Fundamentals of Sinusoidal AC
In an AC circuit, the instantaneous current i and voltage v are expressed as functions of time. The most common form is the sinusoidal wave, represented by the equations i = I₀ sin(ωt) and v = V₀ sin(ωt + φ), where I₀ and V₀ are the peak values (amplitudes), ω is the angular frequency, and φ is the phase difference.
Because the current and voltage fluctuate, we cannot use simple arithmetic averages, as the average current over one complete cycle is zero. Instead, we use the Root Mean Square (RMS) value, which is equivalent to the DC current that would produce the same amount of heat in a resistor. The relationship between the peak value and the RMS value is given by Irms = I₀ / √2 and Vrms = V₀ / √2.
Note: When competitive exams mention “AC voltage” or “AC current” without specifying, they are almost always referring to the RMS values.
Impedance and Phasor Diagrams
To analyze AC circuits, we represent voltage and current as rotating vectors called Phasors. A phasor rotates at an angular frequency ω, and its projection on the vertical axis gives the instantaneous value. This graphical method simplifies the addition of voltages across different components (resistor, inductor, and capacitor) that have different phase relationships.
The total opposition offered by an AC circuit to the flow of current is called Impedance (Z). Unlike resistance, which is constant, impedance depends on frequency. For a series LCR circuit, the impedance is given by the formula Z = √[R² + (Xʟ – Xᴄ)²], where Xʟ = ωL is the inductive reactance and Xᴄ = 1/(ωC) is the capacitive reactance.
- Resistors: Current and voltage are in phase (φ = 0).
- Inductors: Current lags behind voltage by π/2.
- Capacitors: Current leads voltage by π/2.
Resonance in LCR Circuits
Resonance occurs in a series LCR circuit when the inductive reactance equals the capacitive reactance (Xʟ = Xᴄ). At this specific frequency, known as the Resonant Frequency (ω₀), the impedance of the circuit is at its minimum, equal solely to the resistance R. Consequently, the current in the circuit reaches its maximum possible value.
The condition for resonance is derived by setting ωL = 1/(ωC), which gives ω₀ = 1 / √(LC). At this point, the circuit is purely resistive, and the power factor becomes unity. This principle is fundamental in radio tuning, where the circuit is adjusted to resonate at the frequency of the desired broadcast signal.
The Quality Factor (Q-factor) determines the sharpness of the resonance peak. A higher Q-factor implies a narrower bandwidth and a more selective circuit. It is calculated as Q = (1/R) * √(L/C).
Power in AC Circuits
The instantaneous power in an AC circuit is the product of instantaneous voltage and current. However, the Average Power (Pavg) is what matters for practical applications. It is given by the formula Pavg = Vrms * Irms * cos(φ), where cos(φ) is known as the Power Factor.
The power factor represents the ratio of real power to apparent power. If the circuit is purely resistive, φ = 0 and the power factor is 1 (maximum efficiency). If the circuit contains only an inductor or capacitor, φ = π/2 and the power factor is 0, meaning no real power is consumed despite the flow of current—this is often called “wattless current.”
Important Facts / Formulas
| Quantity | Formula |
|---|---|
| RMS Current | Irms = I₀/√2 |
| Inductive Reactance | Xʟ = ωL |
| Capacitive Reactance | Xᴄ = 1/ωC |
| Impedance (LCR) | Z = √(R² + (Xʟ – Xᴄ)²) |
| Resonant Frequency | ω₀ = 1/√(LC) |
| Average Power | P = Vrms Irms cos φ |
Previous Year Question Hints
- Frequency Dependency: Expect questions asking how the current changes if the frequency of the AC source is doubled in a circuit containing only a capacitor versus only an inductor.
- Phase Relationships: A common challenge involves identifying the phase angle between voltage and current in a series LCR circuit given specific values for R, L, and C.
Quick Revision Summary
- AC changes direction periodically; sinusoidal waves are the standard model.
- Always use RMS values for power calculations unless peak values are explicitly requested.
- Phasors are the most efficient way to handle phase differences between V and I.
- Impedance (Z) is the frequency-dependent opposition to current flow.
- Resonance occurs when Xʟ = Xᴄ, leading to maximum current flow.
- The Q-factor defines the sharpness of the resonance; higher Q means higher selectivity.
- Power factor (cos φ) is 1 for purely resistive circuits and 0 for purely reactive circuits.
- Transformers rely on AC and mutual induction to step up or step down voltage.
