Alternating Current (AC) Circuits: Complete Theory, Derivations, and Series Circuits

Fundamentals of Direct Current (DC) and Alternating Current (AC)

• Direct Current (DC):
  • Definition: Direct Current is unidirectional in nature, meaning it does not change its polarity over time.
  • Ideal DC: An ideal Direct Current maintains a completely constant magnitude over time.
  • Pulsating DC: Pulsating Direct Current changes its magnitude over time, but maintains a single, constant polarity direction.
  • Frequency of DC: Equal to zero ($0\,\text{Hz}$). Frequency measures how many times a signal changes polarity per cycle; since DC never changes polarity, its frequency is zero.

• Alternating Current (AC):
  • Definition: Alternating Current is a current that periodically changes its direction and polarity over time.
  • Magnitude: The magnitude of AC is not constant and varies continuously with time.
  • Commercial Frequency: Standard commercial AC has a frequency $f = 50\,\text{Hz}$.
  • Polarity Changes per Cycle: In one complete cycle, AC changes its polarity twice (transitioning from positive to negative, and back from negative to positive).
  • Polarity Changes per Second: In one second, an AC signal operating at $50\,\text{Hz}$ changes its polarity $100\,\text{times}$ ($50 \times 2 = 100\,\text{times}$).
  • Peak Magnitudes per Cycle: In one cycle, the positive maximum magnitude occurs once, and the negative maximum magnitude occurs once.
  • Peak Magnitudes per Second: Overall, the magnitude of $50\,\text{Hz}$ AC reaches a peak maximum $100\,\text{times}$ in one second ($50\,\text{times}$ for positive maximum and $50\,\text{times}$ for negative maximum).

AC Terminology and Waveform Characteristics

• Cycles:
  • A set of complete positive ($+\text{ve}$) and negative ($-\text{ve}$) halves of alternating current is defined as one cycle.

• Frequency ($f$):

  • Definition: The number of complete cycles executed per second.
  • Formula: $f = \frac{1}{T}$
  • Unit: Hertz ($\text{Hz}$) or reciprocal seconds ($\text{s}^{-1}$).

• Time Period ($T$):

  • Definition: The total time required to complete one full cycle.
  • Formula: $T = \frac{1}{f}$
  • Unit: Seconds ($\text{s}$).

• Peak Value ($I_0$ or $V_0$):

  • Definition: The maximum positive ($+I_0$) or negative ($-I_0$) value attained by an alternating quantity during one cycle.

• Peak-to-Peak Value ($I_{pp}$ or $V_{pp}$):

  • Definition: The difference between the maximum positive peak value and the maximum negative peak value.
  • Formulas:
  • $I_{pp} = 2 I_0$
  • $V_{pp} = 2 V_0$
  • Calculation: $+I_0 - (-I_0) = 2 I_0$.
  • Numerical Example: If maximum peak voltage $V_0 = 100\,\text{V}$, peak-to-peak voltage $V_{pp} = 200\,\text{V}$.

• Average Value of AC ($I_{avg}$ / $V_{avg}$):

  • Over One Full Cycle: The average value of alternating current or voltage over one complete cycle is always equal to zero ($I_{avg} = 0$, $V_{avg} = 0$), because the positive half-cycle cancels the negative half-cycle.
  • Over a Half Cycle: The average value over a single half-cycle is non-zero ($I_{avg} \neq 0$, $V_{avg} \neq 0$).
  • Formulas for Half Cycle:
  • $I_{avg} = \frac{2 I_0}{\pi}$
  • $V_{avg} = \frac{2 V_0}{\pi}$

Instantaneous Value and Mathematical Representations

• Instantaneous Value ($I_{ins}$ / $V_{ins}$):

  • Definition: The value of alternating current or voltage at any specific instant of time $t$
  • General Formulas:
  • $I_{ins} = I = I_0 \sin(\theta)$
  • Since $\theta = \omega t = 2 \pi f t = \frac{2 \pi}{T} t$:
    • $I = I_0 \sin(\omega t)$
    • $I = I_0 \sin\left(\frac{2 \pi}{T} t\right)$
    • $I = I_0 \sin(2 \pi f t)$
  • Proportionality: $I \propto \sin(\theta)$

