Series Resistor and Voltage Division

Series Resistor 

Resistors can be connected in series; that is, the current flows through them one after another. The circuit in Figure 1 shows three resistors connected in series, and the direction of current is indicated by the arrow.

 Resistors connected in series.

Voltage division 

   Voltage division is a simple rule which can be used in solving circuits to simplify the solution. Applying the voltage division rule can also solve simple circuits thoroughly. The statement of the rule is simple:

Voltage Division Rule: The voltage is divided between two series resistors in direct proportion to their resistance.


the Ohm's law implies that
v1(t)=R1i(t) (I)
v2(t)=R2i(t) (II)

Applying KVL
−v(t)+v1(t)+v2(t)=0→v(t)=v1(t)+v2(t) .

Thereforev(t)=R1i(t)+R2i(t)=(R1+R2)i(t) .

Hence
i(t)=v(t)R1+R2 .


Substituting in I and II
v1(t)=R1v(t)R1+R2 ,
v2(t)=R2v(t)R1+R2 .


Consequently

v1(t)=R1R1+R2v(t) ,
v2(t)=R2R1+R2v(t) .

which shows that the voltage is divided between two series resistors in direct proportion to their resistance. The rule can be easily extended to circuits with more than two resistors. For example,

Voltage Division among four resistors

  v1(t)=R1R1+R2+R3+R4v(t) ,
v2(t)=R2R1+R2+R3+R4v(t) ,
v3(t)=R3R1+R2+R3+R4v(t) ,
v4(t)=R4R1+R2+R3+R4v(t) .

The voltage division rule can be used solve simple circuits or to simplify solving complicated circuits.