2.2 AC Theory, Reactance, Impedance & Power Factor

AC Theory, Reactance, Impedance & Power Factor

Unlike direct current circuits where opposition to current flow is purely resistive, alternating current (AC) circuits introduce magnetic and electrostatic field phenomena that create reactive opposition. Inductors (motor windings, transformers, ballasts) and capacitors store and release energy cyclically, causing phase shifts between voltage and current. Journeyman electricians must understand waveform metrics, reactive properties, vector impedance, and power factor correction to ensure energy efficiency and code compliance.

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1. AC Sinusoidal Waveforms & Measurement Standards

In North America, commercial and residential electric utilities supply alternating current generated as a pure sine wave alternating at 60 Hertz (Hz) (60 cycles per second).

+-----------------------------------------------------------------------------------------+
|                                 AC SINE WAVE CHARACTERISTICS                            |
|                                                                                         |
|      +V_peak ---"""---.                                                                 |
|               /         \                                                               |
|     +V_rms --+-----------+-- (0.707 × V_peak)                                           |
|             /             \                                                             |
|    0V -----+---------------+---------------+----- 0V (Zero Crossings)                   |
|                             \             /                                             |
|                              +-----------+  -- (-0.707 × V_peak)                        |
|      -V_peak -----------------"""""""""""                                               |
|            |<------- 1 Full Cycle (Period T = 1/60s = 16.67 ms) ------->|               |
+-----------------------------------------------------------------------------------------+

Mathematical Metrics of AC Waveforms:

1. Frequency ($f$): Number of complete cycles completed per second, expressed in Hertz (Hz) ($f = 60\text{ Hz}$ in the USA).
2. Period ($T$): The time required in seconds to complete one full electrical cycle: $T = \frac{1}{f} = \frac{1}{60\text{ Hz}} \approx 0.01667\text{ seconds} = 16.67\text{ milliseconds}$
3. Peak Voltage ($V_{\text{peak}}$): Maximum instantaneous voltage amplitude measured from the zero axis to the positive or negative crest.
4. Peak-to-Peak Voltage ($V_{\text{p-p}}$): Full voltage span from positive peak to negative peak ($V_{\text{p-p}} = 2 \times V_{\text{peak}}$).
5. Root-Mean-Square (RMS / Effective Voltage): The effective value of an AC sine wave that produces the exact same heating effect in a resistor as an equivalent DC voltage: $V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}} \approx 0.7071 \times V_{\text{peak}}$ $V_{\text{peak}} = \sqrt{2} \times V_{\text{rms}} \approx 1.4142 \times V_{\text{rms}}$ Key Trade Rule: All standard electrical test meters (DMMs), panelboard ratings, nameplates, and NEC references specify RMS values unless explicitly stated otherwise.
6. Average Voltage ($V_{\text{avg}}$): The mathematical average of one rectified half-cycle sine wave: $V_{\text{avg}} = \frac{2}{\pi} \times V_{\text{peak}} \approx 0.637 \times V_{\text{peak}} \approx 0.90 \times V_{\text{rms}}$

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2. Inductive Reactance ($X_L$) & The "ELI" Concept

When alternating current passes through a coil (inductor), the expanding and collapsing magnetic field induces a Counter-Electromotive Force (CEMF) that opposes changes in current (Lenz's Law). This reactive opposition is called Inductive Reactance ($X_L$).

+-----------------------------------------------------------------------------------------+
|                               INDUCTIVE REACTANCE FORMULA                               |
|                                                                                         |
|                                    X_L = 2π × f × L                                     |
|                                                                                         |
|   • X_L = Inductive Reactance in Ohms (Ω)                                               |
|   • f   = Frequency in Hertz (Hz)                                                       |
|   • L   = Inductance in Henrys (H)                                                      |
|   • 2π  = 2 × 3.14159 ≈ 6.2832                                                          |
+-----------------------------------------------------------------------------------------+

Phase Relationship in Inductors:

In a purely inductive circuit, the induced CEMF delays the buildup of current. Consequently, Voltage leads Current by $90^\circ$ (or Current lags Voltage by $90^\circ$).

• Mnemonic: "ELI" — E (Voltage) leads I (Current) in an L (Inductor).

Frequency Impact:

Inductive reactance is directly proportional to frequency. As frequency increases, the magnetic flux lines collapse and expand more rapidly, creating higher CEMF and greater inductive reactance ($f \uparrow \implies X_L \uparrow$).

