Resistors can be connected in series connection alone or in parallel connection alone. Some resistor circuits are made from combination of series and parallel networks to develop more complex circuits. These circuits are generally known as Mixed Resistor Circuits. Even though these circuits have combined series and parallel circuits, there is no change in the method of calculating the Equivalent Resistance. The basic rules of individual networks like “same current flows through resistors in series” and “voltage across resistors in parallel is same” are applicable to mixed circuits.
It consists of four resistors R1, R2, R3 and R4 in a mixed resistor circuit combination. The supply voltage is V and the total current flowing in the circuit is I. The current flowing through the resistors R2 and R3 is I1 and the current flowing through the resistor R4 is I2.
Here the resistors R2 and R3 are in series combination. Hence applying the rule of resistors in series combination the equivalent resistance of R2 and R3 is given as
RA = R2 + R3
Here RA is the equivalent resistance of R2 and R3
Now the resistors R2 and R3 can be replaced by a single resistor RA. The resulting circuit is shown below.
Now the resistors RA and R4 are in parallel combination. Hence by applying the rule of resistors in parallel combination the equivalent resistance of RA and R4 is
RB = RA × R4 / (RA + R4)
Here RB is the equivalent resistance of RA and R4
Now we can replace the resistors RA and R4 with a single resistor RB.After replacing the resistors the resulting circuit is shown below.
Now the circuit consists of only two resistors. Here also the resistors R1 and RB are in series combination. Hence by applying the rule of resistors in series the total circuit equivalent resistance is given as
REQ = R1 + RB
Here REQ is the total circuit equivalent resistance. Now the resistors R1 and RB can be replaced by a single resistor REQ.
The final equivalent circuit for above complex circuit is shown below.
Though they look complicated, the mixed resistor circuits can be reduced to a simple circuit consisting of only one voltage source and a single resistor by following the simple rules of resistors in series and resistors in parallel.
Resistors in series and parallel Example
Let us calculate the equivalent resistance for the below circuit which consists of 7 resistors R1 = 4 Ω, R2 = 4 Ω, R3 = 8 Ω, R4 = 10 Ω, R5 = 4Ω, R6 = 2 Ω and R7 = 2Ω.The supply voltage is 5 V.
Now the resistors R6 and R7 are in series combination. If the equivalent resistance of R6 and R7in series is Ra, then
Ra = R6 + R7 = 2+2 = 4Ω
The resulting circuit is reduced to the one shown below.
In the above circuit the resistors Ra and R5 are in parallel combination. Hence the equivalent resistance of Ra and R5 is
Now the resistors R1 and RT are in series. If REQis the total circuit equivalent resistance, then REQ = R1 + RT = 3 + 3 = 6 Ω.
Finally the above complex circuit can be redrawn as follows
The total current in the circuit can be calculated using Ohm’s law
I = V1 / REQ = 6 / 6 = 1 A
Hence any complex resistive circuit consisting of number of resistors connected in combination of both series and parallel combinations can be reduced by first identifying the simple parallel resistor branches and series resistor branches. The equivalent resistance of these simple branches is calculated and the branches are replaced with the equivalent resistor. This process reduces the complexity of the circuit. By continuing this process we can replace a complex resistive circuit with a single resistor.
There are some complex resistive circuits which cannot be reduced to simple circuits by simply applying the rules of series resistive combinations and parallel resistive combinations.Circuits like T-Pad Attenuators and some complex resistive bridge networks are examples of such complex resistive circuits. In order to simplify these complex resistive circuits, a different approach is to be followed.
Some complex resistive circuits can be reduced by using Kirchhoff’s Current Law and Kirchhoff’s Voltage Law.
To find the currents and voltages in a complex resistive circuit just by using Ohm’s law might not possible. For such type of circuits Kirchhoff’s Circuit Laws will be helpful.
Kirchhoff’s Circuits laws are based on the concept of conservation of current and energy in a circuit. There are two Kirchhoff’s Circuit laws. First is Kirchhoff’s Current Law which deals with current at node and the second is Kirchhoff’s Voltage Law which deals with voltage in a closed circuit.
Kirchhoff’s Current Law states that “The current entering a node is equal to the current leaving the node because it has no other place to go and no current is lost in the node.”
In simple words, the Kirchhoff’s Current Law states that the sum of currents entering a node is equal to sum of currents leaving the circuit.
Kirchhoff’s Voltage Law states that “the total voltage in a closed loop is equal to sum of all voltage drops in that loop.”
In simple words, the Kirchhoff’s Voltage Law states that the directed algebraic sum of voltages in a closed loop is equal to zero.
With the help of these two laws the values of currents and voltages in any complex circuit can be calculated.
Still we may have some complex resistive circuits in which it is difficult to identify the equivalent resistance, in such situations we will use Star Delta Transformations of resistors in order to simplify the resistor networks.