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| © Eugene Brennan |
Resistors
are ubiquitous components in electronic circuitry both in industrial
and domestic consumer products. Often, in circuit analysis, we need to
work out the values when two or more resistors are combined. In this
tutorial, we'll work out the formulas for resistors connected in series
and parallel.
Some Revision: A Circuit With One Resistor
In an earlier tutorial, 'How to Understand Electricity: Volts, Amps and Watts Explained on Appliances', you learned that when a single resistor with resistance R ohms was
connected in a circuit with a voltage source V, the current I through
the circuit was given by Ohm's Law:
Ohms Law
I = V/R
Example: A 240 V mains supply is connected to a heater with a resistance of 60 ohms. What current will flow through the heater?
Current = V/R = 240/60 = 4 amps
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| Schematic of a simple circuit. A voltage source V drives a current I through the resistance R. © Eugene Brennan |
Now,
let's add a second resistor in series. Series means that the resistors
are like links in a chain, one after another. We call the resistors R1 and R2.
Because the resistors are linked together, the voltage source V causes the same current I to flow through both of them.
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| Two resistors connected in series. The same current I flows through both resistors. © Eugene Brennan |
There will be a voltage drop or potential difference across both resistors.
Let the voltage drop measured across R1 be V1 and let the voltage measured across R2 be V2, as shown in the diagram below.
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| Voltage drop across resistors connected in series. © Eugene Brennan |
From Ohm's Law, we know that for a circuit with a resistance R and voltage V:
I = V/R
Therefore, rearranging the equation by multiplying both sides by R
IR = V
or switching around
V = IR
So for resistor R1
V1 = IR1
and for resistor R2
V2 = IR2
Kirchoff's Voltage Law
From
Kirchoff's Voltage Law, we know that the sum of voltages around a
closed loop in a circuit adds up to zero. We decide on a convention, so
voltage sources with arrows pointing clockwise from negative to positive
are considered positive and voltage drops across resistors are
negative. So, in our example:
V - V1 - V2 = 0
Rearranging
V = V1 + V2 ................ (i.e., the voltage V equals the sum of the drops across the resistors)
Substitute for V1 and V2 calculated earlier
V = IR1 + IR2 = I(R1 + R2)
Divide both sides by I
V/I = R1 + R2
But from Ohm's Law, we know I = V/R, so rearranging:
V/I = R = total resistance of the circuit. Let's call it Rtotal
Therefore
V/I = Rtotal = R1 + R2
In general, if we have n resistors:
Rtotal = R1 + R2 + ...... Rn
So, to get the total resistance of resistors connected in series, we just add all the values.
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| Formula for resistors connected in series |
Example 1
Five 10k resistors and two 100k resistors are connected in series. What is the combined resistance?
Answer
Resistor values are often specified in kiloohm (abbreviated to "k") or megaohms (abbreviated to "M")
1 kiloohm or 1k = 1000 ohms
1 megaohm or 1M = 1000,000 ohms
So total resistance = sum of the resistances
= 5 x (10k) + 2 x (100k)
= 50k + 200k
= 250k or 250,000 ohms
Example 2
Three 47 ohm, five 1.2k, four 100k and two 3.3M resistors are connected in series. What is the total resistance?
Answer
We
often replace the decimal point in resistor values with the multiplier
to avoid misreading if, e.g., the "dot" gets erased from the value
printed on a component or in documents. So 1.2k becomes 1k2.
So total resistance = sum of the resistances
= 3 x 47 + 5 x 1k2 + 4 x 100k + 2 x 3M3
= 3 x 47 + 5 x 1200 + 4 x 100,000 + 2 x 3,300,000
= 141 + 6000 + 400,000 + 6,600,000
= 7,006,141 ohms
Next,
we'll derive the expression for resistors in parallel. Parallel means
all the ends of the resistors are connected together at one point, and
all the other ends of the resistors are connected at another point.
When
resistors are connected in parallel, the current from the source is
split between all the resistors instead of being the same as was the
case with series connected resistors. However, the same voltage is now
common to all resistors.
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| Two resistors connected in parallel. © Eugene Brennan |
Let the current through resistor R1 be I1 and the current through R2 be I2.
The voltage drop across both R1 and R2 is equal to the supply voltage V.
