Parallel Resistors | Formula Of Parallel Resistors

I have discussed series-connected resistors, now it's time to learn more about parallel-connected resistors. First of all, I would like to recall the definition of parallel circuits.

Two components are connected in such a way that they have two common terminals. A parallel circuit is one in which all components are connected in parallel.

Similarly, two resistors are connected in parallel when they have two common terminals. It is often confusing to recognize parallel circuits. Beginners should redraw the circuits.

Properties of parallel circuits:

• Voltage remains the same in every parallel-connected resistor
• Current divides proportionally to all resistors. The larger the resistance value the less will be the current flowing through the resistor

Formula Of Parallel Resistance:

Figure 1

Let's consider the circuit in figure 1. A voltage source is in parallel with three resistors. As I discussed above, the voltage remains the same in each parallel-connected resistor. The current divides among the resistances. So, with the help of Ohm's law, we can easily evaluate the current flowing through each resistor.

Current through R1:

i1 = v/R1

Current through R2:

i2 = v/R2

Current through R3:

i3 = v/R3

The total current flowing through the circuit:

i = i1 + i2 + i3

i = v/R1 + v/R2 + v/R3

i = v{1/R1 + 1/R2 + 1/R3}

i/v = 1/R1 + 1/R2 + 1/R3

From Ohm's law i/v = 1/Req. Substitute in the above equation

1/Req = 1/R1 + 1/R2 + 1/R3

Req = R1 || R2 || R3

The easiest way to solve the above equation is the reciprocal method. Solve individual fractions first and then simply add them.

Solved Examples:

Example #1:

Determine total or equivalent resistance and the current flows through each resistor with the help of Ohm's law.

Equivalent Resistance: It is easy to solve with the help of the above formula.

1/Req = 1/R1 + 1/R2 + 1/R3

1/Req = 1/1k + 1/2k + 1/3k

Req = 545.5 Ω

Current:

Let,

i1 = current flowing through R1

i2 = current flowing through R2

i3 = current flowing through R3

Voltage remains in all three resistors as they are connected in parallel combination

.

i1 = v1/R1

i1 = 10/1000

i1 = 10mA

i2 = v1/R2

i2 = 10/2000

i2 = 5mA

i3 = v1/R3

i3 = 10/3000

i3 = 3.33mA

Conclusion:

• The total/equivalent resistance of the circuit will be less than the smallest resistance present in the parallel circuit
• If we are continuously adding parallel resistance, the total resistance of the circuit decreases
• The larger the resistance, the lower will be the current. Or you can say current will always flow through the less resistive path
• There is another simpler method of finding current through parallel resistors. This method is known as the Current Divider Rule

Series Resistors | Equivalent Series Resistance | Formula Of Series Resistance

Current & Voltage Through Series Resistors: Analysis of Series Circuits:

Before starting my tutorial on series-connected resistors, I would like to recall series circuits.

“Two elements connected in such a way that they share a single node exclusively.”

Similarly, two resistors are connected in series, when they share a single node exclusively.

Properties Of Series Connected Resistors:

• Current remains the same in every series-connected resistor
• Another property of the series circuit is the voltage division property. The voltage is divided proportionally to all resistors. The larger the resistance value, the greater will be the voltage drop across that resistance.

Formula Of Resistance In Series:

Apply Ohm's law to the circuits below,

Circuit 1:

The voltage drop across R1 is va

va = iR1

The voltage drop across R2 is vb

vb = iR2

The total voltage across circuit the is the sum of voltage drops across R1 and R2

v1 = va + vb … equation 1

Circuit 2:

Voltage across Req

v2 = iReq … equation 2

Observations and Calculations:

Have a close look at both circuits. v1 and v2 are the same that is 10V. I take same the same voltage sources, and the current flowing through the circuit remains the same. The difference is the number of resistances in the circuits. Circuit 1 has R1 and R2. and Circuit 2 has only Req.

Look at circuits 1 and 2,

v1 = v2

va + vb = v2

iR1 + iR2 = iReq

From observations and calculations, we can conclude that

Req = R1 + R2

Figure: Electrically equivalent circuits

Implying that we can replace R1 and R2 with a single equivalent resistor. Similarly, we can replace N series resistors with a single equivalent resistor. This is because Req has the same voltage drop across terminals a & b. Also, current and power relationships in the equivalent circuit will remain the same.

