Alternating Current — Revision Notes

⚡ 30-Second Revision

AC vs DC: — AC reverses direction, DC is unidirectional.\n- **Peak Value ($V_m, I_m$): Maximum value.\n- RMS Value ($V_{rms}, I_{rms}$):** Effective value, $V_{rms} = V_m/\sqrt{2}$, $I_{rms} = I_m/\sqrt{2}$.\n- Average Value (full cycle): Zero for sinusoidal AC.\n- Inductive Reactance: $X_L = \omega L = 2\pi f L$. Current lags voltage by $90^\circ$.\n- Capacitive Reactance: $X_C = 1/(\omega C) = 1/(2\pi f C)$. Current leads voltage by $90^\circ$.\n- Impedance (Series RLC): $Z = \sqrt{R^2 + (X_L - X_C)^2}$.\n- **Phase Angle ($\phi$):** $\tan\phi = (X_L - X_C)/R$.\n- Resonant Frequency: $f_0 = 1/(2\pi\sqrt{LC})$. At resonance, $X_L = X_C$, $Z=R$, $I_{max}$, $\phi=0$, $\cos\phi=1$.\n- Q-factor: $Q = (\omega_0 L)/R = (1/R)\sqrt{L/C}$.\n- Average Power: $P_{avg} = V_{rms} I_{rms} \cos\phi$. $\cos\phi$ is power factor.\n- Apparent Power: $S = V_{rms} I_{rms}$.

2-Minute Revision

Alternating Current (AC) is characterized by its periodically reversing direction and continuously changing magnitude, typically in a sinusoidal pattern. Key parameters include peak value ($V_m, I_m$) and RMS value ($V_{rms} = V_m/\sqrt{2}$), which represents the effective DC equivalent.

The average value over a full cycle is zero. \n\nIn AC circuits, resistors (R) are in phase with current, inductors (L) cause current to lag voltage by $90^\circ$ ($X_L = \omega L$), and capacitors (C) cause current to lead voltage by $90^\circ$ ($X_C = 1/(\omega C)$).

\n\nFor a series RLC circuit, the total opposition is impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$. The phase angle $\phi$ between voltage and current is given by $\tan\phi = (X_L - X_C)/R$. \n\nResonance occurs when $X_L = X_C$, leading to a resonant frequency $f_0 = 1/(2\pi\sqrt{LC})$.

At resonance, impedance is minimum ($Z=R$), current is maximum, and the power factor ($\cos\phi$) is unity. The Q-factor ($Q = (\omega_0 L)/R$) describes the sharpness of resonance. \n\nAverage power dissipated in an AC circuit is $P_{avg} = V_{rms} I_{rms} \cos\phi$, where $\cos\phi$ is the power factor.

Apparent power is $V_{rms} I_{rms}$.

5-Minute Revision

Alternating Current (AC) is the backbone of modern power systems, distinguished by its sinusoidal variation in magnitude and periodic reversal of direction. Unlike DC, AC can be efficiently transmitted over long distances by stepping up/down voltage using transformers.

\n\nKey Definitions: \n* **Peak Value ($V_m, I_m$):** Maximum instantaneous value. \n* **RMS Value ($V_{rms}, I_{rms}$):** Effective value, $V_{rms} = V_m/\sqrt{2}$, $I_{rms} = I_m/\sqrt{2}$. This is what meters read and what's used for power calculations.

\n* Average Value: For a full cycle, it's zero. For a half-cycle, $I_{avg} = 2I_m/\pi$. \n* **Frequency (f) & Angular Frequency ($\omega$):** $f$ is cycles/sec, $\omega = 2\pi f$. \n\nComponent Behavior in AC: \n* Resistor (R): Voltage and current are in phase.

$V_R = IR$. \n* Inductor (L): Current lags voltage by $90^\circ$. Opposition is inductive reactance $X_L = \omega L$. $V_L = IX_L$. \n* Capacitor (C): Current leads voltage by $90^\circ$. Opposition is capacitive reactance $X_C = 1/(\omega C)$.