Therefore, from Ohm's Law
I1 = V/R1
and
I2 = V/R2
Kirchoff's Current Law
From Kirchoff's current we know the current entering a node
(connection point) is equal to the current leaving the node.
Therefore,
I = I1 + I2
Substituting the values derived for I1 and I2 gives us
I = V/R1 + V/R2
= V(1/R1 + 1/R2)
The lowest common denominator (LCD) of 1/R1 and 1/R2 is R1R2 so we can replace the expression (1/R1 + 1/R2) by
R2/R1R2+ R1/R1R2
Switching around the two fractions
= R1/R1R2+ R2/R1R2
and since the denominator of both fractions is the same
= (R1 + R2)/R1R2
Therefore,
I = V(1/R1 + 1/R2) = V(R1 + R2)/R1R2
Rearranging by dividing by dividing both sides of the equation by V and taking the reciprocal of both sides gives us:
V/I = R1R2/(R1 + R2)
But from Ohm's Law, we know V/I = total resistance of the circuit. Let's call it Rtotal.
Therefore,
V/I = Rtotal = R1R2 / (R1 + R2)
So,
for two resistors in parallel, the combined resistance is the product
of the individual resistances divided by the sum of the resistances.
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| Formula for two resistors connected in parallel. |
Example
A 100 ohm resistor and a 220 ohm resistor are connected in parallel. What is the combined resistance?
Answer
For two resistors in parallel, we just divide the product of the resistances by their sum.
So total resistance = 100 x 220 / (100 + 220) = 22000/320 = 8.75 ohms
If
we have more than two resistors connected in parallel, the current I
equals the sum of all the currents flowing through the resistors.
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| Multiple resistors in parallel. © Eugene Brennan |
So for n resistors
I = I1+ I2+ I3. ........... + In
= V/R1+ V/R2+ V/R3+ ............. V/Rn
= V(1/R1+ 1/R2 + V/R3 ........... 1/Rn)
Rearranging
I/V = (1/R1 + 1/R2 + V/R3 ........... 1/Rn)
If V/I = Rtotal then
I/V = 1/Rtotal = (1/R1 + 1/R2 + V/R3 ........... 1/Rn)
So, our final formula is
1/Rtotal = (1/R1 + 1/R2 + V/R3 ........... 1/Rn)
We could invert the right side of the formula to give an expression for Rtotal; however, it's easier to remember the equation for the reciprocal of resistance.
So,
to calculate the total resistance, we calculate the reciprocals of all
the resistances first and sum them together, giving us the reciprocal of
the total resistance. Then, we take the reciprocal of this result,
giving us Rtotal.
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| Formula for multiple resistors in parallel |
Example
Calculate the combined resistance of three 100-ohm and four 200-ohm resistors in parallel.
Answer
Let's call the combined resistance R.
So
1/R = 1/100 + 1/100 + 1/100 + 1/200 + 1/200 + 1/200 + 1/200
We
can use a calculator to work out the result for 1/R by summing all the
fractions and then inverting to find R, but let's try and work it out
"by hand".
So
1/R = 1/100 + 1/100 + 1/100 + 1/200 + 1/200 + 1/200 + 1/200
= 3/100 + 4/200
To
simplify a sum or difference of fractions, we can use a lowest common
denominator (LCD). The LCD of 100 and 200 in our example is 200
Therefore, multiply the top and bottom of the first fraction by 2 giving:
1/R = 3/100 + 4/200 = (2 x 3) / (2 x 100) + 4/200
= 6 / 200 + 4/200
= (6 + 4)/200 = 10/200
and inverting gives R = 200 / 10 = 20 ohms. No calculator needed!
Recommended Books
Introductory Circuit Analysis
by Robert L Boylestad and available from Amazon covers the basics of electricity and circuit
theory and also more advanced topics such as AC theory, magnetic
circuits and electrostatics. It's well illustrated and suitable for high
school students and also first and second-year electric or electronic
engineering students. New and used versions of the hardcover 10th
edition are available on Amazon. Later editions are also available.
References
Boylestad, Robert L. (1968) Introductory Circuit Analysis (6th ed. 1990) Merrill Publishing Company, London, England.
Disclaimer
This
article is accurate and true to the best of the author’s knowledge.
Content is for informational or entertainment purposes only and does not
substitute for personal counsel or professional advice in business,
financial, legal, or technical matters.
© 2020 Eugene Brennan