The equivalent resistance of a series-connected resistor is the algebraic sum of all the individual resistances.

Req = R1 + R2 + R3 + …..RN

Analysis Of Some Confusing Series Circuits:

Circuit 1:

Circuit 1 is a series circuit. All 5 components are serially connected.

Circuit 2:

It is not a series circuit.

Circuit 3:

It is also a series circuit. It contains multiple sources and resistors. All are connected in series with each other.

Circuit 4:

It is not a series circuit.

Solved Examples:

Example 1:

A 12 V source is connected with these resistances: 1kΩ, 2kΩ and 4kΩ. How much current flows through the 4kΩ resistance?

Solution:

Figure: Draw the circuit with the help of given data

Before calculating the current, you have to find out the total or equivalent resistance of the circuit. After evaluating the total or equivalent resistance, apply Ohm's Law to find out the current through the circuit.

The total resistance will be the sum of all individual resistances.

RT = R1 + R2 + R3

RT = (1+2+4) kΩ

RT = 7 kΩ

Current remains the same  in a series circuit

Current through the 4kΩ is:

i = v/RT

i = 12/7

i = 1.7 mA

Example 2:

From the following data, evaluate the resistance value that should be connected in series to limit the load current to 5mA.

RL = load resistance = 250Ω

RS = series resistance =?

IT = total current required by the load resistance

First of all, draw a circuit and label it with the given data.

Figure: Circuit diagram for example 2

RT = total resistance = RS + RL = RS + 250

RS = RT -250

IT = V/RT

5*10-3 = 5/RT

RT = 1000Ω

RS = 750Ω

Conclusion:

I discussed series-connected resistors in detail. As far as practical applications are concerned, resistors in series are frequently used to limit the voltage. Such circuits are called voltage divider circuits. In complex circuits, we frequently use this technique to simplify circuit analysis.
 There is another way of solving series circuits without knowing the current values. This method is known as Voltage Divider Rule.

Measuring A Resistor | Understand Resistor Color Codes

Practical ways to measure resistance:

As a beginner, you might find it difficult. Believe me, it is simple and fascinating. You can measure resistance using a multimeter or with the help of colour codes.

Using Multimeter:

An ohmmeter is an instrument used to measure the value of resistance. Today, you might not find an ohmmeter. Because it is a part of a multimeter.

A multimeter is an instrument that combines several measuring instruments. Usually, you can measure voltage, current and resistance with a multimeter.

A multimeter is a handy instrument for measuring resistances. Adjust the multimeter knob such that it can measure resistance. There are various ranges available like kΩ, MΩ etc. Place the resistor in parallel with the multimeter. You can also check for open and short-circuit.

Touch the probes with resistor legs. If the meter reads “1” or “OL”, it means overload. Select higher ranges. If the meter reads “0.00”, then select lower ranges.

Keep in mind that the resistance may vary due to temperature and tolerance level.

Understand the colour coding:

Even if you are a beginner, you must have seen a resistor. You also observe the different colour bands painted on resistors. Each colour represents a number and every band has a different value. These colour bands help determine the resistance of resistors as well as the tolerance level. Now we will learn how to read the resistor value from these colour codes.

Colour-code is a system of standards for the identification of resistances of resistors. The colours painted on the resistor body are called colour bands.

There are three different types of marking standards for resistors. Some resistors are marked with four bands of colours, some are marked with five bands of colours. We will see each standard in detail in the next section.

The multiplier band is a decimal multiplier.

The tolerance band gives you accuracy. It indicates the difference between the actual value and theoretical value. It is measured in percent. You can measure the actual value with the help of a multimeter. While theoretical value can be determined from the colour codes. The gold band means +/- 5% tolerance. A 1000 ohm resistor with a gold band means its value is between 950-1050 ohms.

TCR stands for the temperature coefficient of resistors. It is defined as the rate of change of resistance of a resistor with the temperature change. It is available in high precision resistors only.

Digits	Colour Codes	Multiplier	Tolerance	TCR
0	Black	100
1	Brown	101	+/- 1%	100
2	Red	102	+/- 2%	50
3	Orange	103		15
4	Yellow	104		25
5	Green	105	+/- 0.5%
6	Blue	106	+/- 0.25%	10
7	Violet	107	+/- 0.1%	5
8	Gray	108
9	White	109
	Silver		+/- 5%
	Gold		+/- 10%

Four band resistors:

Four band resistors are the most commonly used. The placement of colour bands on the resistor is very important.  3 bands are painted on the left, while 4th is on the right. Put a resistor read it from your left, the tolerance band is on your right.