• Worked Problem on Waveform Equations:

  • Question: What is the instantaneous value of current for a sine wave graph completing one full cycle at time $T = 2\,\text{s}$?
  • Options Offered:
  • Option A: $I = I_0 \sin(4 \pi t)$
  • Option B: $I = I_0 \sin(\pi t)$
  • Option C: $I = I_0 \sin(3 \pi t)$
  • Option D: $I = I_0 \sin(6 \pi t)$
  • Step-by-Step Solution:
  • Standard equation: $I = I_0 \sin\left(\frac{2 \pi}{T} t\right)$
  • Substitute $T = 2\,\text{s}$ into the formula: $I = I_0 \sin\left(\frac{2 \pi}{2} t\right) = I_0 \sin(\pi t)$
  • Correct Choice: Option B ($I = I_0 \sin(\pi t)$).

Root Mean Square (RMS) Value of AC

• Definition:
  • The Root Mean Square value ($I_{rms}$) is defined as the square root of the mean of the squared values of alternating current over a given duration.

• Step-by-Step Derivation for Discrete Values:

  • Given three discrete current values: $I_1 = 2\,\text{A}$, $I_2 = 4\,\text{A}$, and $I_3 = 6\,\text{A}$.
  • Step 1 (Square values): $I_1^2 = 4\,\text{A}^2$, $I_2^2 = 16\,\text{A}^2$, $I_3^2 = 36\,\text{A}^2$.
  • Step 2 (Mean of squares): $\overline{I^2} = \frac{4 + 16 + 36}{3}$.
  • Step 3 (Root mean square): $I_{rms} = \sqrt{\frac{4 + 16 + 36}{3}}$.

• Derivation for Sinusoidal AC ($I = I_0 \sin(\omega t)$):

  • Key boundary values in a cycle: $0$ and peak $I_0$
  • Squares of key values: $0^2$ and $I_0^2$
  • Mean of squared values: $\overline{I^2} = \frac{0^2 + I_0^2}{2} = \frac{I_0^2}{2}$
  • Square root of mean: $I_{rms} = \sqrt{\frac{I_0^2}{2}} = \frac{I_0}{\sqrt{2}}$

• Numerical Relationships and Conversion Factors:

  • Formula: $I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0$
  • Percentage representation: $I_{rms} = 70.7\% \text{ of } I_0$
  • Numerical values: $\frac{1}{\sqrt{2}} \approx 0.707$, $\sqrt{2} \approx 1.414$

• Worked Examples:

  • If $I_0 = 100\,\text{A}$, then $I_{rms} = 70.7\% \times 100\,\text{A} = 70.7\,\text{A} \approx 70\,\text{A}$.
  • If $I_0 = 10\,\text{A}$, then $I_{rms} = 7\,\text{A}$.
  • If $I_0 = 50\,\text{A}$, then $I_{rms} \approx 35\,\text{A}$.
  • Problem: Given $I_{rms} = 100\,\text{A}$, calculate peak current I_0$.\n * I_0 = I_{rms} \times \sqrt{2} = 100\,\text{A} \times 1.414 = 141.4\,\text{A}\n\n* **Commercial Household Voltage Specs:**\n * Household rated voltage is an RMS value: V_{rms} = 220\,\text{V}.\n * Peak household voltage value V_0:\n * V_0 = V_{rms} \times \sqrt{2} = 220\,\text{V} \times 1.414 = 311\,\text{V}\n\n* **Physical Meaning / DC Equivalent Concept:**\n * The RMS value of AC is physically defined as that value of steady Direct Current (DC) which generates the exact same heating effect in a given resistor during a given time as the AC signal.\n * Example: An AC signal with peak current I_0 = 10\,\text{A}$($I_{rms} = 7\,\text{A}$) produces the same heating effect as a continuous DC current of$7\,\text{A} flowing through the same resistor.\n\n# Phase and Phase Difference (\phi)\n\n* **Definition of Phase (\phi):**\n * Phase is an angle that specifies the instantaneous state or value of alternating current or voltage at any specific instant of time.\n\n\n\n* **In-Phase Waves:**\n * Both waves increase and decrease simultaneously, reaching their respective peaks and zero crossings at the exact same time.\n * Both waves are in phase.\n * Phase difference: \Delta \phi = 0^\circ - 0^\circ = 0^\circ$or$\Delta \phi = 90^\circ - 90^\circ = 0^\circ.\n * \Delta \phi = 0^\circ\n\n* **Out-of-Phase Waves (180^\circ Phase Difference):**\n * As the magnitude of Wave I increases in the positive direction, the magnitude of Wave II increases in the negative direction simultaneously.\n * Wave I and Wave II are out of phase.\n * Phase difference: \Delta \phi = 180^\circ - 0^\circ = 180^\circ$or$\Delta \phi = 270^\circ - 90^\circ = 180^\circ.\n * \Delta \phi = 180^\circ\n\n* **Leading and Lagging Waves (90^\circ Phase Difference):**\n * When Wave I reaches its positive peak (90^\circ$) at the instant Wave II is at zero ($0^\circ):\n * Phase difference: \Delta \phi = 90^\circ - 0^\circ = 90^\circ$or$\Delta \phi = 180^\circ - 90^\circ = 90^\circ.\n * Wave I leads Wave II by 90^\circ.\n * Wave II lags behind Wave I by 90^\circ.\n\n# Vector and Phasor Representation of Alternating Quantities\n\n* **Conditions for Vector Representation:**\n 1. **Vector Length:** The length of the vector must be equal to or proportional to the peak value (I_0$or$V_0) of the alternating quantity.\n * Example: If peak current I_0 = 7\,\text{A}$, vector length =$7\,\text{cm}.\n 2. **Frequency of Rotation:** The frequency or speed of rotation of the vector must equal the frequency of the alternating quantity (1\,\text{rotation} = 1\,\text{cycle}).\n 3. **Initial Horizontal Alignment:** The vector is aligned horizontally at the reference time instant when the alternating quantity is zero and increasing in the positive direction.\n\n# AC Circuit Analysis: Pure Resistive Circuit\n\n* **Circuit Description:**\n * A circuit consisting of a pure resistor (R) connected in series with an AC voltage source.\n\n\n\n* **Derivation of Current and Voltage:**\n * Applied AC voltage: V = V_0 \sin(\omega t)$, so$V \propto \sin(\omega t)\n * According to Ohm's Law: I \propto V \implies I = \frac{V}{R} \implies V = I R\n * Substituting voltage into Ohm's Law: I = \frac{V_0 \sin(\omega t)}{R} = I_0 \sin(\omega t)\n * Where peak current I_0 = \frac{V_0}{R}.\n * Current equation: I = I_0 \sin(\omega t)$, so$I \propto \sin(\omega t)\n\n* **Phase Relationship:**\n * In a pure resistive circuit, voltage (V$) and current ($I) are in phase.