find the voltage across each device.1. Write the formula. VT = V1 = V2 = V3

2. Substitute numbers. VT = V1 = V2 = V3 =110 V

c. Find the total resistance RT.1. Write the formula. VT = IT x RT

2. Substitute numbers. 110 = 11 x RT

3. Solve for RT. 110/11 = RT = 10

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If the branches of a parallel circuit have the same resistance, then each will draw the same current.

If the branches of a parallel circuit have different resistances, then each will draw a different current. The larger the resistance, the smaller the current drawn.

Using Ohm’s Law in Parallel Circuits

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Fractional Equations

When fractions appear on both sides of the equality sign, eliminate the denominator by multiplying both sides by the least common denominator.

Using the Least Common Denominator to Solve Fractional Equations

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1. Write the equation. 2V/9 = 4/3

2. Multiply both sides by the LCD (9).

9 x 2V/9 = 4/3 x 9

3. Multiply each side separately.

2V = 12

4. Solve for V. V = 12/2 = 6 Ans.

Example

Solution

Find V in the equation 2V/9 = 4/3

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To cross-multiply, the product of the numerator of the first fraction and the denominator of the second is set equal to the product of the numerator of the second fraction and the denominator of the first.

CrossMultiplication

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1. Write the equation. 3/R = 2/5

2. Cross-multiply. 2 x R = 3 x 5

3. Simplify each side. 2R = 15

4. Solve for R. R = 15/2 = 7 1/2 Ans.

Example

Solution

Find the value of R in the equation

3/R = 2/5

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Total Resistance in A Parallel Circuit

where RT is the total resistance in parallel

and R1, R2, and R3 are the branch resistances.

Formula: 1/RT = 1/R1 + 1/R2 + 1/R3 + etc.

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1. Write the formula. 1/RT = 1/R1 +1/R2 +1/R3

2. Substitute numbers. 1/RT = 1/3 + 1/4 + 1/83. Add fractions. 1/RT = 17/244. Cross-multiply. 17 x RT = 1 x 245. Simplify. 17RT = 246. Solve for RT. RT = 24/17 = 1.4

Example

Solution

Find the total resistance of a 3 , a 4 , and an 8 resistor in parallel.

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The total resistance of a number of equal resistors in parallel is equal to the resistance of one resistor divided by the number of resistors.

Total Resistance of a Number of Equal Branches

Formula: RT = R/N

where RT = total resistance of equal resistors in parallel

R = resistance of one of the equal resistors

N = number of equal resistors

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Given: R1 = R2 = R3 = 60

Find: RT = ?

RT = R/N

RT = 60/3 = 20

Example

Solution

Three lamps, each having a resistance of 60 , are connected in parallel. Find the total resistance of the combination.

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Formula RT = (R1 x R2) (R1 + R2)

Two Resistors in Parallel

To find the total resistance of only two resistors in parallel, multiply the resistances and then divide the product by the sum of the resistors.

where RT is the total resistance in parallel, and R1 and R2 are the two resistors in parallel.

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Given: R1 = 4 , R2 = 12

Find: RT = ?

RT = (R1 x R2) (R1 + R2)

RT = (4 x 12) (4 + 12)

RT = 48/16 = 3

Example

Solution

Find the total resistance of a 4 , and a 12 resistor in parallel.

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Total Voltage in a Parallel Circuit

What voltage is needed to send 3 A through a parallel combination of a 3-, a 4-, and a 12 resistance?

Example

Continued

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1. Find the total resistance RT.

1/RT = 1/R1 + 1/R2 + 1/R3

1/RT = 1/3 + 1/4 + 1/12

1/RT = 2/3

2RT = 3

RT = 3/2 = 1.5

2. Find the total voltage VT.

VT = IT x RT

VT = 3 x 1.5 = 4.5 V

Solution

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Division of Current in a Parallel Circuit

Given: Resistors R1 = 16 , R2 = 48 , and R3 = 24, connected in parallel across a voltage line. IT = 12 A.

Example

Find: (1) RT

(2) VT (3) VT, V1, V2, V3

(4) I1, I2, I3

Continued

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1. Find the total resistance RT.

1/RT = 1/R1 + 1/R2 + 1/R3

1/RT = 1/16 + 1/48 + 1/24

1/RT = 1/8 = 8 2. Find the total voltage VT.

VT = IT x RT

VT = 12 x 8 = 96 V

3. Find the branch voltages. VT = V1 = V2 = V3 = 96 V

Solution

Continued

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4. Find the branch currents.V1 = I1 x R1 96 = I1 x 16 = 96/16 = 6 A V2 = I2 x R2

96 = I2 x 48 I2 = 96/48 = 2 A Ans. V3 = I3 x R3

96 = I3 x 24

I3 = 96/24 = 4 A Ans.

5. Check: IT = I1 + I2 + I3

12 = 6 + 2 + 4 12 = 12

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Formula: I1 = R2/(R1 + R2) x IT

Formula: I2 = R1/(R1 + R2) x IT

Division of Current in Two Branches in Parallel

When only two branches are involved, the current in one branch will be only some fraction of the total current. This fraction is the quotient of the second resistance divided by the sum of the resistances.

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Parallel Circuit (Simplifed) Voltage is constant

Why? There are 3 closed

loops in the circuit, which means the voltage though each loop is equivalent to the voltage supplied (like in a series circuit)

Branch currents add to equal total current

1 2 3

1 23 1 2 3

23 2 3

1 1 1 1

where

R R R R

I I I I I I

I I I

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Parallel Equivalent Circuits

231 2 3 2 3 1 23

123 1 2 3123 1 23

1 2 31 2 3

1 2 3 1 2 3

1 1 1 1 1 1 1 1 1 1

1 1 1 1

11 1 1 1 1 1

let so

and

R R R R R R R R R R

R R I I I IR R R R

I I IV I R I I I

R R R R R R

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Kirchhoff’s Current Law

Current into junction = Current leaving junction0Current

Iin I1

I2

I2

I1

Iout

1 2

0

in out

in out

I I I I

I I

The amount of current that enters a junction is The amount of current that enters a junction is equivalent to the current that leaves the junctionequivalent to the current that leaves the junction

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1. Write the formulas for finding IT , VT , and RT in a parallel circuit.

Exercise 1

2. A toaster, waffle iron, and a hot plate are connected in parallel across a 120 V line. The current through the toaster is 3 A, through the waffle iron 5 A, and through the hot plate 4 A. Find: (a) IT; (b) V1 , V2 , V3; (c) RT.

(12 A, 120 V, 10 )

IT = I1 + I2 + I3 + etc.VT = V1 = V2 = V3 = etc.1/RT = 1/R1 + 1/R2 + 1/R3 = etc.

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3. Find the value of R in the equation R/6 = 4/6.

Exercise 1

4. Find V in the equation 4K/10 = 6/1.0.

5. Find the value of L in the equation 18/L = 0.6/17.8.

(4)

(15)

(534)

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2. Three lamps, each having a resistance of 45 , are connected in parallel. Find the total resistance.

Exercise 2

1. Find the total resistance of a 2 , 5 , and an 9 resistors in parallel.

3. Find the total resistance of a 7 , and a 15 resistors in parallel.

(1.23 )

(15 )

(4.77 )

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Exercise 2

4. What voltage is needed to send 4 A through a parallel combination of a 8 , 9 , 17 resistance?

5. Three resistors 12-, 35-, 25 are in parallel. IT = 10 A. Find the current flowing in each branch.

(13.6 V)

(5.49 A, 1.88 A, 2.68 A)

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END OF PRESENTATION