Alternating Current — JEE Mains Physics Formulas and Traps

RMS Values and Timing in AC

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

Common traps

Peak to rms is T/8, not T/4

From the peak, the wave must lose a phase of π/4 to fall to 1/√2 of the peak. That is one eighth of a cycle. A quarter cycle takes it all the way to zero.

Taking the coefficient of t as the frequency

In sin(200πt) the coefficient 200π is ω in rad/s. The frequency is ω/2π = 100 Hz. Do not multiply by 2π again.

Using the peak where the meter reads rms

A meter reading or a rating like '220 V' is an rms value. Multiply by √2 only when the question asks for a peak.

Adding rms values

For 3 + 4√2 sin ωt the rms is √(9 + 16) = 5, not 3 + 4 = 7. Mean squares add; rms values do not.

Averaging i instead of i²

The plain average of a sinusoid over a cycle is zero. The rms comes from the average of the SQUARE, and the root is taken last.

Adding the amplitudes of a sine and a cosine

sin ωt and cos ωt are 90° apart, so their amplitudes combine like the sides of a right triangle: √(I₁² + I₂²), not I₁ + I₂.

Reactance of a Resistor, Inductor and Capacitor

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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}}

Phase of the current in R, L and C

Element	Opposition	Change with frequency	Current compared with voltage	Average power
Pure resistor	R	None	In phase	$V_{rms}I_{rms}$
Pure inductor	$X_L = \omega L$	Grows in proportion to f	Lags by $\dfrac{\pi}{2}$	Zero
Pure capacitor	$X_C = \dfrac{1}{\omega C}$	Falls as $\dfrac{1}{f}$	Leads by $\dfrac{\pi}{2}$	Zero
Ideal L and C in series	$\|X_L - X_C\|$	Falls to zero at resonance	Lags by $\dfrac{\pi}{2}$ if $X_L > X_C$ , leads if $X_C > X_L$	Zero No resistance anywhere, so the gap is exactly 90° whichever reactance wins.
Series LCR	$\sqrt{R^{2} + (X_L - X_C)^{2}}$	Least at resonance	Angle $\phi$ with $\tan\phi = \dfrac{X_L - X_C}{R}$	$V_{rms}I_{rms}\cos\phi$

Current zero while the voltage is at its peak means a 90° gap: no resistance in the circuit.

Common traps

Making the capacitive reactance rise with frequency

Capacitive reactance goes DOWN as f or C goes up. Doubling both divides it by 4.

Multiplying a given ω by 2π

In sin(500t) the 500 is already ω in rad/s. Use 2πf only when the frequency is given in hertz.

Peak or rms

An emf written as E₀ sin ωt gives the PEAK current. An a.c. ammeter shows the rms value, √2 times smaller.

Which one leads in an inductor

In an inductor the VOLTAGE leads the current. Saying 'the current leads' is the capacitor's rule. CIVIL settles it.

Adding 90° to the wrong quantity

Going from current to voltage in an inductor, add π/2. Going from voltage to current, subtract it. For a capacitor it is the other way round.

Taking V/I on a.c. as the reactance

V/I on a.c. is the impedance, which still contains the coil's resistance. Find R on d.c. first, then

X_L = \sqrt{Z^{2} - R^{2}}

.

Power in the inductance

Only the resistance takes power: P = I²R with the rms current. I²Z overstates it.

Average or peak stored energy

½LI² with the rms current is the energy averaged over a cycle; with the peak current it is the maximum. The two differ by a factor of 2.

Series LCR: Impedance, Phase and Power

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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

\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

Common traps

Adding the voltages as numbers

V_R + V_L + V_C

is not the supply voltage. The parts are out of phase, so use

V^{2} = V_R^{2} + (V_L - V_C)^{2}

.

Changing only one reactance with the frequency

When ω changes,

X_L

and

X_C

both change, in opposite directions. Scale both before finding Z.

Mixing peak and rms

Peak voltage over Z gives the peak current; rms over Z gives rms. A 'percent lower' frequency means ω times the remaining fraction: 30% lower is 0.7ω.

Using sin φ for the power factor

The power factor is cos φ = R/Z. The ratio of the net reactance to Z is sin φ.

Losing lead or lag

cos φ is the same for +φ and −φ. Decide lead or lag from which reactance is bigger, or from which equation has the larger phase constant.

Letting the amplitude change the power factor

A new source amplitude changes the current, not the angle. Only a new frequency or a new part changes the power factor.

Peak values in the rms formula

V_{rms}I_{rms}\cos\phi

needs rms values. With peak values, halve the product:

\dfrac{1}{2}V_0I_0\cos\phi

.

Using Z in I²R

Power is

I_{rms}^{2}R

. Writing

I^{2}Z

charges the inductor and capacitor for power they never keep.

Resonance, Quality Factor and Bandwidth

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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

Q = \frac{\omega_0L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} \qquad \Delta\omega = \frac{R}{L} = \frac{\omega_0}{Q}

Common traps

ω or f

\dfrac{1}{\sqrt{LC}}

is ω in rad/s. For hertz, divide by 2π. Check which one the blank asks for.

Putting R into the formula

The resistance changes how tall and wide the resonance peak is, never where it sits.

Losing a power of ten

Convert first:

1\ \mu\text{F} = 10^{-6}\ \text{F}

,

1\ \text{nF} = 10^{-9}\ \text{F}

,

1\ \text{mH} = 10^{-3}\ \text{H}

. Most wrong answers here are off by a power of ten.

Forgetting √2 on the amplitude

A supply quoted as '220 V' is rms. At resonance V/R is then the rms current; the amplitude is √2 times that.

Using L and C to find the resonant current

Once the circuit is at resonance, L and C have cancelled. The current depends only on V and R.

Reading the I–ω curve the wrong way round

On the low-frequency side

X_C

is large, so the circuit is capacitive. On the high side

X_L

wins.

Half power is not half current

At the edges of the band the CURRENT is 1/√2 of its peak. Power goes as I², so the power is half.

Q with f₀ instead of ω₀

Q = ω₀L/R uses the angular frequency. Using f₀ makes Q too small by a factor of 2π.

Bandwidth in rad/s or in Hz

R/L is a bandwidth in rad/s. In hertz it is R/(2πL).

LC Oscillations, Transformers and AC Devices

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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

Common traps

√(C/L) or √(L/C)

From ½CV₀² = ½LI², the current is V₀√(C/L). A large capacitor or a small inductor gives a large current.

Charge fraction and energy fraction

Energy goes as q². Half the charge leaves a quarter of the energy in the capacitor, not half.

Turning the ratio upside down

Voltage follows the turns: more turns, more volts. Current goes the other way. Write

\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}

before putting numbers in.

Forgetting the efficiency on the current

With losses, the output power is η times the input. Find the output current from that power, not from the ideal turns ratio.

Alternating Current formulas

RMS Values and Timing in AC

RMS value, meters and timing on a sine wave

RMS of a sum: d.c. plus a.c., sine plus cosine

Common traps

Reactance of a Resistor, Inductor and Capacitor

Reactance: ωL and 1/ωC

A real coil on d.c. and a.c., and the frequency limits

Phase of the current in R, L and C

Common traps

Series LCR: Impedance, Phase and Power

Impedance and the voltage triangle

Average power, wattless current and the choke coil

Common traps

Resonance, Quality Factor and Bandwidth

The resonant frequency

Current, impedance and power at resonance

Common traps

LC Oscillations, Transformers and AC Devices

LC oscillations

Transformers and a.c. devices

Common traps

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