Comprehensive Study Notes on AC Circuits

Generation of Alternating EMF

Alternating Electro-Motive Force (EMF) is generated through the principle of electromagnetic induction. This can be achieved either by rotating a coil within a stationary magnetic field or by rotating a magnetic field within a stationary coil. In both scenarios, the magnetic field is cut by the conductors, and an EMF is induced in the coil according to Faraday's laws of electromagnetic induction. The magnitude of this induced EMF is determined by the number of turns in the coil, the strength of the magnetic field, and the speed of rotation.

To understand the mathematical derivation, consider a rectangular coil with $N$ turns and area $A\,\text{m}^2$ rotating counter-clockwise at a constant angular velocity of $\omega\,\text{rad/s}$ in a uniform magnetic field. Let $\phi_m$ represent the maximum flux cutting the coil when its axis coincides with the reference axis $XX'$. When the coil is aligned with $XX'$, the flux linkage is at its maximum ($\phi_m$), and when aligned with $YY'$, parallel to the flux lines, the flux linkage is zero. As the coil rotates through an angle $\theta = \omega t$ at any instant $t$, the flux linking with the coil is defined as $\phi = \phi_m \cos(\omega t)$.

According to Faraday's laws, the induced EMF is the negative rate of change of flux linkage. Mathematically, this is expressed as $E = -N \frac{d\phi}{dt}$. Substituting the flux equation gives $E = -N \frac{d(\phi_m \cos(\omega t))}{dt} = N \phi_m \omega \sin(\omega t)$. This results in the standard form $E = E_m \sin(\omega t)$, where $E_m = N \phi_m \omega$ is the maximum value of the induced EMF. In terms of flux density $B_m$, this can also be written as $E_m = N B_m A \omega$. When the coil has turned through $90^\circ$ ($\theta = \pi/2$), the EMF reaches its maximum value $E_m$, whereas at $0^\circ$ or $360^\circ$, the value is zero. A plot of this EMF against time produces a sinusoidal waveform. Similarly, the induced current is expressed as $I = I_m \sin(\omega t)$.

Terms and Values of Alternating Quantities

Several specific terms are used to describe alternating quantities. The peak amplitude, or simply amplitude, is the maximum value (positive or negative) of an alternating quantity, denoted as $V_p$ for voltage and $I_p$ for current. The instantaneous value refers to the magnitude of the quantity at any specific point in time, represented as $v(t)$ or $i(t)$. A waveform is the graphical representation of these instantaneous values plotted against time. A cycle consists of one complete set of both positive and negative values.

Frequency ($f$) is defined as the number of cycles completed per second and is measured in Hertz ($Hz$). The time period ($T$) is the duration required to complete one full cycle, measured in seconds. These are related such that $T = \frac{1}{f}$. The phase ($\theta$) represents the time elapsed since the quantity last passed through the zero-reference point, measured in degrees or radians. For a current waveform, the phase at a specific point can be calculated as $\theta = \omega t = (\frac{2\pi}{T}) \times t$. Phase difference is a comparative measure between two quantities; they are "in phase" if they reach their zero and maximum points simultaneously, regardless of their individual magnitudes.

In the context of phase differences, a "leading" quantity reaches its maximum or zero value earlier than a reference quantity, marked with a plus ($+\text{sign}$). A "lagging" quantity reaches these points later, marked with a minus ($-\text{sign}$). For example, if $V_A = V_m \sin(\omega t)$ is the reference, a leading voltage $V_B$ would be $v_B = V_m \sin(\omega t + \phi)$, while a lagging voltage would be $v_B = V_m \sin(\omega t - \phi)$.

RMS and Average Values

The Root Mean Square (RMS) or effective value is defined based on the heating effect of the current. It is the value of steady direct current ($DC$) that would produce the same amount of heat or do the same work in the same time through the same resistance as the alternating current. Mathematically, for a current $i(t)$ over an interval from $t_1$ to $t_2$, the RMS value is $I_{rms} = \sqrt{\frac{1}{t_2-t_1} \int_{t_1}^{t_2} i^2(t)\,dt}$. For a sinusoidal waveform where $v = V_m \sin(\theta)$, the RMS value is derived as $V_{rms} = \frac{V_m}{\sqrt{2}} \approx 0.707 V_m$. This value is critical in practice because standard AC ammeters and voltmeters are calibrated to display RMS values.

The average value of an alternating quantity is the arithmetic mean of all values over one complete cycle. For symmetrical waveforms, the average value over a full cycle is zero; therefore, the average is specifically calculated over a half-cycle. Mathematically, this is $I_{avg} = \frac{1}{t_2-t_1} \int_{t_1}^{t_2} i(t)\,dt$. For a sinusoidal wave, the average value is derived as $V_{avg} = \frac{2 V_m}{\pi} \approx 0.637 V_m$.

Two factors relate these values. The Crest Factor (or Peak Factor), denoted $k_p$, is the ratio of the maximum value to the RMS value ($k_p = \frac{V_m}{V_{rms}}$). For a sine wave, this is approximately $1.414$. The Form Factor ($k_f$) is the ratio of the RMS value to the average value ($k_f = \frac{V_{rms}}{V_{avg}}$). For a sine wave, the form factor is exactly $1.11$.

Phasor Representation of AC Signals

Alternating quantities can be represented as phasors, which are lines of definite length rotating counter-clockwise at a constant angular velocity $\omega$. The length of the phasor corresponds to the maximum (or peak) value of the quantity. The projection of this rotating phasor onto the Y-axis at any given instant provides the instantaneous value: $i = I_m \sin(\omega t)$. While the length usually represents the peak value, in practical engineering, phasor diagrams often use RMS values for convenience, provided the total length is scaled correctly (multiplied by $\sqrt{2}$ if a sine wave needs to be reconstructed).

