Assignment Sample: AC Circuit Analysis With Phasors

Model Answer

Introduction

Analysing a circuit driven by a sinusoidal alternating current (AC) source directly in the time domain requires solving a differential equation for every reactive element present, which quickly becomes unwieldy for circuits containing both inductance and capacitance. Phasor analysis avoids this by representing each sinusoidal voltage and current as a complex number that captures its magnitude and phase angle relative to a common reference, at a single fixed frequency (Hayt, Kemmerly and Durbin, 2018). Under this representation, resistors, inductors and capacitors are each assigned a complex impedance, and circuit analysis reduces to the same algebraic techniques — Ohm’s law and Kirchhoff’s laws — used for direct current circuits, with complex arithmetic in place of real arithmetic (Alexander and Sadiku, 2016). A series RLC circuit is a particularly instructive case because it contains both an inductor and a capacitor whose reactances work in opposition, so that the circuit’s overall behaviour depends on the operating frequency relative to a single characteristic resonant frequency determined by the inductance and capacitance values. This assignment applies phasor analysis to a series RLC circuit connected to the UK 230 V rms, 50 Hz mains supply, calculating the circuit impedance, current, power factor, and real, reactive and apparent power, and comparing the supply frequency with the circuit’s natural resonant frequency. A phasor can be pictured as a rotating vector whose length is the peak (or rms) magnitude of the sinusoid and whose angle relative to a chosen reference axis is the phase of the sinusoid at time zero; because every quantity in a fixed-frequency linear circuit rotates at the same angular speed, the relative angles between phasors stay constant, which is what allows them to be drawn and manipulated as if they were static complex numbers (Boylestad, 2015). This equivalence follows from Euler’s relation, which allows a real sinusoid v(t) = V_mcos(ωt + θ) to be written as the real part of a complex exponential, V_me^j(ωt+θ) = V_me^jθe^jωt; because every element in the circuit is driven at the same angular frequency ω, the common factor e^jωt can be suppressed throughout the calculation, leaving only the time-invariant complex amplitude V_me^jθ — the phasor itself — to be carried through the analysis (Hayt, Kemmerly and Durbin, 2018). This assignment restricts attention to steady-state, single-frequency operation of a linear circuit.

Approach/Method

In phasor form, the impedance of a resistor is purely real, Z_R = R, while the impedance of an inductor and a capacitor are purely imaginary, Z_L = jωL and Z_C = 1/(jωC) = −j/(ωC), where ω = 2πf is the angular frequency (Boylestad, 2015). For a series RLC circuit, the total impedance is the sum Z = R + j(X_L − X_C), where X_L = ωL and X_C = 1/(ωC) are the inductive and capacitive reactances. This complex impedance can be written in polar form as Z = |Z|∠φ, where |Z| = √(R² + X²) and φ = tan⁻¹(X/R), with X = X_L − X_C (Irwin and Nelms, 2015). The current phasor follows directly from Ohm’s law, I = V/Z, and lags the supply voltage by φ when the circuit is net inductive (X > 0) or leads it by φ when the circuit is net capacitive (X < 0). The real power delivered to the circuit is P = VI cosφ, the reactive power exchanged with the reactive elements is Q = VI sinφ, and the apparent power is S = VI, with cosφ known as the power factor (Nilsson and Riedel, 2018). These three quantities form the sides of the power triangle, with P along the reference axis, Q at right angles to it (upward for an inductive load, downward for a capacitive load), and S as the hypotenuse at angle φ to P, giving a convenient graphical summary of how much of the apparent power drawn from the supply is actually converted to useful work. Finally, the circuit’s resonant frequency, at which X_L = X_C and the circuit behaves as a pure resistance, is found from f₀ = 1/(2π√(LC)). Working with impedances and phasors in both polar and rectangular form is useful in practice: rectangular form, Z = R + jX, is convenient for adding series impedances directly, while polar form, Z = |Z|∠φ, is convenient for multiplying and dividing phasors, since magnitudes multiply and angles add under multiplication and magnitudes divide and angles subtract under division (Irwin and Nelms, 2015). The imaginary unit j in these impedance expressions is best understood as a 90° rotation operator rather than merely an algebraic symbol: multiplying a phasor by j rotates it anticlockwise by 90° without changing its magnitude, which is why the inductor voltage leads its current by a quarter-cycle and the capacitor voltage lags by the same amount (Alexander and Sadiku, 2016). This geometric view also underlies the definition of complex power, S = VI*, where I* is the conjugate of the current phasor: multiplying by the conjugate cancels the current’s phase angle against the voltage’s, leaving a single complex number whose real part is P and whose imaginary part is Q, so that S = P + jQ restates the power triangle in a form that is convenient when several loads must later be combined, since complex powers simply add.