• 1st band represent the first significant value
• 2nd band represent the second significant value
• 3rd band is a multiplier.
• 4th band is a tolerance band. The tolerance band is separated from the others.

You can get your resistor value with the help of the following formula. This is applicable for +/- 5 % tolerance carbon film resistors.

R = (a*10 + b)m +/- tolerance

Where,

a and b are the values of the first and second band,

m is the multiplier band.  

Example:

Look at the figure above, it shows a 4 band resistor. Brown, black, red and gold colour band. How do you know the resistance value?

Brown >> 1 >> 1st significant digit

Black >> 0 >> 2nd significant digit

Red >> 2 >> multiplier

Gold >> +/- 5% >> tolerance

The theoretical resistance of above resistor is 10*102 = 10*100 = 1000 ohms.

The actual value may vary from 950 ohms to 1050 ohms. Evaluate 5% of 1000 ohm, which is 50. The 5% tolerance shows the precision of the resistor.

Five band resistors:

The placement of colour bands on a five-band resistor is such that 4 bands are painted closely while the 5th band is painted such that it is separated by a small distance.

• 1st band represent the first significant value
• 2nd band represent the second significant value
• 3rd band represent the third significant value
• 4th band is a multiplier
• 5th band is a tolerance band. The tolerance band is separated from the others

Example:

Look at the figure above, it shows a 6 band resistor. Brown, blue, black, yellow, brown, red colour band. How do you know the resistance value?

Yellow >> 4 >> 1st significant digit

Violet >> 7 >> 2nd significant digit

Black >> 0 >> 3rd significant digit

Red >> 2 >> multiplier

Brown >> +/- 1% >> tolerance

The theoretical resistance of above resistor is 470*102 = 470*100 = 47000 ohms = 47 Kilo ohms.

To evaluate tolerance level, calculate the 1% of 47000, which is 470. The actual value of the above resistor may vary from 46530 to 47470.

Six band resistors:

The placement of colour bands on a six-band resistor is such that 4 bands are painted closely while the 5th and 6th bands are painted such that they are separated by a small distance.

• 1st band represent the first significant value
• 2nd band represent the second significant value
• 3rd band represent the third significant value
• 4th band is a multiplier
• 5th band is a tolerance band. The tolerance band is separated from the others
• 6th band TCR

Example:

Look at the figure above, it shows a 6 band resistor. Brown, blue, black, yellow, brown, red colour band. How do you know the resistance value?

Brown >> 1 >> 1st significant digit

Blue >> 6 >> 2nd significant digit

Black >> 0 >> 3rd significant digit

Yellow >> 4 >> multiplier

Red >> +/- 2% >> tolerance

Violet >> 5 >> TCR

The theoretical resistance of above resistor is 160*104 = 160*10000 = 1600000 ohms = 1.6 Mega ohms

To evaluate tolerance level, evaluate 2% of 1.6*106, which is 32000. The actual value of the resistor may vary from 1.56*106 ohms to 1.63*106 ohms.

Frequently Asked Questions:

Why don't manufacturers print numerical values on resistors?

Preciously, it is very difficult to print numerical values on a tiny component like resistors. Modern printing technologies are also available to print numerical values on resistors. But the colour-coded resistors are still popular.

A disadvantage of resistor colour codes

Colour blindness is a common problem, colour-coded resistance value might be problematic for such people.

Why do we measure resistance when there is no power?

It is advisable to isolate the resistor you want to measure and switch off the supply. This is because, when you place the multimeter probes to a resistor present in a circuit, it provides a small voltage and current flows through the resistor. With the help of Ohm's law, the resistance multimeter calculates the resistance.

If there is a supply voltage, the measured value of resistance is wrong. Also, if a resistor is placed in a circuit, other components may affect the value of resistance.

Checking for a defective resistor

A resistor can burn easily with over-voltage and is responsible for malfunctioning. A defective resistor can either be short or open internally. Now, we aim to check whether a resistor is defective or not.

Place the multimeter probes to the resistor, if the reading is too high as compared to its rated value then it is open-circuited internally. If the resistance is too low and approaches zero, then it is shorted internally.