StudentAidUnit 17: Alternating Current — Long Questions
12th Class Physics · Unit 17: Alternating Current
A capacitor stores energy in an electric field and, unlike a resistor, allows alternating current to pass while blocking direct current, because it repeatedly charges and discharges as the applied voltage reverses. [Figure: a capacitor of capacitance C connected directly across an AC source V = V₀sinωt.]
Derivation of capacitive reactance Let the applied alternating voltage be V = V₀ sinωt. The charge on the capacitor at any instant is q = CV = CV₀ sinωt. The current is the rate of change of charge: I = Δq/Δt = CV₀ω cosωt. Since cosωt = sin(ωt + π/2), this gives I = ωCV₀ sin(ωt + π/2) = I₀ sin(ωt + π/2), where I₀ = ωCV₀ is the peak current.
Comparing with V = V₀ sinωt, the current is ahead of the voltage by a phase angle of π/2 (90°); equivalently, the voltage lags behind the current by 90°, because the current is maximum when the voltage is changing fastest (at the zero-crossing) and the current is zero when the voltage is at its peak (capacitor momentarily fully charged, no further charge flows).
Capacitive reactance is defined as XC = Vrms/Irms. Since Vrms = 0.707V₀, Irms = 0.707I₀ and I₀ = ωCV₀, we get XC = V₀/(ωCV₀) = 1/(ωC) = 1/(2πfC). XC is inversely proportional to frequency f and capacitance C: at f = 0 (DC), XC → ∞ (capacitor blocks DC), while at high AC frequency, XC is small (capacitor readily passes AC).
An inductor (choke) stores energy in a magnetic field when current flows through it and opposes any change in current by producing a back emf. [Figure: an inductor of inductance L (negligible resistance) connected directly across an AC source V = V₀sinωt.]
Derivation of inductive reactance Let the current through the inductor be I = I₀ sinωt. The back emf set up is V = L(ΔI/Δt). To maintain this current, the applied voltage must equal the back emf: V = L d/dt(I₀ sinωt) = ωLI₀ cosωt. Since cosωt = sin(ωt + π/2), V = ωLI₀ sin(ωt + π/2) = V₀ sin(ωt + π/2), where V₀ = ωLI₀.
Comparing with I = I₀ sinωt, the voltage leads the current by π/2 (90°), i.e., the current lags behind the voltage by 90°. This is because the back emf (and hence the applied voltage needed to drive the current) is largest when the current is changing fastest (at the zero-crossing), and zero when the current is momentarily constant at its peak value.
Inductive reactance is defined as XL = Vrms/Irms. Since Vrms = 0.707V₀, Irms = 0.707I₀ and V₀ = ωLI₀, we get XL = ωLI₀/I₀ = ωL = 2πfL. XL is directly proportional to frequency f and inductance L: at f = 0 (DC), XL = 0 (inductor acts as a plain conductor for DC), while at high AC frequency, XL is large (inductor blocks high-frequency AC).
An RLC-parallel circuit is one in which a resistor R, an inductor L and a capacitor C are all connected in parallel with each other across a common AC source, so that the voltage is the same across all three elements while the source current divides among the three branches. [Figure: resistor, inductor and capacitor connected in parallel across a variable-frequency AC source.]
Key properties (1) Resonance and impedance — resonance occurs at XL = XC, and at resonance the impedance of the parallel circuit is maximum, so it behaves like a purely resistive circuit. (2) Voltage and current — since voltage is common to all branches, it is taken as the reference phasor; at resonance the total (source) current is minimum, even though the individual branch currents IL and IC can each be much larger than this resonance current (current magnification). (3) Admittance (Y), the reciprocal of impedance, is used to analyze the circuit; the total admittance is the sum of the admittances of R, L and C. (4) Frequency and bandwidth — impedance, admittance and the branch currents are all frequency dependent; the resonance frequency is f = 1/(2π√(LC)), and the bandwidth is the range of frequencies over which the circuit operates effectively (used in tuning circuits, signal processing, and induction cooktops). (5) Quality factor (Q) measures the sharpness of the resonance peak and the energy stored compared with energy dissipated; in a parallel RLC-circuit, Q is inversely proportional to the resistance. (6) Power factor varies with frequency and the relative sizes of XL and XC, becoming unity (purely resistive) exactly at resonance.