$V_C = IX_C$. \n\nSeries RLC Circuit: \n* Impedance (Z): Total opposition to current. $Z = \sqrt{R^2 + (X_L - X_C)^2}$. \n* **Phase Angle ($\phi$):** Phase difference between source voltage and current.

$\tan\phi = (X_L - X_C)/R$. If $X_L > X_C$, current lags (inductive circuit). If $X_C > X_L$, current leads (capacitive circuit). \n* Source Voltage: $V_{source} = \sqrt{V_R^2 + (V_L - V_C)^2}$. \n\nExample: A series RLC circuit has $R=60\,\Omega$, $L=0.

2\,\text{H}$,$C=50\,\mu\text{F}$connected to$V=100\sin(500t)\,\text{V}$. \n1.$\omega = 500\,\text{rad/s}$. \n2.$X_L = \omega L = 500 \times 0.2 = 100\,\Omega$. \n3.$X_C = 1/(\omega C) = 1/(500 \times 50 \times 10^{-6}) = 1/(25000 \times 10^{-6}) = 1/(0.

025) = 40\,\Omega$. \n4.$Z = \sqrt{60^2 + (100 - 40)^2} = \sqrt{60^2 + 60^2} = \sqrt{2 \times 60^2} = 60\sqrt{2}\,\Omega$. \n5.$I_m = V_m/Z = 100 / (60\sqrt{2}) = 5/(3\sqrt{2})\,\text{A}$. \n\n**Resonance:** Occurs when$X_L = X_C$.

\n* Resonant frequency: $f_0 = 1/(2\pi\sqrt{LC})$. \n* At resonance: $Z=R$ (minimum), $I_{max}$, $\phi=0$, $\cos\phi=1$. \n* Q-factor: $Q = (\omega_0 L)/R = (1/R)\sqrt{L/C}$. Measures sharpness of resonance.

\n\nPower in AC: \n* Average Power (True Power): $P_{avg} = V_{rms} I_{rms} \cos\phi$. Dissipated only in resistor. \n* **Power Factor ($\cos\phi$):** Ratio of true power to apparent power. Ranges from 0 to 1.

\n* Apparent Power: $S = V_{rms} I_{rms}$. Total power supplied. \n* Reactive Power: $Q = V_{rms} I_{rms} \sin\phi$. Power exchanged between source and reactive components.

