A.C. Circuits: Theoretical Principles and Mathematical Analysis

Introduction to Alternating Current (AC) and Direct Current (DC)

Alternating Current (AC) and Direct Current (DC) represent two distinct ways in which electricity flows through a circuit. Their differences are fundamental to how power is generated, transmitted, and utilized.

• Flow Direction:

  • AC: The direction of the current flow reverses periodically.

  • DC: The current flows in a single, constant direction.

• Voltage Behavior over Time:

  • AC: Voltage varies over time, typically following a sinusoidal, square, or triangular waveform.

  • DC: Voltage remains constant and does not vary with time.

• Generation Methods:

  • AC: Generated primarily by alternators in power plants.

  • DC: Generated by batteries, DC generators, and rectifiers (devices specifically designed to convert AC to DC).

• Voltage Transformation:

  • AC: It is significantly easier to transform voltages using transformers.

  • DC: It is difficult to transform voltages without the use of complex power electronics.

• Transmission Efficiency:

  • AC: Can be transmitted over very long distances with significantly less power loss compared to DC.

  • DC: Suitable for short-distance transmission or specific localized applications.

• Primary Applications:

  • AC: Used for powering homes, offices, industries, large electric motors, and general household appliances.

  • DC: Used for powering electronic devices such as computers, mobile phones, and electric vehicles.

Basics of Alternating Current

Alternating Current is characterized by periodic variations in magnitude and direction. Understanding its properties involves several key parameters:

• Waveforms: While typically sinusoidal, AC signals can also take the form of triangular or square waves when plotted against time.

• Frequency ($f$):

  • This is the number of complete cycles per second.

  • It is measured in Hertz ($Hz$).

  • Common standard frequencies are $50\,Hz$ and $60\,Hz$.

  • One cycle consists of one complete oscillation of the wave.

• Amplitude ($V_o$ or $I_o$):

  • This represents the maximum value (peak value) reached by the voltage or current during its cycle.

• Period ($T$):

  • The time required to complete one full cycle of the AC wave.

  • It is measured in seconds ($s$).

  • The relationship between Period and Frequency is given by: $T = \frac{1}{f}$.

• Angular Frequency ($\omega$):

  • The rate of change of the current or voltage phase, representing how quickly the signal oscillates.

  • It is measured in radians per second ($rad/s$).

  • Formula: $\omega = 2\pi f$ or $\omega = \frac{2\pi}{T}$.

• Phase ($\phi$):

  • Describes the position of the current or voltage relative to a specific reference point in time.

  • It is measured in degrees ($^{\circ}$) or radians ($rad$).

    • Leading Phase: Occurs if the current or voltage reaches its maximum earlier than the reference point.

    • Lagging Phase: Occurs if the current or voltage reaches its maximum later than the reference point.

Mathematical Representations of AC Signals

AC signals are commonly represented using sinusoidal functions or complex numbers (phasors).

Sinusoidal Function

This is the most common approach to describe instantaneous values of voltage ($v(t)$) or current ($i(t)$).

$v(t) = V_0 \sin(\omega t - \phi)$

• $v(t)$: Instantaneous value of voltage at time $t$.

• $V_0$: Peak amplitude.

• $\omega$: Angular frequency.

• $t$: Time.

• $\phi$: Phase angle.

Phasors

Phasors utilize complex numbers to represent AC signals, simplifying the analysis of circuits. They can be written in two forms:

• Rectangular Form: $a + jb$

• Polar Form: $V \angle \phi$

Root-Mean-Square (RMS) and Peak Values

Because AC values change constantly, "effective" values (RMS) are used to compare AC to DC in terms of heating effect.

• Root-Mean-Square Current ($I_{rms}$): The value of the steady (DC) current which, when flowing through a load (resistor), produces heat at the same mean rate as the alternating current.

• Root-Mean-Square Voltage ($V_{rms}$): The value of the direct potential difference (p.d.) which, when applied across a load, produces heat at the same mean rate as the alternating voltage.

