Summary & Master Memory Capsule

Master Formula Sheet

#	Result	Formula
1	rms values	$I = i_m/\sqrt2$, $V = v_m/\sqrt2$ (mains: 220 V rms, 311 V peak)
2	Resistor power	$P = I^2R = IV = V^2/R$ (v, i in phase)
3	Inductive reactance	$X_L = \omega L = 2\pi\nu L$ (current lags 90 degrees)
4	Capacitive reactance	$X_C = 1/\omega C$ (current leads 90 degrees)
5	LCR impedance	$Z = \sqrt{R^2 + (X_C - X_L)^2}$, $i_m = v_m/Z$
6	Phase angle	$\tan\phi = (X_C - X_L)/R$; $\cos\phi = R/Z$
7	Resonance	$\omega_0 = 1/\sqrt{LC}$; $Z = R$; $I_{max} = V/R$; $\phi = 0$
8	Average power	$P = VI\cos\phi = I^2Z\cos\phi = I^2R$
9	Wattless current	pure L or C: $\phi = 90$ degrees, $P = 0$
10	Transformer	$v_s/v_p = N_s/N_p$; ideal: $i_pv_p = i_sv_s$

Units: reactance and impedance in ohms; 1/(rad/s x F) = ohm; rad/s x H = ohm.

The Phase-Relation Table (Memorise Cold)

Circuit	Phase of current vs voltage	Opposition	Average power
Pure R	In phase ($\phi = 0$)	R	$I^2R$ (max)
Pure L	Lags by 90 degrees	$X_L = \omega L$ (rises with $\nu$)	Zero
Pure C	Leads by 90 degrees	$X_C = 1/\omega C$ (falls with $\nu$)	Zero
Series LCR	lags/leads by $\phi$, $\tan\phi = \frac{X_C - X_L}{R}$	Z	$VI\cos\phi$ (in R only)
LCR at resonance	In phase	R (minimum)	$V^2/R$ (max)

Hook: 'L lags, C leads, R pays the bills.' Phasor rules: $V_R \parallel I$; $V_L$ ahead 90; $V_C$ behind 90; add as vectors, never as numbers.

Resonance & Power Facts

• $X_L$ rises and $X_C$ falls with frequency — they cross at $\omega_0 = 1/\sqrt{LC}$: minimum Z, maximum current, $\cos\phi = 1$.
• $V_L$ and $V_C$ cancel exactly at resonance, yet each can be many times the source voltage (voltage magnification $X_L/R$).
• No resonance in RL or RC circuits — cancellation needs both reactances.
• R changes the peak's height ($I_{max} = V/R$) and sharpness, not the resonant frequency.
• Applications: radio/TV tuning (vary C), airport metal detectors (metal shifts L, circuit detunes).
• Instantaneous power oscillates at $2\nu$ (100 Hz on Indian mains); average power needs $\cos\phi$.
• Low power factor at fixed P, V forces large current: line losses $I^2R$ grow as $1/\cos^2\phi$ — fix with parallel capacitors.

Transformer Capsule & Revision Flow

Transformer: mutual induction on a soft-iron core; same $d\phi/dt$ per turn → $v_s/v_p = N_s/N_p$; ideal power balance $i_pv_p = i_sv_s$; step-up trades current for voltage; AC only. Losses → remedies: flux leakage → overlapped windings; winding resistance → thick wire; eddy currents → laminations; hysteresis → soft magnetic core. Grid story: step up at the plant ($I^2R$ loss falls as $V^{-2}$), step down in stages to 240 V.

The chapter in six steps:

1. rms makes AC arithmetic look like DC ($I = i_m/\sqrt2$).
2. Phasors encode phase: R in phase, L lags, C leads — reactances $\omega L$ and $1/\omega C$, both lossless.
3. Series LCR: $Z = \sqrt{R^2+(X_C-X_L)^2}$, $\tan\phi = (X_C-X_L)/R$; voltages add as phasors.
4. Resonance at $1/\sqrt{LC}$: Z = R, current peaks, tuning works.
5. Power: $P = VI\cos\phi$, all of it in R; wattless current at 90 degrees.
6. Transformers move power across the country by trading voltage against current.

Morning-of-exam checklist: 220 V is rms … peak power = 2 x average in a resistor … reactance graphs: $X_L$ straight line, $X_C$ hyperbola … never add rms voltages algebraically … at resonance pf = 1 and Z = R … wattless = 90 degrees … transformer: V up, I down … lamination kills eddy currents. Now go score.