Worked Solution/Analysis

Circuit Data

A series RLC circuit has R = 40 Ω, L = 150 mH and C = 100 μF, connected to a 230 V rms, 50 Hz supply.

Reactances and Impedance

ω = 2πf = 2π(50) ≈ 314.16 rad/s.

X_L = ωL = 314.16 × 0.15 ≈ 47.12 Ω
X_C = 1/(ωC) = 1/(314.16 × 100 × 10⁻⁶) ≈ 31.83 Ω

Net reactance X = X_L − X_C = 47.12 − 31.83 = 15.29 Ω (net inductive, since X > 0).

In rectangular form the total series impedance is simply Z = R + jX = 40 + j15.29 Ω. Converting to polar form, |Z| = √(R² + X²) = √(40² + 15.29²) = √(1600 + 233.8) ≈ 42.83 Ω, at a phase angle φ = tan⁻¹(15.29/40) ≈ 20.9°, so Z = 42.83∠20.9° Ω.

Current, Power Factor and Power

I = V/|Z| = 230/42.83 ≈ 5.37 A rms, lagging the supply voltage by 20.9°. Taking the supply voltage as the reference phasor V = 230∠0° V, the current phasor in polar form is I = 5.37∠−20.9° A. Converting to rectangular form for later use, I = 5.37cos(−20.9°) + j5.37sin(−20.9°) ≈ 5.02 − j1.92 A.

Power factor = cosφ = cos(20.9°) ≈ 0.934 (lagging).

Real power P = VI cosφ = 230 × 5.37 × 0.934 ≈ 1153.6 W.
Reactive power Q = VI sinφ = 230 × 5.37 × sin(20.9°) ≈ 440.3 VAr.
Apparent power S = VI ≈ 1235.1 VA.

As a check, S² should equal P² + Q²: 1153.6² + 440.3² ≈ 1,524,655, and √1,524,655 ≈ 1234.8 VA, which agrees with the directly calculated S to within rounding.

Branch Voltages and Resonance

V_R = IR = 5.37 × 40 ≈ 214.8 V
V_L = IX_L = 5.37 × 47.12 ≈ 253.0 V
V_C = IX_C = 5.37 × 31.83 ≈ 170.9 V

Because V_L and V_C are in anti-phase, the supply voltage is recovered by Kirchhoff’s voltage law as V = √(V_R² + (V_L − V_C)²) = √(214.8² + 82.1²) ≈ 229.9 V, which matches the 230 V supply to within rounding, confirming the phasor calculation. As a further check on internal consistency, computing the complex power directly from S = VI* using the rectangular voltage and current phasors, V = 230 + j0 V and I* = 5.02 + j1.92 A, gives S = (230)(5.02 + j1.92) ≈ 1154.6 + j441.6 VA, whose real and imaginary parts agree with the P and Q values calculated above to within rounding, confirming that the rectangular and polar routes through the calculation are mutually consistent.

The resonant frequency is f₀ = 1/(2π√(LC)) = 1/(2π√(0.15 × 100 × 10⁻⁶)) ≈ 41.1 Hz, which is below the 50 Hz supply frequency, consistent with the circuit being net inductive at the operating frequency (Bird, 2017).

Phasor Diagram (Described)

Drawing the current I along the horizontal reference axis, since it is common to all three series elements, V_R lies in phase with I along that same axis at 214.8 V. V_L leads the current by exactly 90°, so it points vertically upward at 253.0 V, while V_C lags the current by 90°, pointing vertically downward at 170.9 V. Because V_L and V_C act in directly opposite directions, they partially cancel to leave a net reactive phasor of 253.0 − 170.9 = 82.1 V pointing upward, which is then combined at right angles with V_R to give the resultant supply phasor V at 230 V, leading the current I by the circuit’s phase angle of 20.9°.

