Continued Fractions Method

Continued Fractions Method:

This Continued Fractions Method is very popularly used to find the equivalent resistance of a ladder type resistive network.

Continued Fractions Method

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.

Continued Fractions Method

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,

Continued Fractions Method

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.

Continued Fractions Method

This is same as obtained above by series parallel method.

Continued Fractions 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,

Continued Fractions Method

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.

Continued Fractions Method

As two resistors are connected in se­ries, the current flowing through both the resistors is same, i.e. I. Then applying KVL, we get,Continued Fractions Method

Total voltage applied is equal to the sum of voltage drops VR1 and VR2, across R1 and R2 respectively.

Continued Fractions Method

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.

Continued Fractions Method

Continued Fractions Method

Continued Fractions Method

Substituting value of I1 in IT,

Continued Fractions Method

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,

Continued Fractions Method

While in a parallel circuit of two impedances, the current division is given by,

Continued Fractions Method

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.

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