• Measurement Note: Standard AC ammeters and voltmeters are calibrated to read and display RMS values, not peak values.

• Peak Values ($I_o, V_o$): The maximum instantaneous value reached by the current or voltage.

• Instantaneous Values ($I(t), V(t)$): The value of the current or voltage at any specific moment in time ($t$).

Mathematical Relationships

For sinusoidal signals:

• $I_{rms} = \frac{I_o}{\sqrt{2}} = 0.7071 \times I_o$

• $V_{rms} = \frac{V_o}{\sqrt{2}} = 0.7071 \times V_o$

For an AC source connected to an external resistive load ($R$):

• $V_o = I_o R$

• $V_{rms} = I_{rms} R$

• $V = I R$ (Relationship for instantaneous values)

AC Through Pure Passive Components

1. Pure Resistor

• Resistors offer resistance ($R$) to current flow in both AC and DC circuits, measured in Ohms ($\Omega$).

• Phase Relationship: In a pure resistive circuit, current and voltage are in phase ($\phi = 0$).

• Equations:

  • $v(t) = V_o \sin(\omega t)$

  • $i(t) = I_o \sin(\omega t)$, where $I_o = \frac{V_o}{R}$

2. Pure Inductor

• An inductor ($L$) opposes changes in current. This opposition in an AC circuit is called Inductive Reactance ($X_L$).

• Reactance Formula: $X_L = \omega L = 2\pi f L$.

• Phase Relationship: In a pure inductor, the voltage leads the current by $90^{\circ}$ (or the current lags the voltage by $90^{\circ}$).

• Derivation and Equations:

  • Voltage: $V_L(t) = L \frac{dI(t)}{dt}$

  • Current: $i(t) = \frac{1}{L} \int v(t)\,dt$

  • If $v(t) = V_o \sin(\omega t)$, then $i(t) = I_o \sin(\omega t - 90^{\circ})$, where $I_o = \frac{V_o}{X_L}$.

3. Pure Capacitor

• A capacitor ($C$) stores electric charge. In AC circuits, it offers opposition called Capacitive Reactance ($X_C$).

• Reactance Formula: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$.

• Phase Relationship: In a pure capacitor, the current leads the voltage by $90^{\circ}$ (or the voltage lags the current by $90^{\circ}$).

• Derivation and Equations:

  • Charge: $Q(t) = C v(t) = C V_o \sin(\omega t)$

  • Current: $i(t) = \frac{dQ(t)}{dt} = \omega C V_o \cos(\omega t) = I_o \sin(\omega t + 90^{\circ})$

  • Alternatively: $v_c(t) = V_o \sin(\omega t - 90^{\circ})$ if the current is the reference.

  • Peak Relationship: $V_o = I_o X_C$

Series AC Circuits

When components are connected in series, the total opposition to current is called Impedance ($Z$), also measured in Ohms ($\Omega$).

R-L Series Circuit

• Total Voltage ($V_T$): Found using the Pythagorean theorem since $V_R$ and $V_L$ are $90^{\circ}$ out of phase.

  • $V_T = \sqrt{V_R^2 + V_L^2} = I \sqrt{R^2 + X_L^2}$

• Impedance ($Z$): $Z = \sqrt{R^2 + X_L^2}$

• Phase Angle ($\phi$):

  • $\tan(\phi) = \frac{V_L}{V_R} = \frac{X_L}{R}$

  • Voltage leads current by $\phi$.

R-C Series Circuit

• Total Voltage ($V_T$): $V_T = \sqrt{V_R^2 + V_C^2} = I \sqrt{R^2 + X_C^2}$

• Impedance ($Z$): $Z = \sqrt{R^2 + X_C^2}$

• Phase Angle ($\phi$):

  • $\tan(\phi) = \frac{V_C}{V_R} = \frac{X_C}{R}$

  • Voltage lags current by $\phi$.

