Core Concepts
Build AC from first principles
This is not a textbook dump. Every concept is built from why it exists, not just what it is.
By the end of this section, you'll understand:
- Why AC is generated (not DC)
- What RMS really means (and why we don't use peak values)
- How phasors make AC simple
- Why current leads/lags voltage in capacitors/inductors
- What resonance actually is (and why it matters)
1. AC Basics: Why Alternating Current?
Why do we use AC and not DC for power transmission?
Because AC voltage can be stepped up/down easily using transformers. This reduces transmission losses (P_loss = I²R). Lower current means lower loss.
AC is generated when a coil rotates in a magnetic field (generator principle).
Using Faraday's law:
Standard Form:
CBSE loves to ask: "Derive expression for EMF generated by rotating coil."
Key steps: Flux → Faraday's law → Derivative → Sinusoidal form
| Quantity | Equation | Meaning |
|---|---|---|
| AC Voltage | V = V₀ sin(ωt + φ) | Oscillates sinusoidally with time |
| AC Current | I = I₀ sin(ωt + φ) | Current also oscillates |
| Angular Frequency | ω = 2πf | Rate of oscillation (rad/s) |
| Phase | φ | Initial angle at t=0 |
Students write: V = V₀ sin(t)
Correct: V = V₀ sin(ωt)
The argument of sine must be dimensionless. Since ω has units rad/s, ωt is dimensionless.
2. RMS and Average Values
Problem: Peak value exists only for an instant. It doesn't represent the "effective" value.
Solution: Use RMS (Root Mean Square) - the DC equivalent that produces same heating effect.
For sinusoidal AC: I = I₀ sin(ωt)
Step 1: Power dissipated in resistor R:
Step 2: Average power over one cycle:
Step 3: Since ⟨sin²(ωt)⟩ = 1/2:
Step 4: For equivalent DC current I_rms:
Final Result:
JEE twist: They'll give you V_rms and ask for power. Don't multiply by √2!
Remember: Household supply "220V" is already RMS, not peak.
Average over full cycle:
Average over half cycle:
Comparison:
| Value | Formula | Ratio to Peak |
|---|---|---|
| Peak | I₀ | 1.0 |
| RMS | I₀/√2 | 0.707 |
| Average (half cycle) | 2I₀/π | 0.637 |
Wrong: Using V₀ in power formula P = V²/R
Correct: Use V_rms in P = V_rms²/R
Rule: For power calculations, ALWAYS use RMS values.
3. Phasors: The Visual Key to AC
Problem: Adding sinusoidal quantities algebraically is messy.
Solution: Represent them as rotating vectors (phasors).
A phasor is a vector whose projection on vertical axis gives the instantaneous value.
For V = V₀ sin(ωt):
- Draw a vector of length V₀
- Rotating anticlockwise with angular velocity ω
- Its vertical projection = V₀ sin(ωt)
Phase Difference:
If V = V₀ sin(ωt) and I = I₀ sin(ωt + φ):
- φ > 0: Current leads voltage
- φ < 0: Current lags voltage
- φ = 0: In phase
They'll give you phasor diagrams and ask:
- What is the circuit element?
- Calculate phase angle from diagram
- Find resultant voltage/current
Master phasor addition using head-to-tail method.
| Element | Phase Relation | Description |
|---|---|---|
| Resistor (R) | V and I in phase (φ = 0) | Both phasors point same direction |
| Inductor (L) | V leads I by 90° (φ = +90°) | Voltage phasor ahead of current |
| Capacitor (C) | I leads V by 90° (φ = -90°) | Current phasor ahead of voltage |
Memory Trick: "CIVIL"
- C: In Capacitor, I leads V
- L: In Inductor, V leads I
Wrong thinking: "Current leads voltage" means current is larger.
Correct: It means current reaches its peak value earlier in time.
Phase is about timing, not magnitude.
4. Reactance and Impedance
- Resistance (R): Opposes current, dissipates energy (in DC and AC)
- Reactance (X): Opposes current, stores energy (only in AC)
- Impedance (Z): Total opposition in AC circuit (combines R and X)
Why does inductor oppose AC?
Because changing current induces back EMF: ε = -L(dI/dt)
Derivation:
For I = I₀ sin(ωt):
Peak voltage across L:
Define Inductive Reactance:
Key Point: X_L ∝ f (Higher frequency → More opposition)
Why does capacitor oppose AC?
Because it takes time to charge/discharge: Q = CV
Derivation:
For V = V₀ sin(ωt), charge Q = CV₀ sin(ωt)
Current I = dQ/dt = CV₀ω cos(ωt) = CV₀ω sin(ωt + 90°)
Peak current:
Define Capacitive Reactance:
Key Point: X_C ∝ 1/f (Higher frequency → Less opposition)
JEE Main frequently asks: "A circuit has X_L = 100Ω and X_C = 60Ω. What happens when frequency doubles?"
