Continued Fractions Method:
This Continued Fractions Method is very popularly used to find the equivalent resistance of a ladder type resistive network.
Consider a ladder network as shown in the Fig. 1.40. Let us calculate equivalent resistance by series parallel method first.
So (t + u) is in parallel with s. The combination is in series with r. The entire combination is in parallel with q and finally equivalent of the combination is in series with p, to get Req.
Instead of doing these calculations, in Continued Fractions Method, the resistances of the ladder network are written in a particular fraction form. According to this method, Req of the above network can be directly written as,
It can be observed that all the horizontal branch resistances appear as it is, in the fraction from while the vertical branch resistances appear as the reciprocals in this form. Solving this fractional form Req can be obtained.
This is same as obtained above by series parallel method.
Note that if resistance p is absent in the network as shown in the Fig. 1.40(a) then it must be treated as a resistance of zero ohms in the first horizontal branch. And in such case, Continued Fractions Method is written as,
Remember that horizontal branch resistances appear as it is while vertical branch resistances appear in the reciprocal form, in this method.
Voltage Division in Series Circuit of Resistors
Consider a series circuit of two resistors R1 and R2 connected to source of V volts.
Total voltage applied is equal to the sum of voltage drops VR1 and VR2, across R1 and R2 respectively.
So this circuit is a voltage divider circuit. So in general, voltage drop across any resistor, or combination of resistors, in a series circuit is equal to the ratio of that resistance value to the total resistance, multiplied by the source voltage.
Current Division in Parallel Circuit of Resistors
Consider a parallel circuit of two resistors R1 and R2 connected across a source of V volts. Current through R1 is I1 and R2 is I2, while total current drawn from source is IT.
Substituting value of I1 in IT,
In general, the current in any branch is equal to the ratio of opposite branch resistance to the total resistance value, multiplied by the total current in the circuit.
Note : The above results of section 1.13 and 1.14 are equally applicable if there exists in network, the impedances in series or parallel instead of pure resistances RI and R2.
If there exists alternating voltage source of V volts and two impedances Z1 and Z2 in series then voltage division is,
While in a parallel circuit of two impedances, the current division is given by,
It must be remembered that in alternating circuits multiplication and division must be carried out in polar form while addition and subtraction must be carried out in rectangular form.