Alternating Currents: AC, RMS, Transformers, Power

Alternating Current (AC) Basics

What is AC?

• Definition: Alternating current (AC) is an electric current that reverses direction periodically (usually sinusoidally) with time.
• Key Feature: Unlike direct current (DC), AC changes polarity (positive to negative and back) at a fixed frequency (e.g., 50 Hz in Nepal).
• Graph: The voltage $V(t)$ in AC varies as: $$V(t)=V_0\sin (\omega t)$$ where:
  • $V_0$ = peak (maximum) voltage,
  • $\omega =2\pi f$ = angular frequency (rad/s),
  • $f$ = frequency (Hz).

Key Terms

Why Use RMS Value?

• RMS (Root Mean Square) is used because it gives the equivalent DC value that would produce the same power in a resistor.
• Example: If an AC supply has $V_{\text{rms}}=220\text{V}$, the peak voltage is: $$V_0=\sqrt{2}\times 220=311\text{V}$$
• Power in AC: For a resistor $R$, power $P$ is: $$P=\frac{V_{\text{rms}}^2}{R}=I_{\text{rms}}^{2}R$$ (Same formula as DC!)

AC in Resistors, Inductors, and Capacitors

1. Pure Resistor (R)

• Behavior: Voltage and current are in phase (peak at the same time).
• Impedance: $Z_R=R$.
• Power: Always positive (dissipated as heat).

2. Pure Inductor (L)

• Behavior: Voltage leads current by $90^{\circ }$ (phase difference of $\pi /2$).
• Impedance: $Z_L=\omega L=2\pi fL$ (called inductive reactance).
• Power: Zero average power (energy stored and returned to the circuit).

3. Pure Capacitor (C)

• Behavior: Current leads voltage by $90^{\circ }$.
• Impedance: $Z_C=\frac{1}{\omega C}$ (called capacitive reactance).
• Power: Zero average power (energy stored and returned).

LCR Circuits and Resonance

Series LCR Circuit

• Impedance: $Z=\sqrt{R^2+(X_L-X_C{)}^2}$, where:
  • $X_L=\omega L$ (inductive reactance),
  • $X_C=\frac{1}{\omega C}$ (capacitive reactance).
• Resonance: Occurs when $X_L=X_C$, so $Z=R$. At resonance:
  • Current is maximum (since impedance is minimum).
  • Frequency $f_0=\frac{1}{2\pi \sqrt{LC}}$ (resonant frequency).

Applications of Resonance

• Radio Tuning: Circuits are tuned to specific frequencies to pick up signals.
• Power Transmission: Resonance reduces energy loss in AC systems.

Transformers

What is a Transformer?

• A device that steps up or steps down AC voltage using electromagnetic induction.
• Types:
  • Step-up: Increases voltage (e.g., for power transmission).
  • Step-down: Decreases voltage (e.g., for household use).

How It Works

• Primary Coil: Connected to input AC voltage $V_p$.
• Secondary Coil: Output voltage $V_s$ is induced.
• Turns Ratio: $$\frac{V_s}{V_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}$$ where $N_s,N_p$ = number of turns in secondary/primary, $I_p,I_s$ = currents.

flowchart TD
    A["AC Source (V_p)"]
    B["Primary Coil (N_p turns)"]
    C["Iron Core"]
    D["Secondary Coil (N_s turns)"]
    E["Load (V_s, I_s)"]
    A --> B
    B -->|"Magnetic Field"| C
    C --> D
    D --> E

Efficiency of Transformers

• Ideal Transformer: $V_p{I}_p=V_s{I}_s$ (power input = power output).
• Real Transformers: Efficiency $\eta =\frac{\text{Output Power}}{\text{Input Power}}\times 100\%$.
• Losses: Due to copper losses (resistance) and eddy currents (reduced by laminating the core).

Power in AC Circuits

Average Power

For a general AC circuit: $$P_{\text{avg}}=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$$ where $\varphi$ = phase difference between $V$ and $I$.

• $\cos \varphi$ is called the power factor.
• For pure resistor: $\varphi =0$, so $P=V_{\text{rms}}{I}_{\text{rms}}$.
• For pure inductor/capacitor: $\varphi =90^{\circ }$, so $P=0$.