Answer strategy: X_L doubles, X_C halves. Net reactance changes from (100-60=40Ω inductive) to (200-30=170Ω inductive)
For series LCR circuit:
Phase angle:
Three cases:
- X_L > X_C: Inductive circuit (current lags)
- X_C > X_L: Capacitive circuit (current leads)
- X_L = X_C: Resonance (current in phase)
Wrong: Z = R + X_L + X_C (Algebraic addition)
Correct: Z = √[R² + (X_L - X_C)²] (Vector addition)
Why? Because R, X_L, and X_C are not in the same direction in phasor diagram.
5. LCR Circuit Analysis
Given: AC source V = V₀ sin(ωt) connected to R, L, C in series
Voltage across each:
- V_R = IR (in phase with I)
- V_L = IX_L (leads I by 90°)
- V_C = IX_C (lags I by 90°)
Phasor Addition:
V_L and V_C are opposite, so net reactive voltage = (V_L - V_C)
Current:
CBSE asks: "Draw phasor diagram for series LCR circuit when (a) X_L > X_C (b) X_C > X_L"
Key: Always take current as reference phasor (horizontal). Then add voltage phasors.
| Quantity | Formula |
|---|---|
| Impedance | Z = √[R² + (X_L - X_C)²] |
| Current amplitude | I₀ = V₀/Z |
| Phase angle | tan φ = (X_L - X_C)/R |
| Power factor | cos φ = R/Z |
- Calculate X_L = ωL and X_C = 1/(ωC)
- Find net reactance: X = X_L - X_C
- Calculate impedance: Z = √(R² + X²)
- Find current: I = V/Z
- Find phase: tan φ = X/R
6. Resonance in AC Circuits
Condition: When X_L = X_C, the circuit is purely resistive.
Effect: Impedance is minimum (Z = R), current is maximum.
Why it matters: Used in radio tuning, filters, signal amplification.
Condition for resonance:
Substitute:
Solve for ω₀:
Resonant Angular Frequency:
Resonant Frequency:
Because it connects three concepts:
- Frequency dependence of X_L and X_C
- Condition for maximum current
- Dimensional analysis (check: [f] = [T⁻¹])
| Property | Value at Resonance |
|---|---|
| Impedance | Z = R (minimum) |
| Current | I = V/R (maximum) |
| Phase angle | φ = 0 (V and I in phase) |
| Power factor | cos φ = 1 (maximum) |
| V_L and V_C | Can be >> V (voltage magnification) |
Wrong: "At resonance, V_L = V_C = 0"
Correct: "At resonance, V_L = V_C but both can be very large!"
Why? V_L and V_C cancel each other in phasor sum, but individually can be much larger than source voltage.
This is called voltage magnification: Q = V_L/V = V_C/V can be > 100!
Definition: Measures sharpness of resonance
Alternative form:
7. Power in AC Circuits
In DC: P = VI (simple)
In AC: V and I may not be in phase, so instantaneous power oscillates.
We care about average power over one cycle.
Given: V = V₀ sin(ωt) and I = I₀ sin(ωt + φ)
Instantaneous power:
Using trigonometry:
sin A sin B = (1/2)[cos(A-B) - cos(A+B)]
Average over one cycle:
⟨cos(2ωt + φ)⟩ = 0
CBSE awards 3-4 marks for this derivation.
Key steps they look for:
- Write P(t) = V(t)I(t)
- Use product-to-sum formula
- Take time average
- Convert to RMS form
Definition:
Three cases:
| Circuit | φ | cos φ | Power |
|---|---|---|---|
| Pure R | 0° | 1 | Maximum |
| Pure L or C | 90° | 0 | Zero |
| RLC (general) | 0° to 90° | 0 to 1 | Intermediate |
Why low power factor is bad:
High current but low power → More transmission losses (I²R)
Wrong: P = V_rms × I_rms (forgetting cos φ)
Correct: P = V_rms × I_rms × cos φ
Special case: Only for pure resistor, φ = 0 so cos φ = 1
In pure L or C: cos φ = 0, so P = 0
Current flows but no power is consumed (energy oscillates between source and component)
This is called wattless current.
Practical importance:
Power companies don't like wattless current (wastes transmission capacity). They charge extra if power factor is too low.
You've now built AC from scratch. Key takeaways:
- AC is sinusoidal: V = V₀ sin(ωt)
- Use RMS values for power: V_rms = V₀/√2
- Phasors visualize phase relationships
- Reactance depends on frequency: X_L = ωL, X_C = 1/(ωC)
- Impedance combines resistance and reactance: Z = √[R² + (X_L - X_C)²]
- Resonance occurs when X_L = X_C: f₀ = 1/(2π√(LC))
- Power depends on phase: P = V_rms I_rms cos φ