R-L-C Series Circuit

• Total Voltage ($V_T$): $V_T = \sqrt{V_R^2 + (V_L - V_C)^2} = I \sqrt{R^2 + (X_L - X_C)^2}$

• Impedance ($Z$): $Z = \sqrt{R^2 + (X_L - X_C)^2}$

• Net Reactance: The term $(X_L - X_C)$ represents the total reactance of the circuit.

• Phase Angle ($\phi$):

  • $\tan(\phi) = \frac{V_L - V_C}{V_R} = \frac{X_L - X_C}{R}$

• Circuit Behavior Conditions:

  • Inductive Circuit: If $X_L > X_C$, $\phi$ is positive. Current lags voltage.

  • Capacitive Circuit: If $X_C > X_L$, $\phi$ is negative. Current leads voltage.

Examples and Numerical Problems

Basic Resistor Examples

1. AC Mains Reading: An AC voltmeter reads $220\,V_{rms}$ at $60\,Hz$.

   • (i) Maximum voltage: $V_o = V_{rms} \times \sqrt{2} = 220 \times 1.414 \approx 311.1\,V$.

   • (ii) Equation for instantaneous voltage: $v(t) = 311.1 \sin(120\pi t)$.

2. Ammeter Reading: A $20\,\Omega$ resistor is connected to $v = 60 \sin(120\pi t)$.

   • $V_o = 60\,V$; $V_{rms} = \frac{60}{\sqrt{2}} \approx 42.43\,V$.

   • Ammeter read ($I_{rms}$): $I_{rms} = \frac{42.43}{20} = 2.12\,A$.

3. Variable Frequency: A $40\,\Omega$ resistor is across a $15\,V_{rms}$ source.

   • Current at $100\,Hz$: $I_{rms} = \frac{15}{40} = 0.375\,A$.

   • Current at $100\,kHz$: $I_{rms} = \frac{15}{40} = 0.375\,A$ (Note: Resistance is independent of frequency).

Inductor Examples

1. Current in 0.700 H Inductor ($120\,V$ source):

   • (a) at $60\,Hz$: $X_L = 2\pi(60)(0.7) = 263.9\,\Omega$. $I = \frac{120}{263.9} = 0.455\,A$.

   • (b) at $60\,kHz$: $X_L = 2\pi(60,000)(0.7) = 263,893.8\,\Omega$. $I = 0.455\,mA$.

Capacitor Examples

1. Current in 20 µF Capacitor ($120\,V$ source):

   • (a) at $60\,Hz$: $X_C = \frac{1}{2\pi(60)(20 \times 10^{-6})} = 132.6\,\Omega$. $I = \frac{120}{132.6} = 0.905\,A$.

   • (b) at $60\,kHz$: $X_C = 0.1326\,\Omega$. $I = 905\,A$.

Complex Problem Set

• A coil ($L=0.14\,H, R=12\,\Omega$) on $110\,V, 25\,Hz$ line:

  • Reactance $X_L = 2\pi(25)(0.14) = 22\,\Omega$.

  • Impedance $Z = \sqrt{12^2 + 22^2} = 25.06\,\Omega$.

  • Current $I = \frac{110}{25.06} = 4.39\,A$.

  • Phase angle $\phi = \tan^{-1}(\frac{22}{12}) = 61.4^{\circ}$.

  • Power factor ($\cos \phi$): $0.479$.

• R-L-C Problem: Resistance $20\,\Omega$, Inductance $0.2\,H$, Capacitance $100\,\mu F$ across $220\,V, 50\,Hz$.

  • $X_L = 2\pi(50)(0.2) = 62.8\,\Omega$.

  • $X_C = \frac{1}{2\pi(50)(100 \times 10^{-6})} = 31.8\,\Omega$.

  • $Z = \sqrt{20^2 + (62.8 - 31.8)^2} = \sqrt{400 + 961} = 36.9\,\Omega$.

  • Current $I = \frac{220}{36.9} = 5.96\,A$.