AC Circuit Phasor Calculator

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Quick Answer

What is a phasor in AC circuits? A phasor is a rotating vector that represents a sinusoidal voltage or current by its magnitude and phase angle. For AC circuits, total impedance is Z = √(R² + (XL - XC)²) where XL = ωL (inductive reactance) and XC = 1/ωC (capacitive reactance). The power factor is cos(φ) where φ is the phase angle between voltage and current. This simulation visualizes phasor diagrams, impedance triangles, and power triangles for series and parallel RLC circuits.

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Introduction

A motor draws 15 amps from the wall, but only 10 amps worth of useful work gets done. Where did the other 5 amps go? They're bouncing back and forth between the motor's inductance and the power line, doing no useful work but heating wires and loading transformers. The datasheet says 15 amps, but in practice your circuit breaker sees all of it while your electric bill reflects the power factor penalty [1].

Phasors transform the headache of sinusoidal AC analysis into simple vector arithmetic. Instead of wrestling with trigonometric identities every time you need to find current in an RL circuit, you represent voltages and currents as rotating vectors. The magnitude tells you the amplitude, the angle tells you the phase—and complex impedance handles the rest. In my experience troubleshooting industrial power systems, engineers who think in phasors catch problems that spreadsheet jockeys miss entirely.

This simulator lets you explore AC circuit behavior across series and parallel RLC configurations. Watch how voltage and current phasors rotate together (or don't), see the impedance and power triangles form in real time, and observe time-domain waveforms to connect the mathematical abstraction to physical reality. Electrical engineering students will find this essential for circuit analysis courses, while practicing engineers can use it to quickly verify power factor correction calculations.

What Are Phasors?

A phasor is a complex number representing a sinusoidal signal's amplitude and phase. For a voltage v(t) = V_m sin(ωt + φ), the phasor representation is V = V_m∠φ or V_m e^(jφ). The angular frequency ω disappears because all signals in a linear AC circuit share the same frequency—we only need to track relative magnitudes and phases [2].

The signal sees circuit elements differently at different frequencies. A capacitor that looks like an open circuit at DC becomes a virtual short at RF. An inductor that barely affects 60 Hz power creates significant voltage drops at switching frequencies. Phasor analysis captures this frequency-dependent behavior through complex impedance Z = R + jX, where X varies with ω [3].

When you probe this node with an oscilloscope, you're seeing the time-domain representation v(t). The phasor V is the frequency-domain shorthand that makes calculations tractable. Both representations carry the same information—phasors just make the math easier.

How the Simulator Works

Impedance: The AC Resistance

In an ideal world, resistance would tell the whole story. But real circuits have inductors and capacitors that store and release energy cyclically. Impedance Z combines resistance R with reactance X into a single complex quantity [4]:

Series Circuit: Z = R + j(X_L - X_C) = R + j(ωL - 1/ωC)

Parallel Circuit: 1/Z = 1/R + 1/jX_L + jωC

The impedance magnitude |Z| = √(R² + X²) determines how much current flows for a given voltage (I = V/|Z|). The impedance angle θ = arctan(X/R) determines the phase relationship between voltage and current.

AC Circuit Configurations

Series RLC Circuit (Components in series, same current through all):

          I →
    ┌─────────────────────────────────────────┐
    │                                         │
   ╭┴╮     ╭┬╮╭┬╮╭┬╮      ─┴─                │
   │ │ R   ╰┴╯╰┴╯╰┴╯  L   ─┬─  C             │
   ╰┬╯      (Inductor)  (Capacitor)          │
    │                                         │
    └─────────────────────────────────────────┘
                      │
                   ───┴───
                  ~  Vs  ~   V = Vs∠0°
                   ───┬───
                      │
    ──────────────────┴───────────────────────

    Total Impedance: Z = R + jX_L - jX_C = R + j(ωL - 1/ωC)
    Current: I = V / Z (same through all elements)
    Voltage drops: V_R = IR, V_L = jX_L·I, V_C = -jX_C·I

