AC Fundamentals: Sine Waves, Phasors, RMS, Power & Transformers

TAKEAWAYS:

• AC voltage/current varies sinusoidally with time, unlike DC, and is described by peak value (V_m), frequency (f), and phase angle (θ).
• Phasors convert time-domain sine waves into rotating vectors for easier circuit analysis using complex numbers.
• RMS (Root Mean Square) values (e.g., 230 V in Nepal) represent the effective AC voltage/current equivalent to DC.
• Power in AC circuits has two components: real power (P) (useful work) and reactive power (Q) (stored/returned in inductors/capacitors).
• Transformers step up/down AC voltages efficiently using mutual inductance between primary and secondary coils, enabling long-distance power transmission.
• Resonance in RLC circuits occurs when inductive and capacitive reactances cancel, maximizing current or voltage at a specific frequency.

1. Introduction to Alternating Current (AC)

AC is the dominant form of electrical power distribution worldwide due to its efficiency in transmission and ease of voltage transformation. Unlike DC, AC reverses direction periodically, typically at 50 Hz (Nepal) or 60 Hz (USA).

Why AC Over DC?

Sine Wave Characteristics

A sinusoidal AC voltage is defined by:

• Peak value (V_m): Maximum voltage (e.g., 325 V for 230 V RMS in Nepal).
• Frequency (f): Cycles per second (Hz). Nepal uses 50 Hz.
• Period (T): Time for one complete cycle ($T=\frac{1}{f}$).
• Phase angle (θ): Position in the cycle (0° to 360°).

Worked Example: Nepal Grid Voltage The Nepal Electricity Authority (NEA) supplies 230 V RMS, 50 Hz AC. Calculate:

1. Peak voltage ($V_m$): $$V_m=V_{RMS}\times \sqrt{2}=230\times 1.414=325.22\text{V}$$
2. Period ($T$): $$T=\frac{1}{f}=\frac{1}{50}=0.02\text{seconds (20 ms)}$$

2. Phasor Representation of AC Quantities

Phasors simplify AC circuit analysis by representing sine waves as rotating vectors in the complex plane. A phasor has:

• Magnitude: Peak value ($V_m$ or $I_m$).
• Angle (θ): Phase shift relative to a reference (e.g., cosine wave).

Phasor Conversion Rules

For a voltage $v(t)=V_m\sin (\omega t+\theta )$:

• Phasor form: $V=V_m\angle \theta$ (polar) or $V=V_m(\cos \theta +j\sin \theta )$ (rectangular).
• Frequency domain: $\omega =2\pi f$ (angular frequency).

Mermaid Diagram: Phasor Addition

flowchart TD
    A["Phasor V1: 10∠30°"] -->|"Add"| B["Phasor V2: 5∠-45°"]
    B --> C["Resultant Phasor: V = V1 + V2"]
    C --> D["Convert back to time-domain: v(t) = V_m sin(ωt + θ)"]

Worked Example: Phasor Addition (Pathao’s Traffic Light System) Pathao’s delivery drivers rely on traffic light synchronization using phasors. Suppose two signals have voltages:

• $V_1=120\angle 0°$ V (reference)
• $V_2=80\angle 60°$ V (phase-shifted) Find the resultant phasor $V_R$:

1. Convert to rectangular: $$V_1=120+j0,V_2=80\cos 60°+j80\sin 60°=40+j69.28$$
2. Add: $$V_R=(120+40)+j(0+69.28)=160+j69.28$$
3. Convert back to polar: $$|V_R|=\sqrt{160^2+69.28^2}=173.2\text{V},\theta ={\tan }^{-1}\left(\frac{69.28}{160}\right)=23.2°$$ Thus, $V_R=173.2\angle 23.2°$ V.

3. RMS and Average Values

RMS (Root Mean Square) Value

RMS represents the equivalent DC value that dissipates the same power in a resistor. For a sine wave: $$V_{RMS}=\frac{V_m}{\sqrt{2}}\approx 0.707V_m$$ Example: Nepal’s grid is 230 V RMS, so $V_m=325$ V.

Average Value

The average value of a full sine wave over one cycle is zero (symmetrical about the x-axis). For half-cycle: $$V_{avg}=\frac{2V_m}{\pi }\approx 0.637V_m$$

Comparison Table: AC Waveform Values

4. Power in AC Circuits

Power in AC circuits has three types:

1. Real Power (P): Actual power consumed (watts, W). $$P=V_{RMS}{I}_{RMS}\cos \varphi$$ where $\varphi$ is the phase angle between voltage and current.
2. Reactive Power (Q): Power stored/returned by inductors/capacitors (volt-amperes reactive, VAR). $$Q=V_{RMS}{I}_{RMS}\sin \varphi$$
3. Apparent Power (S): Total power (volt-amperes, VA). $$S=V_{RMS}{I}_{RMS}=\sqrt{P^2+Q^2}$$

Power Triangle

flowchart TD
    A["Apparent Power (S)"] --> B["Real Power (P)"]
    A --> C["Reactive Power (Q)"]
    B -->|"P = S cosφ"| D["Phase Angle (φ)"]
    C -->|"Q = S sinφ"| D

