Many a times
we come across circuits in which there are multiple resistors connected in
series with each other. There, the question arises –
How to solve
this circuit? How to simplify the circuit?
In this
post, I will be discussing two things:
- How to add resistances in series?
- Principle of voltage division.
Consider the
following circuit:
In this
circuit, there are three resistances connected in series which are connected to
a voltage source. Suppose current I flows through this circuit, then the
voltage across individual resistances are given by –
\[{{V}_{1}}=I{{R}_{1}}\text{, }{{V}_{2}}=I{{R}_{2}}\text{, }{{V}_{3}}=I{{R}_{3}}\]
Applying
Kirchhoff’s voltage law to the circuit,
\[I{{R}_{1}}+I{{R}_{2}}+I{{R}_{3}}-V=0\]
\[\Rightarrow
V=I({{R}_{1}}+{{R}_{2}}+{{R}_{3}})\]
\[Or\text{ }V=I{{R}_{eq}}\]
Thus,
equivalent resistance (Req) of this circuit is given by,
Req= R1+R2+R3
In general,
for n number of resistors in series, Req = R1+R2+R3+……+Rn
\[{{\operatorname{R}}_{eq}}=\sum\limits_{i=1}^{n}{{{R}_{n}}}\]
This is the
expression for equivalent resistance in series. In simple words, equivalent
resistance in series can be obtained by simply adding all the values of the
resistors present in the circuit.
WHAT IF WE NEED TO CALCULATE VOLTAGE ACROSS A PARTICULAR RESISTOR?
For this,
again take a look at the equation obtained before by using KVL, i.e.
\[V=I({{R}_{1}}+{{R}_{2}}+{{R}_{3}})\]
\[\Rightarrow
I=\frac{V}{({{R}_{1}}+{{R}_{2}}+{{R}_{3}})}\]
Now, voltage
across R1 = V1 = IR1
Putting the
value of I, gives ${{V}_{1}}=\left(
\frac{{{R}_{1}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}} \right)V$
Similarly, ${{V}_{2}}=\left(
\frac{{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}} \right)V$
And ${{V}_{3}}=\left(
\frac{{{R}_{3}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}} \right)V$
This is
known as the principle of voltage division and the circuit illustrated in this
post is voltage divider circuit.
In general,
if a voltage divider has N resistors in series with a voltage source V, the nth
resistor Rn will have a voltage drop of
\[{{V}_{n}}=\left(
\frac{{{R}_{n}}}{{{R}_{1}}+{{R}_{2}}+........{{R}_{n}}} \right)V\]
In this
formula we can observe that for finding the voltage across any resistance we do
not require the current through that resistance. We need to know the values of
all resistors and the value of the voltage source. This method of voltage
division can be used as a shortcut for finding out individual voltages quickly!