Prelims Revision Notes

Alternating Current (AC) - NEET Revision Notes\n\n1. Basic Definitions & Values:\n* AC: Current/voltage varies sinusoidally with time, reversing direction periodically. $V = V_m \sin(\omega t + \phi_V)$, $I = I_m \sin(\omega t + \phi_I)$.\n* Peak Value ($V_m, I_m$): Maximum instantaneous value.\n* RMS Value ($V_{rms}, I_{rms}$): Effective value, equivalent to DC for power dissipation.\n * For sinusoidal AC: $V_{rms} = V_m/\sqrt{2} \approx 0.707 V_m$; $I_{rms} = I_m/\sqrt{2} \approx 0.707 I_m$.\n* Average Value: Over a full cycle, average of sinusoidal AC is zero. Over a half-cycle: $V_{avg} = 2V_m/\pi \approx 0.637 V_m$; $I_{avg} = 2I_m/\pi \approx 0.637 I_m$.\n* Angular Frequency ($\omega$): $\omega = 2\pi f = 2\pi/T$. (Units: rad/s)\n\n2. AC Circuits with Pure Components:\n* Resistor (R):\n * Voltage and current are in phase ($\phi = 0$).\n * $V_R = I_R R$.\n * Power factor $\cos\phi = 1$.\n* Inductor (L):\n * Current lags voltage by $90^\circ$ (or $\pi/2$ rad). (Voltage leads current)\n * Inductive Reactance: $X_L = \omega L = 2\pi f L$. (Units: Ohms, $\Omega$)\n * $V_L = I_L X_L$.\n * Average power consumed is zero.\n* Capacitor (C):\n * Current leads voltage by $90^\circ$ (or $\pi/2$ rad). (Voltage lags current)\n * Capacitive Reactance: $X_C = 1/(\omega C) = 1/(2\pi f C)$. (Units: Ohms, $\Omega$)\n * $V_C = I_C X_C$.\n * Average power consumed is zero.\n * Mnemonic: CIVIL (Capacitor: Current Leads Voltage; Inductor: Voltage Leads Current)\n\n3. Series RLC Circuit:\n* Impedance (Z): Total effective opposition to current.\n * $Z = \sqrt{R^2 + (X_L - X_C)^2}$.\n* Current: $I_{rms} = V_{rms}/Z$; $I_m = V_m/Z$.\n* Phase Angle ($\phi$): Phase difference between source voltage and current.\n * $\tan\phi = (X_L - X_C)/R$.\n * If $X_L > X_C$: Circuit is inductive, current lags voltage ($\phi > 0$).\n * If $X_C > X_L$: Circuit is capacitive, current leads voltage ($\phi < 0$).\n * If $X_L = X_C$: Circuit is resistive, current is in phase with voltage ($\phi = 0$).\n* Voltage across components: $V_{source} = \sqrt{V_R^2 + (V_L - V_C)^2}$.\n\n4. Resonance in Series RLC Circuit:\n* Condition: $X_L = X_C$.\n* Resonant Frequency ($\omega_0, f_0$):\n * $\omega_0 = 1/\sqrt{LC}$.\n * $f_0 = 1/(2\pi\sqrt{LC})$.\n* At Resonance:\n * Impedance $Z = R$ (minimum value).\n * Current $I_{max} = V_{rms}/R$ (maximum value).\n * Phase angle $\phi = 0$.\n * Power factor $\cos\phi = 1$ (unity).\n * $V_L = V_C$ (magnitudes are equal, but still $180^\circ$ out of phase).\n* Q-factor (Quality Factor): Measures sharpness of resonance.\n * $Q = (\omega_0 L)/R = 1/(\omega_0 C R) = (1/R)\sqrt{L/C}$.\n\n5. Power in AC Circuits:\n* Instantaneous Power: $P(t) = V(t)I(t)$.\n* Average Power (True Power): Power actually dissipated (in resistor).\n * $P_{avg} = V_{rms} I_{rms} \cos\phi$. (Units: Watts, W)\n* Power Factor ($\cos\phi$): Ratio of true power to apparent power.\n * $\cos\phi = R/Z$.\n* Apparent Power (S): Total power supplied by source.\n * $S = V_{rms} I_{rms}$. (Units: Volt-Amperes, VA)\n* Reactive Power (Q): Power exchanged between source and reactive components.\n * $Q = V_{rms} I_{rms} \sin\phi$. (Units: Volt-Ampere Reactive, VAR)\n\n6. LC Oscillations:\n* Energy oscillates between electric field of capacitor ($U_E = \frac{1}{2}CV^2$) and magnetic field of inductor ($U_B = \frac{1}{2}LI^2$).\n* Total energy is conserved in an ideal LC circuit.\n* Angular frequency of oscillation: $\omega = 1/\sqrt{LC}$.\n\n7. Transformer:\n* Works on mutual induction, only with AC.\n* Ideal Transformer: $V_S/V_P = N_S/N_P = I_P/I_S$. (S = secondary, P = primary)\n* Efficiency ($\eta$): $\eta = (P_{out}/P_{in}) \times 100\% = (V_S I_S \cos\phi_S) / (V_P I_P \cos\phi_P) \times 100\%$.

Vyyuha Quick Recall

To remember phase relationships in AC circuits, use 'CIVIL': \n\nCapacitor: Current In Voltage Is Leading. (Current leads Voltage) \nInductor: In Voltage Is Leading Current. (Voltage leads Current)