\n * Phase difference: \Delta \phi = 0^\circ\n * Vector diagram: Both voltage and current vectors point in the same direction along the horizontal line.\n\n* **Opposition to Current:**\n * \text{Opposition} = \frac{\text{Potential Difference}}{\text{Current}} \implies R = \frac{V}{I}\n * Unit: Ohm (\Omega).\n * **Frequency Dependence:** Resistance R does not depend on the frequency of the AC source.\n\n* **Power Loss / Dissipation:**\n * Formula: P = I V\n * **Positive Power (+P$):** Represents power consumed/dissipated as heat (power spent,$\text{Kharcha हुआ}).\n * **Negative Power (-P$):** Represents power stored/gained (power stored,$\text{Stored हुआ}).\n * **Power in Resistor:**\n * Positive half-cycle: (+I) \times (+V) = +P\n * Negative half-cycle: (-I) \times (-V) = +P\n * Power is always positive (+P) throughout both half-cycles.\n * **Key Conclusion:** Power is never stored in a resistor; power is continuously dissipated as heat.\n\n# AC Circuit Analysis: Pure Capacitive Circuit\n\n* **Definitions:**\n * **Capacitor:** A device that stores energy/charge in the form of an electric field.\n * **Capacitance (C):** The ability of a capacitor to store charge/electric energy.\n * Formula: Q \propto V \implies Q = C V \implies C = \frac{Q}{V}\n * Unit: Farad (\text{F}$), where$\text{Farad} = \frac{\text{Coulomb}}{\text{Volt}}.\n\n* **Circuit Description:**\n * A circuit consisting of a pure capacitor (C) connected in series with an AC source.\n\n* **Derivation of Current:**\n * Applied voltage: V = V_0 \sin(\omega t)\n * Charge on capacitor: Q = C V = C V_0 \sin(\omega t)\n * Current I = \frac{dQ}{dt} = \frac{d}{dt}\left(C V_0 \sin(\omega t)\right) = C V_0 \frac{d}{dt}\left(\sin(\omega t)\right)\n * Derivative rule: \frac{d}{dt}\left(\sin(\omega t)\right) = \omega \cos(\omega t)\n * I = C V_0 \omega \cos(\omega t) = (\omega C V_0) \cos(\omega t) = I_0 \cos(\omega t)\n * Where peak current I_0 = \omega C V_0 = \frac{V_0}{1 / (\omega C)}.\n * Using identity \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right)$:$I = I_0 \sin\left(\omega t + \frac{\pi}{2}\right)\n\n* **Phase Relationship:**\n * Current leads voltage by 90^\circ$(or$\frac{\pi}{2}$radians), or voltage lags behind current by$90^\circ\n * Phase difference: \Delta \phi = 90^\circ\n * At time t = 0$($\omega t = 0^\circ$): Voltage$V = V_0 \sin(0^\circ) = 0$, while current$I = I_0 \cos(0^\circ) = I_0 (maximum).\n\n* **Power Dissipation:**\n * Power dissipation P = 0 over a full cycle. Energy stored during charging quarter-cycles is completely returned during discharging quarter-cycles.\n\n* **DC Blocking Property:**\n * A capacitor blocks Direct Current (DC) and allows Alternating Current (AC) to flow through.\n\n* **Capacitive Reactance (X_C):**\n * **Definition:** Opposition offered by a capacitor to the flow of alternating current.\n * **Formulas:**\n * X_C = \frac{V_{rms}}{I_{rms}} = \frac{\text{Volt}}{\text{Ampere}} = \text{Ohm}\,(\Omega)\n * X_C = \frac{1}{\omega C} = \frac{1}{2 \pi f C} = \frac{T}{2 \pi C}\n * **Proportionality Relationships:**\n * X_C \propto \frac{1}{\omega}\n * X_C \propto \frac{1}{f}\n * X_C \propto T$(directly proportional to time period$T)\n * X_C \propto \frac{1}{C}\n\n* **Conceptual Question:**\n * **Question:** When frequency of the AC source increases, current through a capacitor...\n * **Options:** A) Increases, B) Decreases, C) Unchanged, D) All are correct.\n * **Explanation:** Since f \uparrow \implies X_C \downarrow$, lowering opposition causes current$I \uparrow.\n * **Correct Option:** A (Increases).\n\n# AC Circuit Analysis: Pure Inductive Circuit\n\n* **Definitions:**\n * **Inductor:** A passive electrical device used to store electrical energy in the form of a magnetic field.\n * **Inductance (L$):** The ability of an inductor to store energy in its magnetic field. Unit: Henry ($\text{H}).