Evaluation

The circuit draws current that lags the supply voltage by 20.9°, confirming that it behaves as a net inductive load at 50 Hz because the supply frequency lies above the resonant frequency of 41.1 Hz, where X_L exceeds X_C. A lagging power factor of 0.934 is fairly close to unity but would still attract a reactive power penalty from a commercial electricity supplier if this were an industrial load, since the apparent power of 1235.1 VA drawn from the supply is noticeably larger than the real power of 1153.6 W actually converted to useful work (Robbins and Miller, 2012). In practice, such a load would typically be corrected towards unity power factor by connecting an additional capacitor in parallel with the load to supply part of the reactive power locally, reducing the current drawn from the supply for the same real power delivered. For example, to raise the power factor from 0.934 to 0.98 lagging, the required reduction in reactive power is Q_c = P(tanφ₁ − tanφ₂), where φ₁ = 20.9° and φ₂ = cos⁻¹(0.98) ≈ 11.5°. This gives Q_c ≈ 1153.6 × (0.382 − 0.203) ≈ 207 VAr, corresponding to a parallel correction capacitance of C = Q_c/(V²ω) ≈ 207/(230² × 314.16) ≈ 12.5 μF — a modest addition that would noticeably reduce the current drawn from the supply for the same useful power delivered to the load. The Kirchhoff’s voltage law check, in which the phasor sum of V_R, V_L and V_C reproduces the 230 V supply voltage, provides useful confidence that the impedance and current calculations are internally consistent, since an arithmetic error in any of the reactances would typically show up as a mismatch at this final check (Hughes et al., 2016). Two further practical qualifications are worth noting. UK grid frequency is tightly regulated but not held at exactly 50 Hz at every instant, and because both X_L and X_C depend on frequency, a small deviation shifts the operating point slightly relative to the 41.1 Hz resonance calculated above, though the effect on this circuit’s impedance and power factor is negligible at typical deviations of a few tenths of a hertz (Hughes et al., 2016). The calculation also assumes a purely sinusoidal supply; background harmonic distortion from other non-linear loads on the same network introduces additional currents at multiples of 50 Hz, and because X_L and X_C scale differently with frequency, the circuit’s response to each harmonic differs from its response at the fundamental, which is why power quality standards specify acceptable distortion limits rather than assuming an ideal sinusoid (IET, 2018). The analysis assumes an idealised circuit in which the inductor and capacitor are purely reactive; in a real component the inductor in particular would have a small series winding resistance that would slightly alter the calculated impedance, current and power factor, and a full laboratory exercise would normally include measuring this parasitic resistance separately. It is also worth noting the practical significance of the resonant frequency: if the supply frequency were reduced to 41.1 Hz, the reactances would cancel exactly, the impedance would fall to its minimum value of R = 40 Ω, and the current would rise to its maximum value for a given supply voltage — a property exploited deliberately in tuned filter and oscillator circuits (Edminister and Nahvi, 2013).

Conclusion

Phasor analysis of the series RLC circuit gave an impedance of 42.83 Ω at a phase angle of 20.9°, a current of 5.37 A rms, a lagging power factor of 0.934, a real power of 1153.6 W, a reactive power of 440.3 VAr and an apparent power of 1235.1 VA. Because the 50 Hz supply frequency sits above the circuit’s resonant frequency of 41.1 Hz, the circuit is net inductive, and the branch voltage calculations were shown to be internally consistent by recovering the 230 V supply via Kirchhoff’s voltage law. The exercise demonstrates how representing sinusoidal quantities as phasors reduces AC circuit analysis to straightforward algebra with complex numbers, while still yielding results, such as the power factor and resonant frequency, that have direct practical significance for the design and operation of AC electrical systems (Alexander and Sadiku, 2016). The additional power-factor-correction calculation showed that a modest 12.5 μF capacitor connected in parallel with the load would raise the power factor from 0.934 to 0.98 lagging, illustrating how the same phasor framework used to analyse a circuit can also be used directly to design a practical improvement to it.

References

• Alexander, C.K. and Sadiku, M.N.O. (2016) Fundamentals of Electric Circuits. 6th edn. New York: McGraw-Hill.
• Bird, J. (2017) Electrical and Electronic Principles and Technology. 6th edn. Abingdon: Routledge.
• Boylestad, R.L. (2015) Introductory Circuit Analysis. 13th edn. Harlow: Pearson.
• Edminister, J.A. and Nahvi, M. (2013) Schaum’s Outline of Electric Circuits. 6th edn. New York: McGraw-Hill.
• Hayt, W.H., Kemmerly, J.E. and Durbin, S.M. (2018) Engineering Circuit Analysis. 9th edn. New York: McGraw-Hill.
• Hughes, E., Hiley, J., Brown, K. and Smith, I.M. (2016) Hughes Electrical and Electronic Technology. 12th edn. Harlow: Pearson.
• Irwin, J.D. and Nelms, R.M. (2015) Basic Engineering Circuit Analysis. 11th edn. Hoboken: Wiley.
• Nilsson, J.W. and Riedel, S.A. (2018) Electric Circuits. 11th edn. Harlow: Pearson.
• Robbins, A.H. and Miller, W.C. (2012) Circuit Analysis: Theory and Practice. 5th edn. Boston: Cengage Learning.
• IET (2018) Requirements for Electrical Installations: BS 7671. 18th edn. Stevenage: Institution of Engineering and Technology.