Worked Example: NTC’s Power Bill Calculation The Nepal Transmission Company (NTC) charges based on real power (kWh). A factory uses:

• $V_{RMS}=400$ V
• $I_{RMS}=15$ A
• Power factor ($\cos \varphi$) = 0.8 (lagging) Calculate:

1. Real power ($P$): $$P=400\times 15\times 0.8=4800\text{W}=4.8\text{kW}$$
2. If the factory runs for 8 hours/day, daily energy: $$E=P\times t=4.8\times 8=38.4\text{kWh}$$
3. Monthly cost (assuming Rs. 12/kWh): $$\text{Cost}=38.4\times 30\times 12=\text{Rs. }13,824$$

5. Transformers: The Backbone of AC Power Systems

Transformers step up/down AC voltages using electromagnetic induction, enabling efficient power transmission.

How a Transformer Works

1. Primary winding: Connected to input voltage ($V_1$).
2. Secondary winding: Output voltage ($V_2$) induced via mutual inductance.
3. Turns ratio: $$\frac{V_1}{V_2}=\frac{N_1}{N_2}$$ where $N_1$, $N_2$ are primary/secondary turns.

Applications in Nepal

Worked Example: NTC Transmission Line NTC transmits power at 400 kV to reduce losses. A substation steps it down to 11 kV for distribution. If the primary voltage is 400 kV and the turns ratio is $N_1/N_2=40$, find:

1. Secondary voltage ($V_2$): $$V_2=\frac{V_1\times N_2}{N_1}=\frac{400,000}{40}=10,000\text{V}=11\text{kV}$$
2. Current ratio (assuming ideal transformer, $I_1{V}_1=I_2{V}_2$): $$I_2=I_1\times \frac{V_1}{V_2}=I_1\times 40$$ (Current steps down by the same factor as voltage steps up.)

6. Resonance in RLC Circuits

Resonance occurs when inductive reactance ($X_L$) = capacitive reactance ($X_C$): $$X_L=X_C⟹2\pi fL=\frac{1}{2\pi fC}⟹f=\frac{1}{2\pi \sqrt{LC}}$$ At resonance:

• Impedance is minimum (for series RLC) or maximum (for parallel RLC).
• Current is maximum in series RLC (used in tuning circuits).

Real-World Example: Radio Tuning

• Allo FM/99 FM radios use RLC circuits to select specific frequencies (e.g., 98.5 MHz).
• At resonance ($f=98.5$ MHz), the circuit maximizes current for that station, blocking others.

## In the Real World

1. eSewa & Khalti Payments

   • AC Fundamentals: Payment gateways use transformers to isolate and regulate voltages for secure transactions. For example, a step-down transformer converts 230 V AC to 5 V DC for microcontrollers processing payments.
   • Phasors: Signal processing in fraud detection algorithms relies on Fourier transforms (a phasor-based technique) to analyze transaction patterns.

2. Nepal Stock Exchange (NEPSE) Data Transmission

   • AC Power: NEPSE’s servers run on uninterruptible power supplies (UPS), which use inverters (AC to DC) and transformers to maintain stable 230 V AC during grid fluctuations.
   • Resonance: High-frequency trading systems use RLC filters to suppress noise in stock price signals.

3. Pathao’s GPS & Traffic Systems

   • AC Sensors: Pathao’s vehicles use AC-powered sensors (e.g., accelerometers) to detect traffic conditions. These sensors output sinusoidal signals analyzed via phasor techniques to calculate speed/direction.
   • Transformers: Onboard electronics convert 230 V AC (from the grid) to low-voltage DC for GPS modules.

## Exam Tip

1. Memorize Key Formulas:

   • $V_{RMS}=\frac{V_m}{\sqrt{2}}$, $P=V_{RMS}{I}_{RMS}\cos \varphi$, $f=\frac{1}{2\pi \sqrt{LC}}$.
   • Transformer turns ratio: $\frac{V_1}{V_2}=\frac{N_1}{N_2}$.

2. Phasor Diagrams:

   • Always draw voltage/current phasors for RL/RC circuits. For inductive loads, voltage leads current; for capacitive loads, current leads voltage.

3. Numerical Problems:

   • NTC/Daraz-style questions: Calculate power, RMS values, or transformer ratios. Assume ideal conditions unless stated otherwise.
   • Resonance: Given $L$ and $C$, always solve for $f$ first.

4. Real-World Applications:

   • Link answers to Nepal’s grid (230 V, 50 Hz), eSewa payments, or NEPSE data systems. Examiners love context!

5. Avoid Common Mistakes:

   • DC vs. AC: Never use DC formulas (e.g., $P=VI$) for AC without $\cos \varphi$.
   • Units: Always use RMS for voltages/currents unless specified otherwise.
   • Signs in Phasors: Current lags voltage in inductive circuits; leads in capacitive circuits.

Final Note: AC Fundamentals is the bridge between DC circuits and advanced power systems. Master phasors, RMS, and transformers—these are the tools engineers use daily in power plants, electronics, and smart grids. Practice with NTC’s voltage data and eSewa’s payment systems to solidify concepts!