Parallel RLC Circuit (Components in parallel, same voltage across all):

         I_total
            ↓
    ┌───────┬───────┬───────┬───────┐
    │       │       │       │       │
    │      ╭┴╮     ╭┬╮    ─┴─      │
    │      │ │     ╰┴╯    ─┬─      │
    │      │ │ R  ╭┬╮╭┬╮   │ C     │
    │      │ │    ╰┴╯╰┴╯   │       │
    │      ╰┬╯     │ L   ──┴──     │
    │       │      │               │
    └───────┴──────┴───────┴───────┘
                   │
                ───┴───
               ~  Vs  ~   V = Vs∠0°
                ───┬───
                   │
    ───────────────┴───────────────────

    Admittance: Y = 1/R + 1/jX_L + jωC = G + jB
    Total Current: I = V·Y
    Branch currents: I_R = V/R, I_L = V/jX_L, I_C = jωC·V

Power Factor Correction Setup (Capacitor added to inductive load):

                 I_source
                    ↓
    ┌───────────────┬───────────────┐
    │               │               │
    │    ┌──────────┴──────────┐    │
    │    │     Motor Load      │  ─┴─
    │    │  (R + jX_L)         │  ─┬─  C_pfc
    │    │   Inductive         │    │
    │    └──────────┬──────────┘    │
    │               │               │
    └───────────────┴───────────────┘
                    │
                 ───┴───
                ~  Vs  ~
                 ───┬───
                    │
    ────────────────┴───────────────

    Without C_pfc: PF = cos(θ), lagging
    With C_pfc: Reactive currents cancel → PF ≈ 1
    Size C_pfc to supply: Q_c = Q_motor - Q_target

Power in AC Circuits

AC power analysis involves three quantities that form the power triangle [5]:

• Real Power (P): Watts—does actual work, heats resistors, turns motors
• Reactive Power (Q): VAR—sloshes between sources and reactive elements
• Apparent Power (S): VA—what the utility must supply to the circuit

The relationships: S² = P² + Q², and Power Factor = P/S = cos(θ).

Ground isn't always at zero potential, and apparent power isn't always what you pay for. Industrial customers face power factor penalties because reactive current loads infrastructure without doing productive work.

The Power Factor Problem

A power factor of 0.8 lagging means 80% of the apparent power does useful work while 20% cycles uselessly through inductive loads. Motors, transformers, and fluorescent lighting ballasts are notorious offenders [6].

Power factor correction adds capacitance in parallel with inductive loads. The capacitor's leading reactive power cancels the inductor's lagging reactive power, bringing the total power factor closer to unity. The simulator's "PF Corr" preset demonstrates this principle—adjust the capacitance to watch the power factor improve.

Learning Objectives

After completing this simulation, you should be able to:

1. Convert between phasor notation (polar) and rectangular complex form
2. Calculate impedance magnitude and angle for series and parallel RLC circuits
3. Determine voltage-current phase relationships from circuit configuration
4. Compute real, reactive, and apparent power from voltage and current phasors
5. Explain why power factor matters for electrical system efficiency
6. Size capacitors for power factor correction in inductive loads

Exploration Activities

Activity 1: Phasor Phase Relationships

Objective: Understand how reactive elements affect current phase

Steps:

1. Set R = 100 Ω, L = 100 mH, C = 0 μF, f = 60 Hz (pure RL)
2. Observe the phasor diagram—current lags voltage
3. Now set L = 0 mH, C = 100 μF (pure RC)
4. Current now leads voltage
5. Adjust both L and C to see partial cancellation

Expected Result: Inductance causes lagging current, capacitance causes leading current. When X_L = X_C, the circuit is resonant and current is in phase with voltage.