Power Factor

• Definition: Ratio of real power to apparent power.
• Improvement: Adding capacitors in inductive circuits can bring the power factor closer to 1 (better efficiency).

AC Generators

How AC is Generated

• A coil rotates in a magnetic field, inducing an AC voltage (Faraday’s Law).
• Frequency: Depends on the speed of rotation and number of poles. $$f=\frac{nP}{120}$$ where $n$ = revolutions per minute (rpm), $P$ = number of poles.

flowchart LR
    A["Rotating Coil"] -->|"in"| B["Magnetic Field"]
    B -->|"induces"| C["AC Voltage (V = V₀ sin(ωt))"]
    C --> D["Output Terminals"]

Applications

• Power Plants: Generate AC at high voltages (e.g., 11 kV, 33 kV).
• Household Appliances: Use step-down transformers to get 220 V AC.

Exam Tip: Key Points to Remember

1. AC vs. DC:
   • AC changes direction; DC does not.
   • AC is easier to step up/down using transformers.
2. RMS Value:
   • Always use $V_{\text{rms}}$ for power calculations.
   • $V_{\text{rms}}=\frac{V_0}{\sqrt{2}}$.
3. Phase Differences:
   • Resistor: $V$ and $I$ in phase.
   • Inductor: $V$ leads $I$ by $90^{\circ }$.
   • Capacitor: $I$ leads $V$ by $90^{\circ }$.
4. Transformers:
   • $\frac{V_s}{V_p}=\frac{N_s}{N_p}$.
   • Used for power transmission (step-up) and distribution (step-down).
5. Resonance:
   • Occurs at $f_0=\frac{1}{2\pi \sqrt{LC}}$.
   • Current is maximum at resonance.
6. Power Factor:
   • $P=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$.
   • Unity power factor ($\cos \varphi =1$) is ideal.

NEB Board-Style Questions

Short Answer Questions

1. Define RMS value of AC. Why is it important?

   • Answer: RMS (Root Mean Square) value is the equivalent DC value that produces the same power in a resistor. It is important because it gives the effective voltage/current for power calculations.

2. What is the phase difference between voltage and current in a pure inductive circuit?

   • Answer: $90^{\circ }$ (voltage leads current).

3. How does a transformer work? Give one application.

   • Answer: A transformer uses electromagnetic induction to step up or step down AC voltage. Application: Power transmission (step-up transformers increase voltage to reduce energy loss).

Numerical Problems

4. An AC supply has a peak voltage of 311 V. Calculate its RMS voltage.

   • Solution: $$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}=\frac{311}{\sqrt{2}}=220\text{V}$$

5. A 100-turn coil is connected to a 220 V AC supply. If the secondary coil has 500 turns, what is the output voltage?

   • Solution: $$\frac{V_s}{V_p}=\frac{N_s}{N_p}⟹V_s=\frac{500}{100}\times 220=1100\text{V}$$

6. In an LCR circuit, $L=0.5\text{H}$, $C=20\mu \text{F}$, and $R=100\Omega$. Find the resonant frequency.

   • Solution: $$f_0=\frac{1}{2\pi \sqrt{LC}}=\frac{1}{2\pi \sqrt{0.5\times 20\times 10^{-6}}}\approx 15.9\text{Hz}$$

Long Answer Questions

7. Explain the working principle of an AC generator. How is the frequency of the generated AC related to the speed of rotation?

   • Answer: An AC generator works on Faraday’s law of electromagnetic induction. A coil rotates in a uniform magnetic field, and the induced EMF is given by: $$V=V_0\sin (\omega t)$$ The frequency $f$ of the generated AC is related to the rotational speed $n$ (rpm) and the number of poles $P$ by: $$f=\frac{nP}{120}$$ Higher rotation speed or more poles increase the frequency.

8. Describe the construction and working of a step-down transformer. Why are transformers essential in power transmission?

   • Answer: A step-down transformer has more turns in the primary coil than the secondary coil. When an AC voltage is applied to the primary, a changing magnetic field is produced in the iron core, inducing a lower voltage in the secondary coil (due to fewer turns). Transformers are essential in power transmission because they:
     • Reduce energy loss by stepping up voltage for transmission (lower current = less $I^{2}R$ loss).
     • Step down voltage for safe household use (e.g., 220 V from 11 kV).

Summary Table: AC Components