Resistors in Series


When just two elements connect at a single node, they are said to be in series. Series-connected circuit elements carry the same current. The resistors shown in the following figure are connected in series:



It can be shown that these resistors carry the same current by applying Kirchoff's current law to each node in the circuit. The series interconnection in the above figure requires that:

is = i1 = −i2 = i3 = i4 = i5

which implies that if any of the six currents are known, then they are all known. Therefore, the figure can be redrawn, retaining the identity of the single current is.



To find is, Kirchoff's voltage law is applied for the single closed loop. Defining the voltage across each resistor as a drop in the direction of is gives:

-vs + isR1 + isR2 + isR3 + isR4 + isR5

or:

Eq1 vs = is(R1 + R2 + R3 + R4 + R5)

The significance of Eq1 for calculating is is that the seven resistors can be replaced by a single resistor whose numerical value is the sum of the individual resistors, that is:

Req = R1 + R2 + R3 + R4 + R5

and

vs = isReq

Thus the figure above can be redrawn as follows:



In general, if k resistors are connected in series, the equivalent single resistor has a resistance equal to the sum of the k resistances, or:

Req =
k
i = 1
Ri = R1 + R2 + ⋅⋅⋅ + Rk

Note that the resistance of the equivalent resistor is always larger than that of the largest resistor in the series connection.

Another way to think about this concept of an equivalent resistance is to visualize the string of resistors as being inside a black box. An electrical engineer uses the term black box to imply an opaque container; that is, the contents are hidden from view. The engineer is then challenged to model the contents of the box by studying the relationship between the voltage and current at its terminals. Determining whether the box contains k resistors or a single equivalent resistor is impossible. The following figure illustrates this method of studying the circuit previously shown: