Resistor in Series Circuit, Series Connected Resistor

Resistor are to be connected in series when they are daisy chained together in a single line resulting in a common current flowing through them.

Individual resistors can be connected together in either a series connection, a parallel connection or combinations of both series and parallel, to produce more complex resistor networks whose equivalent resistance is the mathematical combination of the individual resistors connected together.

A resistor is not only a fundamental electronic component that can be used to convert a voltage to a current or a current to a voltage, but by correctly adjusting its value a different weighting can be placed onto the converted current and/or the voltage allowing it to be used in voltage reference circuits and applications.

Resistors in series or complicated resistor networks can be replaced by one single equivalent resistor, R_EQ or impedance, Z_EQ and no matter what the combination or complexity of the resistor network is, all resistors obey the same basic rules as defined by Ohm’s Law and Kirchhoff’s Circuit Laws.

Resistors in Series

Resistors are said to be connected in “Series”, when they are daisy chained together in a single line. Since all the current flowing through the first resistor has no other way to go it must also pass through the second resistor and the third and so on. Then, resistors in series have a Common Current flowing through them as the current that flows through one resistor must also flow through the others as it can only take one path.

Then the amount of current that flows through a set of resistors in series will be the same at all points in a series resistor network. For example:

In the following example the resistors R₁, R₂ and R₃ are all connected together in series between points A and B with a common current, I flowing through them.

Series Resistor Circuit

As the resistors are connected together in series the same current passes through each resistor in the chain and the total resistance, R_T of the circuit must be equal to the sum of all the individual resistors added together. That is

and by taking the individual values of the resistors in our simple example above, the total equivalent resistance, R_EQ is therefore given as:

R_EQ = R₁ + R₂ + R₃ = 1kΩ + 2kΩ + 6kΩ = 9kΩ

So we see that we can replace all three individual resistors above with just one single “equivalent” resistor which will have a value of 9kΩ.

Where four, five or even more resistors are all connected together in a series circuit, the total or equivalent resistance of the circuit, R_T would still be the sum of all the individual resistors connected together and the more resistors added to the series, the greater the equivalent resistance (no matter what their value).

This total resistance is generally known as the Equivalent Resistance and can be defined as;  “a single value of resistance that can replace any number of resistors in series without altering the values of the current or the voltage in the circuit“. Then the equation given for calculating total resistance of the circuit when connecting together resistors in series is given as:

Series Resistor Equation

R_total = R₁ + R₂ + R₃ + ….. R_n etc.

Note then that the total or equivalent resistance, R_T has the same effect on the circuit as the original combination of resistors as it is the algebraic sum of the individual resistances.

One important point to remember about resistors in series networks to check that your maths is correct. The total resistance ( R_T ) of any two or more resistors connected together in series will always be GREATER than the value of the largest resistor in the chain. In our example above  R_T = 9kΩ where as the largest value resistor is only 6kΩ.

Series Resistor Voltage

The voltage across each resistor connected in series follows different rules to that of the series current. We know from the above circuit that the total supply voltage across the resistors is equal to the sum of the potential differences across R₁ , R₂ and R₃ , V_AB = V_R1 + V_R2 + V_R3 = 9V.

Using Ohm’s Law, the voltage across the individual resistors can be calculated as:

Voltage across R₁ = IR₁ = 1mA x 1kΩ = 1V

Voltage across R₂ = IR₂ = 1mA x 2kΩ = 2V

Voltage across R₃ = IR₃ = 1mA x 6kΩ = 6V

giving a total voltage V_AB of ( 1V + 2V + 6V ) = 9V which is equal to the value of the supply voltage. Then the sum of the potential differences across the resistors is equal to the total potential difference across the combination and in our example this is 9V.

The equation given for calculating the total voltage in a series circuit which is the sum of all the individual voltages added together is given as:

Then series resistor networks can also be thought of as “voltage dividers” and a series resistor circuit having N resistive components will have N-different voltages across it while maintaining a common current.

By using Ohm’s Law, either the voltage, current or resistance of any series connected circuit can easily be found and resistor of a series circuit can be interchanged without affecting the total resistance, current, or power to each resistor.

Resistors in Series : Example No 1

Using Ohms Law, calculate the equivalent series resistance, the series current, voltage drop and power for each resistor in the following resistors in series circuit.

Then for the circuit above, R_T = 60Ω, I_T = 200mA, V_S = 12V and P_T = 2.4W

Voltage Divider Circuit

We can see from the above example, that although the supply voltage is given as 12 volts, different voltages, or voltage drops, appear across each resistor within the series network. Connecting resistors in series like this across a single DC supply has one major advantage, different voltages appear across each resistor producing a very handy circuit called a Voltage Divider Network.

This simple circuit splits the supply voltage proportionally across each resistor in the series chain with the amount of voltage drop being determined by the resistors value and as we now know, the current through a series resistor circuit is common to all resistors. So a larger resistance will have a larger voltage drop across it, while a smaller resistance will have a smaller voltage drop across it.

The series resistive circuit shown above forms a simple voltage divider network were three voltages 2V, 4V and 6V are produced from a single 12V supply. Kirchhoff’s Voltage Law states that “the supply voltage in a closed circuit is equal to the sum of all the voltage drops (I*R) around the circuit” and this can be used to good effect.

Voltage Divider Network

The circuit shown consists of just two resistors, R₁ and R₂ connected together in series across the supply voltage V_in. One side of the power supply voltage is connected to resistor, R₁, and the voltage output, V_out is taken from across resistor R₂. The value of this output voltage is given by the corresponding formula.

If more resistors are connected in series to the circuit then different voltages will appear across each resistor in turn with regards to their individual resistance R (Ohms Law I*R) values providing different but smaller voltage points from one single supply.

We can also apply the same idea to a group of resistors in the series chain. For example if we wanted to find the voltage drop across both R2 and R3 together we would substitute their values in the top numerator of the formula and in this case the resulting answer would give us 5 volts (2V + 3V).

In this very simple example the voltages work out very neatly as the voltage drop across a resistor is proportional to the total resistance, and as the total resistance, (R_T) in this example is equal to 100Ω or 100%, resistor R1 is 10% of R_T, so 10% of the source voltage V_S will appear across it, 20% of V_S across resistor R2, 30% across resistor R3, and 40% of the supply voltage V_S across resistor R4. Application of Kirchhoff’s voltage law (KVL) around the closed loop path confirms this.

Now lets suppose that we want to use our two resistor potential divider circuit above to produce a smaller voltage from a larger supply voltage to power an external electronic circuit. Suppose we have a 12V DC supply and our circuit which has an impedance of 50Ω requires only a 6V supply, half the voltage.

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