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Parallel Resistance

Resistors in parallel may take a few more calculations.

When there are parallel paths, some current will flow through one
path and some through another path(s). The voltage across a group of parallel
items is the same for each item. Ohms Law tells us that for any given voltage
across a circuit, or section of a circuit, that voltage will equal the product
of the circuit current, times the circuit resistance.
If the current increases, then the resistance must decrease by some amount to
keep the product the same. So in the formula E= I x R the current is inversely
proportional to the resistance. We also know that if there are two resistors
of different values in parallel, the path with the lower value of resistance to the current flowing
will have the more current flowing through it.

With resistors in parallel we need to calculate the circuits equivalent
resistance before we can calculate the total current flowing through the set
of resistors. There are two formulas for finding the total resistance of
parallel resistors in a circuit. A shorthand formula for just 2 resistors
in parallel and the general formula for any number of resistors
in parallel circuit.

In this circuit diagram there are two resistors labeled R1 and R2 wired
in parallel. Some of the current will flow through R1
and some other current will flow through R2. One question that comes to mind
is how much current flows through each path? The voltage across each resistor is the same.

Given: 9 volt supply, R1 is 510 Ohms, R2 is 1000 ohms.

Find: Current flowing through the R1 path and through the R2 path?

Step 1: Find the total resistance.

SHORTHAND FORMULA: (works for just two resistors)

(R1 X R2) / (R1 + R2) The product, divided by the sum.

(510 X 1000) / (510 + 1000 ) OR (510,000/1510) OR 337 ohms

GENERAL FORMULA:

1 / ( (1/R1) + (1/R2) + and so on (1/Rx) )

Applying the general formula to this problem - R1 and R2

( 1 / ( 0.00196 + .001 ) ) OR (1/0.00296) OR 338 Ohms

NOTE: The Total resistance of resistors in parallel will always

be less than the smallest value resistor.

Step 2: Find the total current. I = E / R

I = 9(V) / 337(Ohms) = 0.0267 A, rounded to 27 ma

Step 3: Find the current through each path

Percent = Total Resistance / resistor value.

For R1 337 / 510 = 66 %

For R2 337 / 1000 = 34 %

Total current times percent of current or

.027 X .66 = rounded to 0.018A or 18 ma for R1

.027 x .34 = rounded to .009A or 9 ma for R2

Take a moment to think about these answers.

One path has twice the resistance to current flow, so it

is reasonable that it will only have half the current flow.

In this circuit diagram three resistors labeled R1, R2 and R3 wired

in parallel. In this circuit the current is split into three paths.

Given:

9 volt supply

R1 is 1000 Ohms or 1k ohm

R2 is 2700 ohms or 2.7k ohm

R3 is 3300 ohms or 3.3k ohm

Find: Current flow in each path: for R1, R2 and R3.

Step 1: Calculate the Total Resistance

General formula:

1 / ((1/R1) + (1/R2) + (1/R3))

1 / (0.001 + 0.00037 + 0.0003) OR 1/0.00167 OR 598.8 Ohms

Step 2: Calculate Total Current

General Current Formula: I = E / R

9(volts) / 600(ohms rounded off) = 15 ma

Step 3: Find the current through each path

Percent = Total Resistance / resistor value.

For R1 598 / 1000 = 60 %

For R2 598 / 2700 = 22 %

For R3 598 / 3300 = 18 %

Path current

R1 0.015 x .60 = 9 ma

R2 0.015 x .22 = 3.3 ma

R3 0.015 x .18 = 2.7 ma

Check the work we just did:

R1 is the lease resistance and the most current.

R2 and R3 currents seem reasonable.

Combination Circuits

Resistors in combination circuits take more time but are not all that hard to
calculate. Bringing series and parallel circuits all together they function as
one combination circuit. The process is to simplify and redrawn, over and over
until it is solved.

Given: 9 volt supply, All 6 Resistors are 1000 Ohms

Find: Voltages at test points A, B and C with respect to ground.

NOTE: With the voltage and all resistor values given, the total

R-equivalent can be calculated. With those totals the

total current can be calculated.

Here are the steps.

Step 1:

Solve the parallel set R5 and R6

Work: R56 = (1000x1000) / (1000 + 1000) = 500 Ohms

R-equivalent R56 = 500 Ohms

After redrawing the circuit with the new R56

resistor in place, it should look like this.

Step 2:

Solve the series set R4 and R56

Work R456 = 1000 + 500 = 1500 Ohms

R-equivalent R456 = 1500 Ohms

After redrawing the circuit with the new R456

resistor in place, it should look like this.

Step 3:

Solve the series set R2 and R3

Work R23 = 1000 + 1000 = 2k Ohms

R-equivalent R23 = 2k Ohms

After redrawing the circuit with the new R23

resistor in place, it should look like this.

Step 4:

Solve the parallel set R23 and R456

Work R2-6 ( 1500 x 2000 ) / (1500 + 2000 ) = 857 ohms

R-equivalent R2-6 = 857 Ohms

After redrawing the circuit with the new R2to6

resistor in place, it should look like this.

Step 5: Solve the series set R1 and R2-6 and this

is the real R-equivalent across the source power supply.

Work: Total R = 1000 + 857 = 1857 Ohms

Step 6: Find the total circuit current.

Work: Current = 9 V / 1857 R = 4.85 mA

Step 7: Find the voltage at point A with respect to the battery

minus terminal (-)

Work: Supply voltage minus R1 voltage drop.

9V – ( 1000 X 0.00485) = 4.15 V

Step 8: Find the voltage at point B with respect to the battery

minus terminal (-)

Known: Voltage at point A is 4.15 volts.

R23 is a voltage divider across the 4.15 volts.

Voltage at point B is half voltage at point A or 2.075 volts

Step 9: Find the voltage at point C with respect to the battery

minus terminal (-)

Known: Voltage at point A = 4.15

R4 and R56 is voltage divider across the 4.15 volts.

R56 is one third the resistance of the divider.

Voltage at point C = 4.15(V) times .33 = 1.39 V

Option 2: Using R456 and voltage A,

calculate current through R456, then voltage across R56

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