Chapter 7 - Alternating Current

DC - Direct Current: Imagine a one-way road where cars always go in one direction with constant speed. Battery gives DC current flows from positive to negative constant direction constant magnitude if circuit steady.

AC - Alternating Current: Imagine a swing going to and fro, or sea waves going forward backward, or you are mopping floor to and fro. AC current direction reverses periodically. In India AC frequency 50 Hz means current changes direction 100 times per second (50 cycles forward 50 backward). Equation i = I0 sinωt where I0 peak max current, ω=2πf angular frequency.

Why we cannot measure average AC over full cycle? Full cycle average is zero because positive half cancels negative half like to and fro swing average displacement zero. So we use two other averages:

Half-cycle average: Average over half cycle I_avg = 2I0/π = 0.637 I0. This is average of positive half only like average speed of swing in one direction.

RMS - Root Mean Square: Most important for power. RMS is effective value that produces same heat as DC. Analogy: If AC and DC produce same brightness in same bulb, RMS of AC equals DC value. Formula I_rms = I0/√2 = 0.707 I0. Similarly V_rms = V0/√2.

Example: Household says 220V AC. This is RMS value. Peak value V0 = V_rms × √2 = 220×1.414=311V. So actual voltage swings from +311V to -311V but effective heating effect is 220V DC.

Day-to-day: When you pay electricity bill, meter measures RMS power because heat produced depends on RMS not peak.

When AC passes through pure resistor R, voltage and current are in phase - they reach peak together like two friends walking together hand in hand. Equation V=V0 sinωt I=I0 sinωt same phase no phase difference. Phasor diagram both arrows same direction.

Daily Analogy: Resistor is like friction in water pipe or roughness on road. Whether water flows forward or backward, friction always produces heat and opposes flow. Similarly resistor always produces heat regardless current direction to and fro AC heat produced both halves. Resistance R same for AC and DC.

Power: Instantaneous power P=VI fluctuates but average over cycle P_avg=V_rms I_rms = I_rms² R = V_rms²/R all electrical energy converted to heat. This is why heater, bulb work on AC same as DC.

Example: 100W bulb at 220V RMS, resistance R=V²/P=220²/100=484Ω same whether AC or DC.

When AC passes through pure inductor coil L, inductor opposes change in current due to self induction e=-L dI/dt. So current lags behind voltage by 90°. Equation V=V0 sinωt I=I0 sin(ωt-90°) current 90° behind voltage. Phasor diagram voltage arrow 90° ahead of current arrow.

Daily Analogy: Inductor is like heavy flywheel or heavy cart. When you try to push heavy cart, it takes time to start moving - velocity lags behind force. Similarly inductor has electrical inertia, current lags behind voltage. Heavy flywheel resists change in motion, inductor resists change in current.

Inductive reactance X_L = ωL = 2πfL ohm, opposition to AC. Depends on frequency f. For DC f=0 X_L=0 inductor acts as short circuit zero resistance wire. For AC high frequency high X_L high opposition. So inductor blocks high frequency AC but passes DC.

Power average zero over full cycle because energy stored in magnetic field quarter cycle then returned to source next quarter, no net heat loss. Current flows but no power loss called wattless current. Like pushing heavy swing, you give energy to swing it goes up then returns energy when coming down net zero over full cycle but motion exists.

Example: Choke coil used in tubelight is inductor with small resistance reduces AC current without much heat loss better than resistor which would waste heat.

When AC passes through pure capacitor C, capacitor blocks DC but passes AC. Current leads voltage by 90° voltage lags behind current 90°. Equation V=V0 sinωt I=I0 sin(ωt+90°) current 90° ahead. Phasor diagram current arrow 90° ahead of voltage arrow.

Daily Analogy: Capacitor is like flexible rubber membrane blocking water pipe. Water cannot flow continuously through membrane DC blocked, but if you push water to and fro AC, membrane stretches and relaxes allowing to-and-fro motion current flows. When you push membrane, water moves instantly but pressure builds slowly voltage lags. So current leads voltage. Like trampoline you jump (current) then membrane stretches and builds tension (voltage) later.

Capacitive reactance X_C=1/ωC=1/2πfC ohm opposition to AC. For DC f=0 X_C infinite open circuit blocks DC. For AC high frequency low X_C passes high frequency easily low frequency less. So capacitor blocks DC passes AC opposite to inductor.

Power average zero over full cycle energy stored in electric field between plates quarter cycle returned next quarter no net heat loss wattless current like inductor but electric field storage.

