Locus Diagrams

A locus diagram (also called a circle diagram) is a graphical method used in AC circuit analysis to show how the tip of a phasor (typically current or impedance) traces a path in the complex plane as a circuit parameter, such as resistance R, inductance L, or capacitance C, is varied continuously from zero to infinity. This approach converts what would otherwise be a set of repeated calculations into a single geometric curve.

Locus diagrams are particularly useful for understanding resonance, power factor correction, and frequency response of RLC circuits. In GATE and university exams, they appear in problems asking for the shape of the current locus, the condition for maximum current, or the power factor at various points on the locus.

Core Concept: Why Loci are Circles

Consider a series R-L circuit with a fixed sinusoidal voltage V. The current phasor is I = V/(R + jωL). As R varies from 0 to infinity, the denominator traces a line parallel to the real axis in the impedance plane (since only the real part R changes, while jωL is fixed). When you take the reciprocal of a line that does not pass through the origin in the complex plane, the result is a circle. This geometric property is the reason why current loci in RLC circuits with one variable parameter are always circular arcs or full semicircles.

For a series R-L circuit with variable R, the locus of the current phasor is a semicircle with diameter equal to V/(ωL) lying along the real axis. When R = 0, the current is purely imaginary with maximum magnitude V/(ωL). When R tends to infinity, the current approaches zero. The top of the semicircle corresponds to the condition where R = ωL, at which point the power factor angle is 45 degrees and the active power delivered is maximum.

For a series R-C circuit with variable R, the locus is again a semicircle, but now the diameter lies along the negative imaginary axis direction (capacitive current leads voltage), and the locus is in the fourth quadrant of the current plane. The maximum active power point is again at the top of the semicircle where R = 1/(ωC).

Mathematical Expression

For a series circuit with impedance Z = A + jB where A is the variable parameter (like R) and B is fixed (like ωL), the current I = V/Z. Separating into real and imaginary parts and eliminating A produces the equation of a circle in the I-plane. Specifically, for I = Ix + jIy, the locus satisfies the equation:

Ix² + (Iy + V/(2B))² = (V/(2B))²

This is the equation of a circle centered at (0, -V/(2B)) with radius V/(2B) in the (Ix, Iy) plane. The circle passes through the origin (corresponding to R → ∞) and has its topmost point at (0, 0) and its lowest point at (0, -V/B). The diameter of the circle is V/B = V/(ωL) for an RL circuit, which is the maximum current magnitude (at R = 0).

Practical Understanding

Locus diagrams are used in power engineering to analyze the effect of varying load on line current and power factor. When the load resistance changes (such as in a variable load connected to a transmission line with series inductance), the current locus is a circle. Engineers can read off power factor and current magnitude graphically for any load condition without repeating calculations.

In RLC resonance analysis, the locus of current with variable frequency can also be drawn. At resonance the current is purely real (in phase with voltage). The locus in this case is still circular. The point on the locus corresponding to resonance is where the phasor is purely horizontal (maximum real part).

Given:
Series R-L circuit: V = 100 V (rms), L = 0.1 H, f = 50 Hz, R varies

Why this formula applies:
Current locus for variable R in series RL is a semicircle.
Diameter = V/(ωL). Maximum active power at R = ωL.

Formula:
ωL = 2π × f × L
Diameter of locus = V / (ωL)
At max power: R = ωL, I = V / (√2 × ωL), pf = cos(45°) = 0.707

Substitution:
ωL = 2π × 50 × 0.1 = 31.4 Ω

Diameter = 100 / 31.4 = 3.18 A

At max active power: R = 31.4 Ω
I = 100 / √(31.4² + 31.4²)
  = 100 / (31.4 × √2)
  = 100 / 44.4 = 2.25 A

Active power = I² × R = 2.25² × 31.4 = 159 W

Final Answer:
Locus diameter = 3.18 A
Max active power condition: R = 31.4 Ω
Current at max power = 2.25 A at pf = 0.707 lagging

Exam Tip: For a series RL circuit with variable R, the current locus is a semicircle with diameter V/(ωL). The point of maximum active power (not maximum current) is at the top of the semicircle where R = ωL and pf = 0.707. Maximum current occurs at R = 0, not at maximum power. Do not confuse these two conditions.

Key Mechanism Points

• The current locus for a series RL circuit with variable R is a semicircle with diameter V/(ωL). Maximum current is at R = 0, maximum active power is at the top of the semicircle where R = ωL.
• The current locus for variable L (with fixed R) is a straight vertical line starting at V/R (when L = 0) and moving downward as L increases. This is because only the imaginary part of impedance changes.
• Variable C in a series RC circuit produces a semicircle in the upper half of the current plane (leading side). When C = 0 the current is zero; when C is very large the current approaches V/R.
• The geometric property underlying all circle loci: the reciprocal of a straight line not through the origin in the complex plane maps to a circle. This is the mathematical basis.
• Maximum power transfer from locus: draw a perpendicular from the origin to the diameter of the locus circle. The foot of the perpendicular gives the point of maximum active power on the locus.

Quick Revision

• Variable R in RL series: locus is a semicircle with diameter V/(ωL) in the lagging (lower) half plane.
• Variable L in RL series: locus is a vertical straight line (only imaginary part of Z changes).
• Variable C in RC series: locus is a semicircle in the leading (upper) half plane with diameter V/R as C → ∞.
• Maximum active power point on semicircle is at the top, where the reactive and resistive parts of impedance are equal.
• All circle loci arise because the reciprocal of a linear function of a parameter in the complex plane is circular.
• Exam trap: Maximum current and maximum active power are different points on the locus. Do not equate them.
• Locus diameter always equals V divided by the fixed reactive impedance element.