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3. Capacitive Reactance ($X_C$) & The "ICE" Concept

A capacitor consists of two conductive plates separated by an insulating dielectric material. When connected to an AC source, charge carriers continuously accumulate on and discharge from the plates, creating opposition to voltage changes called Capacitive Reactance ($X_C$).

+-----------------------------------------------------------------------------------------+
|                               CAPACITIVE REACTANCE FORMULA                              |
|                                                                                         |
|                                  X_C = 1 / (2π × f × C)                                 |
|                                                                                         |
|   • X_C = Capacitive Reactance in Ohms (Ω)                                              |
|   • f   = Frequency in Hertz (Hz)                                                       |
|   • C   = Capacitance in Farads (F)  [1 µF = 1 × 10⁻⁶ F]                                |
|   • 2π  = 2 × 3.14159 ≈ 6.2832                                                          |
+-----------------------------------------------------------------------------------------+

Phase Relationship in Capacitors:

When AC voltage is applied, maximum current flows initially to charge the uncharged plates, while voltage across the plates is zero. As voltage peaks, current drops to zero. Consequently, Current leads Voltage by $90^\circ$.

• Mnemonic: "ICE" — I (Current) leads E (Voltage) in a C (Capacitor).
• Combined Mnemonic: "ELI the ICE man".

Frequency Impact:

Capacitive reactance is inversely proportional to frequency. At higher frequencies, the plates charge and discharge before significant opposing charge builds up, reducing reactance ($f \uparrow \implies X_C \downarrow$).

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4. Total Impedance ($Z$) & Series RLC Circuits

Impedance ($Z$) is the total opposition to alternating current flow, combining pure Resistance ($R$) and net Reactance ($X$). Because inductive reactance and capacitive reactance operate $180^\circ$ out of phase with each other, they cancel each other directly: $X_{\text{net}} = X_L - X_C$

Because resistance and net reactance operate at a $90^\circ$ phase angle, they must be added vectorially using the Pythagorean theorem.

+-----------------------------------------------------------------------------------------+
|                                 IMPEDANCE TRIANGLE FORMULAS                             |
|                                                                                         |
|               Z = √[ R² + (X_L - X_C)² ]          or          Z = √[ R² + X_net² ]      |
|                                                                                         |
|   • Z = Total Impedance in Ohms (Ω)                                                     |
|   • R = Pure Resistance in Ohms (Ω) [Horizontal Base]                                   |
|   • X_net = (X_L - X_C) in Ohms (Ω) [Vertical Leg]                                      |
|   • Phase Angle θ = arctan( X_net / R )                                                 |
|   • Circuit Current: I = E / Z                                                          |
+-----------------------------------------------------------------------------------------+

Electrical Resonance in Series RLC Circuits:

Resonance occurs at the specific frequency where inductive reactance exactly equals capacitive reactance ($X_L = X_C$):

• Net reactance becomes zero: $X_{\text{net}} = X_L - X_C = 0\ \Omega$.
• Total impedance drops to its absolute minimum value: $Z = \sqrt{R^2 + 0^2} = R$.
• Circuit current reaches its absolute maximum: $I = \frac{E}{R}$.
• Power factor reaches $1.0$ (Unity) with zero phase shift ($\theta = 0^\circ$).
• Resonant Frequency Formula: $f_r = \frac{1}{2\pi\sqrt{LC}}$.

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5. The Power Triangle & Power Factor Analysis

In AC circuits with reactive components, current and voltage are out of phase, creating three distinct forms of power represented by the Power Triangle.

+-----------------------------------------------------------------------------------------+
|                                 THE AC POWER TRIANGLE                                   |
|                                                                                         |
|                    /|                                                                   |
|                   / |                                                                   |
|   Apparent Power /  | Reactive Power (Q)                                                |
|       (S)       /   | (VAR / kVAR)                                                      |
|     [in VA]    /    | [Non-working reactive field power]                                |
|               /     |                                                                   |
|              /θ_____|                                                                   |
|           True Power (P) (Watts / kW)                                                   |
|           [Useful working thermal / mechanical power]                                   |
|                                                                                         |
|   1. True Power (P)     = E × I × cos θ  [Units: Watts (W) or Kilowatts (kW)]           |
|   2. Reactive Power (Q) = E × I × sin θ  [Units: Volt-Amps Reactive (VAR / kVAR)]       |
|   3. Apparent Power (S) = E × I          [Units: Volt-Amps (VA) or Kilovolt-Amps (kVA)] |
|   4. Vector Sum:          S = √( P² + Q² )                                              |
+-----------------------------------------------------------------------------------------+

