Locus Diagram in AC circuits

Locus Diagram of AC Circuit

Locus diagrams in electrical engineering show graphically the frequency dependence of admittances and impedances.

Before we start, a brief explanation of the following terms used in this chapter. In electrical engineering:

• The susceptance(B) is the imaginary part of admittance (Y), where the real part is conductance (G). In SI units, admittance is measured in siemens.
• The reciprocal of admittance is impedance (Z), where the imaginary part is reactance (B) and the real part is resistance (R).

Locus diagram of  series R-L circuits

At f = 0 Hz, the inductive reactance according to X_bL = j ꙍL has the value 0 Ω - thus the imaginary part of the impedance is zero. If the frequency increases, the inductive reactance also increases. → the pointer tip of the impedance moves on a straight line parallel to the imaginary axis.

Locus diagram of series R-L circuits - impedance

You already recognize one disadvantage of the Locus Diagram of a RL Series Circuit: For big frequencies, i.e. ꙍ = 2πf, the reactive component becomes very large and thus the straight line of the locus curve also. A solution would be the representation of the admittance, i.e. the reciprocal of the impedance:

Locus Diagram for Impedance and Admittance of a RL series circuit

For f = 0 Hz, only the ohmic resistance is effective in an RL-series circuit. As the frequency increases, the inductive reactance increases and the magnitude of the admittance Y decreases. The phase thereby moves from 0° to -90° with the pointer tip for Y moving on a semicircle.

Locus diagram of  series R-C circuits

The reactance of a capacitance is calculated as X_C = 1 / (2πfC). According to this, the capacitive reactance, i.e. impedance at DC voltage, i.e. for f = 0 Hz is infinitely large, comparable to an interruption. If the frequency increases, the capacitance becomes more and more conductive. Because the voltage V_C at the capacitor lags behind the current by 90°, the pointer of the locus curve for RC series is in the negative imaginary range.

Locus diagram of series R-C circuits

The locus diagram of the admittance is again a semicircle in shape, but now in the positive imaginary part. Since the admittance is the reciprocal of the impedance, the sign of the imaginary part must reverse when the reciprocal is formed.

Impedance and Admittance Locus diagram of RC in series

Locus diagram of  series RLC circuits

With an RLC series connection, two effects come together:

• At f = 0 Hz, the inductive reactance X_L = 0 Ω, while the capacitive reactance tends to infinity.
• At f → ∞ Hz the inductive reactance also tends towards infinity; the capacitive reactance, on the other hand, approaches zero.
• At resonant frequency, the following condition applies: X_C = X_L, i.e. both reactances cancel each other out, with the result that only the ohmic resistance appears at the terminals of this series circuit RLC.

Locus Diagram - RLC in series, Resonance