Phasors allow for the simultaneous representation of multiple signals of the same frequency. Since they rotate at the same speed, their relative positions remain fixed. For example, if an EMF leads a current by angle $\alpha$, they are drawn with that fixed angular displacement. Mathematically, phasors are represented in four forms: Rectangular ($V = X \pm jY$), Trigonometric ($V = V(\cos(\phi) \pm j\sin(\phi))$), Exponential ($V = V e^{\pm j\phi}$), and Polar ($V = V \angle \pm \phi$). Magnitude is calculated as $|V| = \sqrt{X^2 + Y^2}$ and phase angle as $\phi = \tan^{-1}(\frac{Y}{X})$.

Steady-State Analysis of Basic AC Elements

In a purely resistive circuit, the current is exactly in phase with the applied voltage. The instantaneous power consists of a constant DC component ($\frac{V_m I_m}{2}$) and a fluctuating component at twice the supply frequency. The average power consumed is $P_{avg} = V_{rms} I_{rms}$, and the power factor ($\cos(\phi)$) is unity ($1$).

In a purely inductive circuit, the current lags the voltage by exactly $90^\circ$ ($\pi/2$). This occurs because the inductor opposes changes in current. The inductive reactance is $X_L = \omega L = 2\pi f L\,\Omega$. At $DC$ ($f=0$), an inductor acts as a short circuit. The instantaneous power fluctuates between positive and negative values, meaning energy is stored in the magnetic field and then returned to the source. Consequently, the average power consumption is zero. The power factor is zero lagging.

In a purely capacitive circuit, the current leads the voltage by exactly $90^\circ$ ($\pi/2$). The capacitive reactance is $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}\,\Omega$. At $DC$, a capacitor acts as an open circuit ($X_C = \infty$). Similar to the inductor, the average power consumption is zero because energy is cycled between the source and the electrostatic field. The power factor is zero leading.

Series AC Circuits (RL, RC, and RLC)

In a series R-L circuit, the total voltage is the phasor sum of the voltage across the resistor ($V_R = IR$) and the inductor ($V_L = jIX_L$). The current lags the applied voltage by an angle $\phi$, where $0^\circ < \phi < 90^\circ$. The complex impedance is $Z = R + jX_L$, with magnitude $|Z| = \sqrt{R^2 + X_L^2}$ and phase $\phi = \tan^{-1}(\frac{X_L}{R})$. The power factor is $\cos(\phi) = \frac{R}{Z}$. Power in these circuits is divided into Active Power ($P = VI \cos(\phi)$), Reactive Power ($Q = VI \sin(\phi)$), and Apparent Power ($S = VI$).

In a series R-C circuit, the current leads the applied voltage by an angle $\phi$. The complex impedance is $Z = R - jX_C$, with magnitude $|Z| = \sqrt{R^2 + X_C^2}$ and phase $\phi = \tan^{-1}(\frac{X_C}{R})$. The power factor is leading. Similar to the RL circuit, the components of power include active, reactive, and apparent power, forming a power triangle.

In a series R-L-C circuit, the net reactance is $X = X_L - X_C$. If $X_L > X_C$, the circuit is inductive and the current lags. If $X_L < X_C$, the circuit is capacitive and the current leads. The total impedance is $Z = R + j(X_L - X_C)$. The phase angle transitions from leading to lagging depending on which reactance dominates.

Resonance in Series R-L-C Circuits

Resonance occurs in a series R-L-C circuit when the inductive reactance equals the capacitive reactance ($X_L = X_C$). At this specific frequency, the net reactance becomes zero, and the circuit's impedance is purely resistive ($Z = R$). The resonant frequency is given by $f_0 = \frac{1}{2\pi \sqrt{LC}}$. At resonance, the current is at its maximum value ($I_0 = \frac{V}{R}$), and the power factor is unity. Because the circuit allows maximum current flow at this frequency, it is often called an "acceptor circuit."

Voltage magnification occurs during resonance, where the voltages across the inductor ($V_L$) and capacitor ($V_C$) are equal and opposite but can be much larger than the source voltage. The Quality Factor ($Q_0$) measures this magnification and the sharpness of the resonance, defined as $Q_0 = \frac{\omega_0 L}{R} = \frac{1}{R} \sqrt{\frac{L}{C}}$. Bandwidth is the range of frequencies over which the current is at least $70.7\%$ of its maximum value ($\frac{I_0}{\sqrt{2}}$), calculated as $BW = f_2 - f_1 = \frac{R}{2\pi L}$.

Three-Phase AC Circuits

A three-phase system utilizes three separate windings displaced by $120^\circ$ from each other. This configuration generates three alternating voltages of equal magnitude and frequency but shifted in phase by $120^\circ$. These are typically named the R, Y, and B phases. A system is considered "balanced" if the phase voltages and currents are equal in magnitude and exactly $120^\circ$ apart, and if the loads on each phase are identical in magnitude and power factor.

There are two primary ways to connect three-phase windings: Star ($Y$) and Delta ($\Delta$). In a balanced Star Connection, the three phases share a common neutral point. The line current is equal to the phase current ($I_L = I_{ph}$), while the line voltage is $\sqrt{3}$ times the phase voltage ($V_L = \sqrt{3}V_{ph}$). This is also referred to as a three-phase, four-wire system.

In a balanced Delta Connection, the windings are connected in a loop (terminating one winding into the start of the next). In this configuration, the line voltage is equal to the phase voltage ($V_L = V_{ph}$), while the line current is $\sqrt{3}$ times the phase current ($I_L = \sqrt{3}I_{ph}$). This is known as a three-phase, three-wire system.