Alternating Current: LCR, Resonance & Transformers

AC Voltage and Current

An alternating voltage is described by V(t) = $V_{0}$ sin($\omega$t), where $V_{0}$ is the peak (amplitude) voltage and $\omega$ = 2$\pi$f is the angular frequency. The corresponding current depends on the circuit element. Peak value $V_{0}$ is the maximum instantaneous voltage.
The root-mean-square (RMS) value $V_{rms}$ = $\frac{V_{0}}{\sqrt{2}}$ is the equivalent DC voltage that dissipates the same power in a resistor.
Similarly, $I_{rms}$ = $\frac{I_{0}}{\sqrt{2}}$.
For household AC (India): $V_{rms}$ = 220 V, so $V_{0}$ = 220$\sqrt{2}$ = 311 V, f = 50 Hz, $\omega$ = 100*$\pi$ rad/s.

Phase Relations and Phasors

In a pure resistor, voltage and current are in phase. In a pure inductor, voltage leads current by $\frac{\pi}{2}$ (remember: ELI — voltage E leads current I in inductor L). In a pure capacitor, current leads voltage by $\frac{\pi}{2}$ (remember: ICE — current I leads voltage E in capacitor C). Phasor diagrams represent sinusoidal quantities as rotating vectors, making phase relationships visual. The projection of the phasor on the vertical axis gives the instantaneous value.

Reactance and Impedance

Inductive reactance: $X_{L}$ = $\omega$L = 2$\pi$fL (increases with frequency — inductors oppose rapid changes).
Capacitive reactance: $X_{C}$ = 1/($\omega$C) = 1/(2$\pi$fC) (decreases with frequency — capacitors pass high frequencies). Both have units of ohms. Impedance Z is the total opposition to AC flow. For a series LCR circuit:

Z = $\sqrt{R^{2} + ({X}_{L} - {X}_{C}}$²)

The phase angle between voltage and current: tan($\phi$) = ($X_{L}$ - $X_{C}$)/R. If $X_{L}$ > $X_{C}$, the circuit is inductive (voltage leads). If $X_{C}$ > $X_{L}$, the circuit is capacitive (current leads).

Series LCR Circuit

Series LCR Circuit Diagram:

In a series LCR circuit driven by V = $V_{0}$ sin($\omega$t), the current I = $V_{0}$/Z * sin($\omega$t - $\phi$).
The voltage across each element: $V_{R}$ = IR (in phase with I), $V_{L}$ = I$X_{L}$ (leads I by $\frac{\pi}{2}$), $V_{C}$ = I$X_{C}$ (lags I by $\frac{\pi}{2}$).
The phasor sum: V = $\sqrt{{V}_{R}^2 + ({V}_{L} - {V}_{C}}$²). Note that $V_{L}$ and $V_{C}$ can individually exceed the source voltage V (voltage magnification).

Phasor Diagram for Series LCR Circuit ($X_{L}$ > $X_{C}$):

Resonance

At resonance, $X_{L}$ = $X_{C}$, so $\omega_{0}$*L = 1/($\omega_{0}$*C), giving the resonant frequency:

$\omega_{0}$ = 1/$\sqrt{LC}$, or $f_{0}$ = 1/(2*$\pi$*$\sqrt{LC}$)

Resonance Curve — Impedance vs Frequency:

At resonance: Z = R (minimum impedance), current is maximum ($I_{0}$ = $V_{0}$/R), phase angle $\phi$ = 0 (voltage and current in phase), and the circuit behaves as a pure resistor.
The quality factor Q = $\omega_{0}$L/R = 1/($\omega_{0}$CR) = (1/R)$\sqrt{L/C}$ measures the sharpness of resonance.
Higher Q means a narrower resonance peak and greater voltage magnification: $V_{L}$ = $V_{C}$ = Q*V at resonance.
Bandwidth = $\omega_{0}$/Q = R/L.

Power in AC Circuits

Instantaneous power p(t) = VI oscillates at 2$\omega$.
Average power: P = $V_{rms}$ * $I_{rms}$ * cos($\phi$), where cos($\phi$) is the power factor.
For pure R: cos($\phi$) = 1, P = $V_{rms}^{2}$/R.
For pure L or C: cos($\phi$) = 0, P = 0 (wattless current). The current component Icos($\phi$) is the active (power) component; Isin($\phi$) is the reactive (wattless) component.
Apparent power = $V_{rms}$ * $I_{rms}$ (in VA).
Real power = apparent power x power factor.
Wattless current = $I_{rms}$ * sin($\phi$) dissipates no power but increases the current drawn from the source.

Transformers

Transformer — Primary and Secondary Coils:

A transformer transfers AC energy between circuits through mutual induction.
For an ideal transformer: $\frac{V_{s}}{V_{p}}$ = $\frac{N_{s}}{N_{p}}$ = k (turns ratio), and $\frac{I_{s}}{I_{p}}$ = $\frac{N_{p}}{N_{s}}$ = 1/k (power conservation: $V_{p}$$I_{p}$ = $V_{s}$$I_{s}$). Step-up: $N_{s}$ > $N_{p}$ (increases voltage, decreases current). Step-down: $N_{s}$ < $N_{p}$ (decreases voltage, increases current). Real transformer losses: copper losses ($I^{2}$R in windings), iron/core losses (eddy currents and hysteresis in core), flux leakage.
Efficiency = $\frac{P_{output}}{P_{input}}$ x 100%. Laminated soft iron cores minimize eddy current losses. Transformers only work with AC — DC produces no changing flux.

The key problem-solving concept is impedance analysis using phasors: draw the phasor diagram, identify phase relationships, and compute Z and $\phi$ to find current and power.

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