Activity 2: Impedance Triangle Construction

Objective: Visualize how R and X combine to form Z

Steps:

1. Select the "Z Triangle" visualization tab
2. Set R = 50 Ω, L = 100 mH, C = 0 at 60 Hz
3. Note X_L = 2π(60)(0.1) = 37.7 Ω
4. Verify |Z| = √(50² + 37.7²) = 62.6 Ω
5. Verify θ = arctan(37.7/50) = 37.0°

Expected Result: The impedance triangle visually confirms the Pythagorean relationship between R, X, and |Z|.

Activity 3: Power Triangle Analysis

Objective: Connect power factor to the power triangle

Steps:

1. Switch to "Power" visualization
2. Use motor preset (V = 120 V, R = 50 Ω, L = 100 mH)
3. Verify P = I²R and S = VI
4. Calculate Q = √(S² - P²) and verify against display
5. Add capacitance and watch Q decrease

Expected Result: The power triangle shrinks as reactive power decreases. At unity power factor, S = P and Q = 0.

Activity 4: Power Factor Correction

Objective: Size a capacitor for power factor correction

Steps:

1. Select "PF Corr" preset (industrial motor scenario)
2. Note initial power factor (lagging due to inductance)
3. Gradually increase C while watching power factor
4. Find the C value that achieves PF ≈ 0.95
5. Record the reactive power reduction

Expected Result: A properly sized capacitor reduces reactive power demand without affecting real power consumption.

Real-World Applications

1. Industrial Motors - Induction motors have inherently lagging power factors (0.7-0.85), requiring capacitor banks for correction to avoid utility penalties

2. Power Grid Management - Utilities use synchronous condensers and static VAR compensators to regulate system power factor and voltage

3. Audio Amplifier Design - Speaker crossover networks use LC combinations where phase relationships determine frequency response

4. Radio Frequency Systems - Antenna matching networks require precise impedance control to maximize power transfer

5. Electric Vehicle Chargers - Power factor correction circuits ensure high efficiency and compliance with grid codes

Reference Data

Challenge Questions

Beginner:

1. If V = 120 V and Z = 60∠30° Ω, what is the current magnitude?

Intermediate: 2. A motor draws 10 A at 0.8 PF lagging from 240 V. Calculate P, Q, and S.

Advanced: 3. What capacitance corrects the motor in Question 2 to 0.95 PF at 60 Hz?

Expert: 4. Derive the condition for resonance in a parallel RLC circuit.

Research: 5. How do harmonic distortions affect power factor measurements and correction strategies?

Common Mistakes to Avoid

1. Confusing RMS and Peak Values - Phasor magnitudes typically represent RMS values. V_peak = V_rms × √2

2. Wrong Sign Convention - Inductive reactance is positive (+jX_L), capacitive is negative (-jX_C) in standard convention

3. Series vs Parallel Formulas - Don't add impedances in parallel circuits; add admittances instead

4. Ignoring Frequency Dependence - Reactances change with frequency. A circuit's behavior at 60 Hz differs from 400 Hz

FAQ

Q: Why do we use complex numbers for AC analysis? A: Complex numbers elegantly capture both magnitude and phase in a single quantity. Euler's formula e^(jθ) = cos(θ) + j·sin(θ) connects sinusoidal time functions to rotating phasors, making circuit analysis algebraic rather than trigonometric [7].

Q: What's the difference between power factor and efficiency? A: Power factor (PF = P/S) measures how much of the apparent power does useful work. Efficiency (η = P_out/P_in) measures how much input power becomes useful output. A motor can have high PF but low efficiency (losses to heat) or vice versa.

Q: Why is lagging power factor more common than leading? A: Most industrial loads (motors, transformers) are inductive. Capacitive loads are less common except for power factor correction capacitors and long transmission lines under light load [8].

Q: How do utilities measure power factor? A: Digital power meters sample voltage and current simultaneously, computing true power (average of v×i) and apparent power (V_rms × I_rms). The ratio gives power factor, accounting for harmonics that analog meters miss.

Q: Can power factor exceed 1.0? A: No. Power factor = cos(θ), which is bounded between 0 and 1. However, displacement power factor (fundamental frequency only) can differ from true power factor when harmonics are present.