Alternating Current — JEE Mains Physics

Formulas (2)

• RMS value, meters and timing on a sine wave · RMS value and time on the wave
  $$I_{rms} = \frac{I_0}{\sqrt{2}} \qquad t = \frac{\Delta(\omega t)}{\omega}$$
• RMS of a sum: d.c. plus a.c., sine plus cosine · Mean squares add
  $$I_{rms} = \sqrt{I_{dc}^{2} + \frac{I_0^{2}}{2}} \qquad I_{rms}^{2} = \frac{1}{T}\int_0^{T} i^{2}\,dt$$

Watch out for (6)

• Peak to rms is T/8, not T/4
  → RMS value, meters and timing on a sine wave
• Taking the coefficient of t as the frequency
  → RMS value, meters and timing on a sine wave
• Using the peak where the meter reads rms
  → RMS value, meters and timing on a sine wave
• Adding rms values
  → RMS of a sum: d.c. plus a.c., sine plus cosine
• Averaging i instead of i²
  → RMS of a sum: d.c. plus a.c., sine plus cosine
• Adding the amplitudes of a sine and a cosine
  → RMS of a sum: d.c. plus a.c., sine plus cosine

Formulas (2)

• Reactance: ωL and 1/ωC · Reactances
  $$X_L = \omega L = 2\pi fL \qquad X_C = \frac{1}{\omega C} = \frac{1}{2\pi fC}$$
• A real coil on d.c. and a.c., and the frequency limits · Coil on d.c. and on a.c.
  $$R = \frac{V_{dc}}{I_{dc}} \qquad Z = \frac{V_{rms}}{I_{rms}} = \sqrt{R^{2} + X_L^{2}}$$

Reference tables (1)

Watch out for (8)

• Making the capacitive reactance rise with frequency
  → Reactance: ωL and 1/ωC
• Multiplying a given ω by 2π
  → Reactance: ωL and 1/ωC
• Peak or rms
  → Reactance: ωL and 1/ωC
• Which one leads in an inductor
  → Phase of the current in R, L and C
• Adding 90° to the wrong quantity
  → Phase of the current in R, L and C
• Taking V/I on a.c. as the reactance
  → A real coil on d.c. and a.c., and the frequency limits
• Power in the inductance
  → A real coil on d.c. and a.c., and the frequency limits
• Average or peak stored energy
  → A real coil on d.c. and a.c., and the frequency limits

Formulas (3)

• Impedance and the voltage triangle · Impedance and voltages
  $$Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \qquad V^{2} = V_R^{2} + (V_L - V_C)^{2}$$
• Phase angle and power factor · Phase angle and power factor
  $$\tan\phi = \frac{X_L - X_C}{R} \qquad \cos\phi = \frac{R}{Z}$$
• Average power, wattless current and the choke coil · Average power
  $$P = V_{rms}I_{rms}\cos\phi = \frac{1}{2}V_0I_0\cos\phi = I_{rms}^{2}R$$

Watch out for (8)

• Adding the voltages as numbers
  → Impedance and the voltage triangle
• Changing only one reactance with the frequency
  → Impedance and the voltage triangle
• Mixing peak and rms
  → Impedance and the voltage triangle
• Using sin φ for the power factor
  → Phase angle and power factor
• Losing lead or lag
  → Phase angle and power factor
• Letting the amplitude change the power factor
  → Phase angle and power factor
• Peak values in the rms formula
  → Average power, wattless current and the choke coil
• Using Z in I²R
  → Average power, wattless current and the choke coil

Formulas (3)

• The resonant frequency · Resonant frequency
  $$\omega_0 = \frac{1}{\sqrt{LC}} \qquad f_0 = \frac{1}{2\pi\sqrt{LC}}$$
• Current, impedance and power at resonance · At resonance
  $$Z = R \qquad I_0 = \frac{V_0}{R} \qquad P_{max} = \frac{V_{rms}^{2}}{R}$$
• Quality factor and bandwidth · Quality factor and bandwidth
  $$Q = \frac{\omega_0L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} \qquad \Delta\omega = \frac{R}{L} = \frac{\omega_0}{Q}$$

Watch out for (9)

• ω or f
  → The resonant frequency
• Putting R into the formula
  → The resonant frequency
• Losing a power of ten
  → The resonant frequency
• Forgetting √2 on the amplitude
  → Current, impedance and power at resonance
• Using L and C to find the resonant current
  → Current, impedance and power at resonance
• Reading the I–ω curve the wrong way round
  → Current, impedance and power at resonance
• Half power is not half current
  → Quality factor and bandwidth
• Q with f₀ instead of ω₀
  → Quality factor and bandwidth
• Bandwidth in rad/s or in Hz
  → Quality factor and bandwidth

Formulas (2)

• LC oscillations · LC oscillator
  $$\omega = \frac{1}{\sqrt{LC}} \qquad I_{max} = \frac{Q_0}{\sqrt{LC}} = V_0\sqrt{\frac{C}{L}}$$
• Transformers and a.c. devices · Transformer
  $$\frac{V_s}{V_p} = \frac{N_s}{N_p} \qquad V_sI_s = \eta\,V_pI_p$$

Watch out for (4)

• √(C/L) or √(L/C)
  → LC oscillations
• Charge fraction and energy fraction
  → LC oscillations
• Turning the ratio upside down
  → Transformers and a.c. devices
• Forgetting the efficiency on the current
  → Transformers and a.c. devices