RC-series circuit With resistor R and capacitor C in series, the same rms current Irms flows through both. The voltage across R, VR = Irms R, is in phase with the current, while the voltage across C, VC = Irms XC, lags the current by 90°. Since these two voltage phasors are at right angles, the applied (resultant) voltage is Vrms = √(VR² + VC²) = Irms√(R² + XC²). Since Z = Vrms/Irms, the impedance of an RC-series circuit is Z = √(R² + XC²), where XC = 1/(ωC) = 1/(2πfC). The phase angle by which current leads voltage is θ = tan⁻¹(XC/R) = tan⁻¹[1/(ωCR)].
RL-series circuit With resistor R and inductor L in series, the same rms current Irms flows through both. The voltage across R, VR = Irms R, is in phase with the current, while the voltage across L, VL = Irms XL, leads the current by 90°. The resultant applied voltage is Vrms = √(VR² + VL²) = Irms√(R² + XL²). Hence the impedance of an RL-series circuit is Z = √(R² + XL²), where XL = ωL = 2πfL. The phase angle by which current lags voltage is θ = tan⁻¹(XL/R).
Full-wave rectification is the process in which both halves (positive and negative) of an alternating input voltage are converted into a unidirectional (one-way) current through a load resistance, unlike half-wave rectification which uses only one half-cycle.
Circuit Full-wave rectification is achieved using a bridge rectifier, in which four diodes D1, D2, D3 and D4 are connected in a closed-loop "bridge" configuration. The AC supply is connected to one diagonal of the bridge (terminals A and B) and the load resistance RL is connected across the other diagonal.
Positive half-cycle (0 → T/2) terminal A is positive with respect to B. Diodes D1 and D3 become forward biased and conduct in series, while D2 and D4 are reverse biased and block. Current flows through RL in one fixed direction.
Negative half-cycle (T/2 → T) terminal A becomes negative and B positive. Now D2 and D4 become forward biased and conduct, while D1 and D3 switch off. Even though the supply polarity has reversed, the current through RL still flows in the same direction as before, because of the way the bridge diodes are arranged.
Since current flows through the load in the same direction during both halves of the input cycle, the output across RL is a unidirectional but pulsating voltage, with two pulses for every one cycle of the input (so the output ripple frequency is twice the input frequency, fripple = 2f). This pulsating output can be smoothed using a filter capacitor connected in parallel with the load, which charges when the rectified voltage rises and discharges when it falls, giving a much steadier DC output. Such full-wave bridge rectifiers (often followed by a smoothing capacitor) are commonly used in mobile phone chargers and other devices that need a stable DC supply.
An RLC-series circuit consists of a resistor R, an inductor L and a capacitor C connected in series with an AC source of variable frequency, so the same current flows through all three elements. [Figure: R, L, C in series with an AC source, together with the impedance phasor diagram showing XL = ωL and XC = 1/ωC directed opposite to each other, and R along the current direction.]
Impedance Since XL and XC act in opposite directions (voltage across L leads the current by 90°, voltage across C lags the current by 90°), the net reactance is (XL − XC), and the impedance of the circuit is Z = √(R² + (XL − XC)²). When frequency is high, XL ≫ XC and the circuit behaves like an RL-series circuit; when frequency is low, XC ≫ XL and the circuit behaves like an RC-series circuit.
Resonance At a particular frequency fr, XL becomes exactly equal to XC; the reactances cancel and the circuit becomes purely resistive. Setting ωL = 1/(ωC) gives the resonance (angular) frequency ωr = 1/√(LC), i.e., fr = 1/(2π√(LC)).
Key properties at/around resonance: (1) Impedance and resonance — at resonance Z is minimum and equal to R. (2) Current — at resonance, current is maximum since impedance is minimum. (3) Phase angle — at resonance the phase angle is zero, voltage and current are in phase, and the power factor is 1. (4) Damping — resistance introduces damping of the oscillations; higher R means faster decay of oscillations (analogous to a damped mass-spring system with a shock absorber). (5) Tuning and filtering — RLC-series circuits at resonance are widely used in tuning radio receivers and as band-pass filters, allowing a chosen frequency range to pass while attenuating others. (6) Voltage amplification — at resonance, the voltage across L and across C can individually be much larger than the applied source voltage, because of the large circulating current, even though they cancel each other in the net applied voltage. (7) Power dissipation — the true power dissipated is P = VI cosθ, maximum (equal to apparent power) only when θ = 0 (resonance, purely resistive behaviour); the product VI itself (ignoring phase) is called the apparent power.