Solving Parallel Circuits Vector or Phaser Method

Solving Parallel Circuits Vector or Phasor Method

In this article we will discuss on Solving Parallel Circuits Vector or Phasor Method to solve the parallel AC circuit.

Consider the circuits shown in Figure (A). Here, two reactors A and B have been joined in parallel across an R.M.S. supply of V volts. The voltage across two parallel branches A and B is the same, but currents through them are different.

For Branch A, ;

I₁ = V/Z₁ ;

cos φ₁ = R₁/Z₁ or φ₁ = cos⁻¹(R₁/Z₁)

Current I₁ lags behind the applied voltage by φ₁ (Figure (B)).

For Branch B,

; I₂ = V/Z₂ ; cos φ₂ = R₂/Z₂ or φ₂ = cos⁻¹(R₂/Z₂)

Resultant Current I

The resultant circuit current I is the vector sum of the branch currents I₁ and I₂ and can be found by (i) using parallelogram law of vectors, as shown in Figure (B).

(ii) resolving I₂ into their X- and Y-components (or active and reactive components respectively) and then by combining these components, as shown in Figure (C) & (D).

Method (ii) is preferable, as it is quick and convenient.

With reference to Figure (C) we have

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Sum of the active components of I₁ and I₂

= I₁ cos φ₁ + I₂ cos φ₂

Sum of the reactive components of I₁ and I₂

= I₂ sin φ₂ − I₁ sin φ₁

If I is the resultant current and φ its phase, then its active and reactive components must be equal to these X-and Y-components respectively [Figure (D)]

I cos φ = I₁ cos φ₁ + I₂ cos φ₂ and I sin φ = I₂ sin φ₂ − I₁ sin φ₁

and

If tan φ is positive, then current leads and if tan φ is negative, then current lags behind the applied voltage V. Power factor for the whole circuit is given by