Power Factor Definition:

Power Factor (PF) is the ratio of True Power (useful work performed) to Apparent Power (total electrical capacity delivered by the utility): $\text{PF} = \frac{\text{True Power } (P)}{\text{Apparent Power } (S)} = \cos\theta$

• Unity Power Factor ($1.0$ or $100%$): Voltage and current are perfectly in phase ($\theta = 0^\circ$). All delivered power performs real work ($P = S$).
• Lagging Power Factor: Current lags voltage (inductive loads like induction motors, transformers, welding equipment, and magnetic ballasts).
• Leading Power Factor: Current leads voltage (capacitive loads or overexcited synchronous motors).

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6. Power Factor Correction Calculations for Journeyman Electricians

Operating commercial or industrial facilities with low lagging power factors results in severe operational penalties:

1. Excess Line Current: For a given kW load, lower PF forces higher current through cables and switchgear ($I = \frac{P}{E \times \text{PF}}$).
2. Increased $I^2R$ Cable Losses: Higher line current causes excessive thermal heating in distribution conductors.
3. Increased Voltage Drop: Higher current drops feeder voltage across long distribution runs.
4. Utility Surcharges: Electric utilities impose severe billing penalties when industrial customers operate below a target threshold (typically $0.90$ or $0.95\text{ PF}$).

+-----------------------------------------------------------------------------------------+
|                    POWER FACTOR CORRECTION STEP-BY-STEP WORKFLOW                        |
|                                                                                         |
|   [STEP 1: Determine Existing True Power and Reactive Power]                            |
|   - P = kW (remains constant because work does not change)                              |
|   - S_initial = P / PF_initial                                                          |
|   - kVAR_initial = √[ S_initial² - P² ]                                                 |
|                                     |                                                   |
|                                     v                                                   |
|   [STEP 2: Calculate Target Apparent Power and Reactive Power]                          |
|   - S_target = P / PF_target                                                            |
|   - kVAR_target = √[ S_target² - P² ]                                                   |
|                                     |                                                   |
|                                     v                                                   |
|   [STEP 3: Determine Sizing of Shunt Capacitor Bank]                                    |
|   - kVAR_capacitors = kVAR_initial - kVAR_target                                        |
|                                     |                                                   |
|                                     v                                                   |
|   [STEP 4: Calculate Feeder Current Reduction]                                          |
|   - Line Current Reduction = (S_initial / E) - (S_target / E)                           |
+-----------------------------------------------------------------------------------------+

Comprehensive Worked Correction Problem:

An industrial manufacturing facility draws $120\text{ kW}$ of real power from a $480\text{V}$ single-phase distribution feeder with an uncorrected lagging power factor of $0.60$.

1. Analyze Uncorrected Conditions:
   • Initial Apparent Power: $S_1 = \frac{P}{\text{PF}_1} = \frac{120\text{ kW}}{0.60} = 200\text{ kVA}$.
   • Initial Feeder Current: $I_1 = \frac{S_1}{E} = \frac{200,000\text{ VA}}{480\text{V}} = 416.67\text{ A}$.
   • Initial Reactive Power: $Q_1 = \sqrt{200^2 - 120^2} = \sqrt{40,000 - 14,400} = \sqrt{25,600} = 160\text{ kVAR}$.
2. Analyze Corrected Target Conditions ($0.95\text{ PF}$ Lagging):
   • Target Apparent Power: $S_2 = \frac{P}{\text{PF}_2} = \frac{120\text{ kW}}{0.95} = 126.32\text{ kVA}$.
   • Target Feeder Current: $I_2 = \frac{S_2}{E} = \frac{126,316\text{ VA}}{480\text{V}} = 263.16\text{ A}$.
   • Target Reactive Power: $Q_2 = \sqrt{126.32^2 - 120^2} = \sqrt{15,956.74 - 14,400} = \sqrt{1,556.74} \approx 39.46\text{ kVAR}$.
3. Calculate Required Capacitor Rating ($Q_C$):
   • $Q_C = Q_1 - Q_2 = 160\text{ kVAR} - 39.46\text{ kVAR} = 120.54\text{ kVAR}$.
   • Installing a $120\text{ kVAR}$ power factor correction capacitor bank reduces feeder current from $416.67\text{ A}$ down to $263.16\text{ A}$—releasing $153.51\text{ A}$ ($36.8%$) of feeder and transformer capacity while eliminating utility penalty surcharges.