Mastering Locus Diagrams: Understanding Circuit Analysis with Phasors

 LOCUS DIAGRAMS

If a circuit with a variable impedance is supplied at a constant voltage and at a constant frequency, the locus of the end-point of the current phasor will be a circle and will be termed as current circle diagram of the circuit.

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Circle diagrams are widely used in electric circuit analysis. 

 R-L Series Circuit. 

The locus of the end point of current phasor in an R-L series circuit with constant inductive reactance X, and a variable resistance R is found to be a semicircle with diameter equal to V/XL  

and having coordinates: { 0, -V/2XL  } .

Current in such a circuit will maximum when R = 0 and will  be equal to V/XL  amperes. 

Power in such a circuit will be maximum when R = XL and will be equal to V²/2XL  watts.

Current corresponding to maximum power is given as  I = V √2 XL amperes.

 Power factor corresponding to maximum power will be cos 45° i.e. 0.707 (lagging).

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R-C Series Circuit. 

The locus of the end point of current phasor in an R-C series circuit with constant capacitive reactance Xc and a variable resistance R is found to be a semicircle with diameter equal to V/Xc and having coordinates.  {0, V /2Xc}

  Maximum power in the circuit would occur when phase angle is 45° or when R = X and is given as V2/2X, watts. Current corresponding to maximum power will be V/√2 Xc amperes. and 

Power factor corresponding to maximum power will be cos 45° i.e. 0.707 (leading).

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Constant Resistance and Variable Reactance Circuits. 

If the resistance of the circuit is constant and reactance is variable, the locus of the tip of the current phasor will be semicircles of diameter V/R and having coordinates (V/2R, 0). If variable reactance is capacitive one, current will lead the applied voltage phasor V and the locus of the tip of the current phasor will be a semicircle lying above phasor OV.

If variable reactance X is inductive one, current will lag behind the applied voltage phasor OV and locus of the tip of the current phasor will be a semicircle lying below the phasor OV.

Circuit current will be maximum when X=0 and is equal to V/R. Since the current and power factor, both will be of maximum value when variable reactance is zero, therefore, power supplied will be maximum when variable reactance is zero.

Maximum power P max =V²/R

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