Example: Capacitor used in fan to give phase difference to start motor, also used to block DC while passing AC signals in electronic circuits like audio systems.

When resistor R, inductor L, capacitor C connected in series with AC source, total opposition is impedance Z=√(R²+(X_L-X_C)²) ohm. Current I0=V0/Z. Phase angle between V and I tanφ=(X_L-X_C)/R φ positive if inductive X_L>X_C voltage leads current, negative if capacitive X_C>X_L current leads voltage.

Phasor diagram: V_R in phase with I same direction, V_L 90° ahead I, V_C 90° behind I, net reactive voltage V_L-V_C. Total V = √(V_R²+(V_L-V_C)²).

Daily Analogy - Resonance: Imagine child on swing. If you push swing at random times, swing does not go high. But if you push at natural frequency of swing every time swing comes to you, amplitude becomes huge small push large motion resonance. Similarly LCR circuit has natural frequency ω0=1/√LC where X_L=X_C inductive and capacitive cancel each other impedance minimum Z=R current maximum I0=V0/R large. At resonance V_L and V_C equal and opposite cancel net voltage across L and C zero but individually large can be more than source voltage voltage magnification.

Resonance frequency f0=1/(2π√LC) Hz. At resonance power maximum because Z minimum. Resonance curve I vs f shows peak at f0 sharpness depends on R. Quality factor Q=ω0L/R=1/(ω0CR)=V_L/V_R at resonance magnification factor sharpness of resonance. High Q sharp peak narrow resonance good for tuning radio.

Example: Radio tuning - you change capacitor C to make circuit resonate at desired station frequency that station amplified others rejected.

Average power in AC P_avg=V_rms I_rms cosφ where φ phase between V and I cosφ called power factor. It tells how effectively power is used.

Pure R φ=0° cosφ=1 P max all power dissipated heat useful.

Pure L or C φ=90° cosφ=0 P=0 no power dissipated despite current flowing current flows but energy stored in magnetic/electric field quarter cycle returned next quarter net zero over full cycle called wattless current current flows but no power loss no heat like ideal swing you push but energy returns.

Daily Analogy: Power factor is like pushing a swing at right time vs wrong time. If you push swing when it moves away from you in phase max power swing goes high efficient. If you push when swing coming towards you 90° out of phase you waste effort no power swing does not go high despite pushing current flows but no power wattless.

Choke coil is inductor with small R used to reduce AC current in tubelight without much heat loss better than resistor which would waste heat I²R loss. Choke has high X_L=ωL reduces current but power loss small because small R cosφ small power loss mostly wattless.

Power factor correction: Many factories have inductive loads motors with low power factor lagging φ positive cosφ small more current needed for same power more I²R loss in wires higher electricity bill. Capacitor added in parallel with inductive load makes net X_L-X_C small φ near 0 cosφ near 1 improves power factor reduces current needed reduces bill. Like adding helper to push swing at right time correcting timing.

Transformer is device to change AC voltage using mutual induction. It has two coils primary and secondary wound on same laminated iron core to increase flux linking. Primary AC current produces changing magnetic flux in core which links secondary coil induces emf in secondary e2=-N2 dφ/dt.

Voltage ratio V2/V1=N2/N1 turns ratio. If N2>N1 V2>V1 step-up transformer increases voltage decreases current. If N2

Daily Analogy: Transformer is like bicycle gearbox. Gearbox changes speed and torque but power remains same (ignoring friction) speed×torque = power constant. Similarly transformer changes voltage and current but power same ideally V1 I1=V2 I2 power conserved. Step-up transformer like high gear high speed low torque high voltage low current, step-down like low gear low speed high torque low voltage high current. Power same product constant.

Ideal transformer power same P1=P2 V1 I1=V2 I2 no loss. Real transformer has losses: copper loss I²R in windings heat, iron losses hysteresis loop area energy loss per cycle and eddy currents in core heat. Efficiency η=P_out/P_in less than 100% typically 95-98% large transformers.

Uses: Power transmission - electricity generated at low voltage high current, transmitted at high voltage low current using step-up transformer to reduce I²R loss in long wires loss ∝ I²R so low I low loss, then at city step-down transformer to 220V safe for homes. This is why AC preferred over DC for long distance because AC voltage can be easily transformed DC cannot easily.

Laminated core reduces eddy currents thin sheets insulated stacked increasing resistance reducing eddy loss.