Series Resistive Circuits

I. Basics of Series Resistive Circuits

         I -->        - - -      I -->

                            R1        R2        R3       R4       R5                    Rn

   

In the Figure above we see several resistors attached "in series" that is: one after another.

The current I flowing through the first resistor R1 flows also through resistor R2, R3, R4, R5

and through resistor Rn where n represents the total number of resistors in the circuit.

Hence our first principle for Series Circuits is that the current I is the same throughout the circuit, or :

 

1.  I is constant throughout the circuit.

 

If we apply Ohm's Law for each individual resistor in the circuit, we will conclude that the voltage

through the first resistor is V1 = I*R1, the voltage through the second resistor is V2 = I*R2 and so on

all the way through the n-th resistor for which Vn = I*Rn.

Hence our second principle for Series Circuits is:

 

2.  For any resistor Rx in the circuit the voltage drop across that resistor is Vx = I*Rx.

 

The total amount of resistance in the circuit is the sum of all the individual resistances and by Ohm's Law :

V = I * Rt  or in other words the total voltage drop is the current times the total resistance in the circuit.

Hence our third principle for Series Circuits is:

 

3.  The total resistance of the circuit is the sum of the individual resistances, or:

     Rtotal = R1 + R2 + R3 + R4 + R5 + ... + Rn

 

and or fourth principle is:

 

4.  The total voltage in the circuit is given by Ohm's Law as :

    Vtotal = I* Rtotal.

 

Finally, we note that the total voltage drop in the circuit can also be expressed as the sum of all the

individual voltages, for which we derive our 5th principle:

 

Kirchhoff's Voltage Law also known as KVL:

 

5.  The total voltage in the circuit is the sum of the individual voltages:

    Vtotal = V1 + V2 + V3 + V4 + V5 +  ... + Vn

 

Circuit Example:

 

     The resistors R1, R2 and R3 are

 

    100Ω, 50Ω and 20Ω respectively.

 

    The current I is 2A.  Find out:

 

    the voltages V, V1, V2, V3.

 

 

 

The voltage V1 is the voltage through resistor R1.  By Ohm's Law: V1 =  I * R1 = 2A *100Ω = 200V.

 

Similarly, V2 = I *R2 = 2A * 50Ω = 100V  and  V3 = I * R3 = 2A * 20Ω = 40V.

 

By principle 5 we can use KVL so that V = V1 + V2 + V3 = 200V + 100V + 40V = 340V.

 

Alternatively, another way of obtaining V is by first getting the total Resistance Rt = 100Ω + 50Ω + 20Ω

 

or Rt = 170Ω by principle 3.  Then use principle 4 to get  V = I * Rt = 2A * 170Ω = 340V. !!!

 

 

II. Voltage Dividers

 

                                                 

 

                  V1                           V2                           V3                      V4

 

 

Since the current throughout a Series circuit is constant, by Ohm's Law I = V1/R1 = V2/R2 =V3/R3 =V4/R4

 

and also I = Vtotal/Rt  or  I = V/Rt.  Hence, we can say that V/Rt = V1/R1 = V2/R2 = V3/R3 = V4/R4 or

 

more simply said for any resistor Rx we have that:

 

    Vx/Rx = V/Rt   and Therefore:   Vx = Rx * V/Rt

 

In simple words, we can state that the voltage across any resistance Rx is given by the ratio of that resistance

 

to the total resistance Rt times the total voltage V across the circuit.  This is called The Voltage Divider Rule.

                     

 

Circuit Example:

 

        We are given the values of the resistors

 

        and the total voltage as V=10V.

 

        We want to find out the voltages V1,

 

        V2 and V3 without knowing I

 

 

By the Voltage Divider Rule we can find V1, V2 and V3 without knowing the value of I !!!

 

Thus:  V1 = V * R1/Rt  (Note that Rt = 1KΩ + 5KΩ + 2KΩ = 8KΩ)  and V1 = 10* 1/8 =1.25V.

 

Similarly: V2 = V* R2/Rt = 10 * 5/8  = 6.25V  and  V3 = V * R3/Rt = 10 * 2/8 = 2.5V.

 

Note that the highest the resistance value the higher the voltage drop.  Also, Notice that by KVL:

 

V1 + V2 + V3 = 1.25V + 6.25V + 2.5V  = 10V = Vtotal !!!

 

 

III. Power in Series Circuits

 

 

So far, we have seen that for the resistances in series we need to add them up to get the total resistance.

 

In the same way we need to add the voltages in series to get the total voltage.  Well, it is no different for power:

 

Here we also need to add the power through each resistor in series to obtain the total power.  This statement

 

follows from the fact that whatever power is delivered by the battery or power supply must be spent through the

 

circuit, in this case through the resistors in the circuit.

 

Hence, we can say that the total power  Pt = P1 + P2 +P3  where P1 is the power through R1, P2 is the

 

power through P2 and P3 is the power through P3.  Also, the basic power formula  P = V*I  holds for either

 

power across any resistor (with V substituted with V1, V2 or V3) or for the total power.

 

Moreover, the power formulas for power across the resistors are equally valid for series circuits.

 

Analysis of the circuit above:    I = V/Rt  = 10/8K = 1.25 mA;  Pt = V*I = 10*1.25mA = 12.5mW;

 

and  P1 = I2*R1 = (1.25mA)2 * 1KOhm = 1.56mW;

 

P2 = I2 *R2 = (1.25mA)2 * 5KOhms = 7.81mW;

 

and  P3 = I2 *R3 = (1.25mA)2 * 2KOhms = 3.13mW.

 

Finally, to doublecheck:  P1 + P2 + P3 = 1.56mW + 7.81mW + 3.13mW =  12.5mW which is Ptotal !!!

 

 

 

DC Electronics

Go Back to School Site

HOME