AC Voltage and Current — Revision Notes

⚡ 30-Second Revision

Instantaneous Voltage/Current — $V = V_0 sin(omega t + phi)$, $I = I_0 sin(omega t + phi')$
Peak Value — $V_0, I_0$ (maximum amplitude)
Angular Frequency — $omega = 2pi f$
Frequency — $f = 1/T$
RMS Value — $V_{rms} = V_0/sqrt{2} approx 0.707 V_0$, $I_{rms} = I_0/sqrt{2} approx 0.707 I_0$
Average Value (Half Cycle) — $V_{avg} = 2V_0/pi approx 0.637 V_0$, $I_{avg} = 2I_0/pi approx 0.637 I_0$
Average Value (Full Cycle) — $0$
Phase Difference — $Delta phi = |phi - phi'|$

2-Minute Revision

Alternating Current (AC) is characterized by voltage and current that periodically reverse direction, typically following a sinusoidal pattern. The instantaneous values are given by $V = V_0 sin(omega t + phi)$ and $I = I_0 sin(omega t + phi')$, where $V_0$ and $I_0$ are peak values, $omega = 2pi f$ is the angular frequency, and $phi, phi'$ are phase angles.

The frequency $f$ (in Hz) is the number of cycles per second. The average value of AC over a full cycle is zero due to symmetry. However, the average value over a half cycle is $2V_0/pi$ or $2I_0/pi$.

The most important concept is the Root Mean Square (RMS) value, which represents the effective AC value for power dissipation, equivalent to a DC value. For sinusoidal AC, $V_{rms} = V_0/sqrt{2}$ and $I_{rms} = I_0/sqrt{2}$.

Household voltage (e.g., 220V) is an RMS value. AC is preferred for power transmission because transformers can easily step up or step down its voltage, minimizing transmission losses.

5-Minute Revision

AC voltage and current are time-varying quantities that periodically reverse direction, most commonly in a sinusoidal fashion. The general equations are $V = V_0 sin(omega t + phi_V)$ and $I = I_0 sin(omega t + phi_I)$.

Here, $V_0$ and $I_0$ are the peak (maximum) values, $omega$ is the angular frequency ($2pi f$), and $phi_V, phi_I$ are the initial phase angles. The frequency $f$ (in Hz) dictates the rate of oscillation.

For example, if $V = 311 sin(100pi t)$, then $V_0 = 311,\text{V}$ and $omega = 100pi,\text{rad/s}$, leading to $f = omega/(2pi) = 50,\text{Hz}$.

Crucially, the average value of AC over a full cycle is zero. This is because the positive and negative halves cancel out. However, the average value over a half cycle is non-zero, given by $V_{avg} = 2V_0/pi$ and $I_{avg} = 2I_0/pi$. For instance, if $V_0 = 100,\text{V}$, then $V_{avg} approx 63.7,\text{V}$ over a half cycle.

The Root Mean Square (RMS) value is the 'effective' value of AC. It's the DC equivalent that would produce the same heating effect. For sinusoidal AC, $V_{rms} = V_0/sqrt{2}$ and $I_{rms} = I_0/sqrt{2}$.

This means if your household supply is $220,\text{V}$ (RMS), the peak voltage is $V_0 = 220sqrt{2} approx 311,\text{V}$. AC meters typically measure RMS values. The phase difference ($Delta phi = phi_V - phi_I$) indicates whether voltage leads or lags current, or if they are in phase.

AC's primary advantage is its easy voltage transformation using transformers, enabling efficient long-distance power transmission.

Prelims Revision Notes

1
Instantaneous Values — $V = V_0 sin(omega t + phi)$ and $I = I_0 sin(omega t + phi')$. These are the values at any given instant $t$.
2
Peak Values ($V_0, I_0$) — Maximum values attained by voltage/current. These are the amplitudes of the sinusoidal waves.
3
Frequency ($f$) — Number of cycles per second (Hz). Standard in India is $50,\text{Hz}$.
4
Angular Frequency ($omega$) — $omega = 2pi f$. Used in the sinusoidal equations.
5
Time Period ($T$) — Time for one complete cycle. $T = 1/f$.
6
RMS Values ($V_{rms}, I_{rms}$)

* Represents the effective value of AC, equivalent to DC for power dissipation. * $V_{rms} = V_0/sqrt{2} approx 0.707 V_0$ * $I_{rms} = I_0/sqrt{2} approx 0.707 I_0$ * Household voltage ($220,\text{V}$) is an RMS value.

1
Average Values ($V_{avg}, I_{avg}$)

* Over a full cycle: $V_{avg} = 0$, $I_{avg} = 0$ (due to symmetry). * Over a half cycle: $V_{avg} = 2V_0/pi approx 0.637 V_0$, $I_{avg} = 2I_0/pi approx 0.637 I_0$.

1
Phase and Phase Difference — The term $(omega t + phi)$ is the phase. The difference $Delta phi = phi_V - phi_I$ indicates if voltage leads, lags, or is in phase with current. For purely resistive circuits, $Delta phi = 0$.
2
Advantages of AC — Easy voltage transformation (step-up/step-down) using transformers for efficient long-distance transmission and distribution. Simpler generation and motor design.
3
Conversions — Remember $cos(\theta) = sin(\theta + pi/2)$ and $sin(\theta) = cos(\theta - pi/2)$ for phase comparisons.

Vyyuha Quick Recall

To remember the RMS and Average values:

Really Means Square: $V_{rms} = V_0 / \sqrt{2}$ (RMS is 'Root Mean Square', and it's $V_0$ divided by $\sqrt{2}$)

Always Very Good Half: $V_{avg} = 2V_0 / \pi$ (Average over Half cycle is $2V_0/\pi$)

Full Cycle Zero: Average over Full Cycle is $0$.