\n\n* **Circuit Description:**\n * A circuit consisting of a pure inductor (L) connected in series with an AC source.\n\n* **Derivation of Voltage:**\n * Self-induced EMF equation: e = L \frac{dI}{dt} \implies V = L \frac{dI}{dt}\n * Let current I = I_0 \sin(\omega t)$, so$I \propto \sin(\omega t)\n * Substituting I$into voltage formula:$V = L \frac{d}{dt}\left(I_0 \sin(\omega t)\right) = L I_0 \frac{d}{dt}\left(\sin(\omega t)\right) = L I_0 \omega \cos(\omega t)\n * V = (\omega L I_0) \cos(\omega t) = V_0 \cos(\omega t)\n * Where peak voltage V_0 = \omega L I_0\n * Using identity \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right)$:$V = V_0 \sin\left(\omega t + \frac{\pi}{2}\right)$, so$V \propto \cos(\omega t)\n\n\n\n* **Phase Relationship:**\n * Voltage leads current by 90^\circ$(or$\frac{\pi}{2}$radians), or current lags behind voltage by$90^\circ\n * Phase difference: \Delta \phi = 90^\circ\n\n* **Inductive Reactance (X_L):**\n * **Definition:** Opposition offered by an inductor to the flow of alternating current.\n * **Formulas:**\n * X_L = \frac{V_{rms}}{I_{rms}} = \frac{\text{Volt}}{\text{Ampere}} = \text{Ohm}\,(\Omega)\n * X_L = \omega L = 2 \pi f L = \frac{2 \pi L}{T}\n * **Proportionality Relationships:**\n * X_L \propto \omega\n * X_L \propto f\n * X_L \propto \frac{1}{T}\n * X_L \propto L\n\n* **Power Dissipation:**\n * Power loss over a complete cycle in a pure inductor is zero (P = 0$). Power alternates between positive ($+P$) and negative ($-P) in consecutive quarter-cycles as energy is stored in and returned from the magnetic field.\n\n* **Frequency Effect on Resistance Question:**\n * **Question:** If frequency of AC increases, then resistance R...\n * **Options:** A) Increases, B) Decreases, C) Unchanged, D) None.\n * **Explanation:** Resistance does not depend upon the frequency of the AC source.\n * **Correct Option:** C (Unchanged).\n\n# RC Series Circuit Analysis\n\n* **Circuit Description:**\n * A circuit that consists of a resistor (R$) and a capacitor ($C) connected in series with an AC voltage source.\n\n* **Series Circuit Characteristics:**\n * Current I_{rms} remains same/identical through all series components.\n * Potential differences across resistor (V_R$) and capacitor ($V_C) are different.\n\n* **Individual Voltage Vectors:**\n * Across Resistor (R$): Voltage$V_R$is in phase with$I_{rms}$($\Delta \phi = 0^\circ).\n * Across Capacitor (C$): Voltage$V_C$lags current$I_{rms}$by$90^\circ$($\Delta \phi = 90^\circ).\n\n* **Combined Phase Relationship:**\n * Current I_{rms}$leads total voltage$V_{rms}$by phase angle$\theta$, where$0^\circ < \theta < 90^\circ\n\n* **Derivation of Impedance (Z):**\n * **Impedance Definition:** Combined opposition offered by resistor and capacitor to the flow of alternating current.\n * Applying Pythagorean theorem to voltage triangle (H^2 = B^2 + P^2):\n * (V_{rms})^2 = (V_R)^2 + (V_C)^2\n * Substituting V_{rms} = I_{rms} Z$,$V_R = I_{rms} R$, and$V_C = I_{rms} X_C:\n * (I_{rms} Z)^2 = (I_{rms} R)^2 + (I_{rms} X_C)^2\n * I_{rms}^2 Z^2 = I_{rms}^2 R^2 + I_{rms}^2 X_C^2\n * Dividing both sides by I_{rms}^2$gives:$Z^2 = R^2 + X_C^2\n * Z = \sqrt{R^2 + X_C^2}\n * Substituting X_C = \frac{1}{\omega C} = \frac{1}{2 \pi f C}:\n * Z = \sqrt{R^2 + \left(\frac{1}{\omega C}\right)^2} = \sqrt{R^2 + \left(\frac{1}{2 \pi f C}\right)^2}\n\n* **Impedance Diagram and Phase Angle (\theta):**\n * General vector conversion: A_x = A \cos(\theta)$,$A_y = A \sin(\theta)$,$\tan(\theta) = \frac{A_y}{A_x} \implies \theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)\n * For RC Impedance Diagram:\n * \tan(\theta) = \frac{X_C}{R} = \frac{V_C}{V_R}\n * Phase angle equation: \theta = \tan^{-1}\left(\frac{X_C}{R}\right) = \tan^{-1}\left(\frac{1}{\omega C R}\right) = \tan^{-1}\left(\frac{1}{2 \pi f C R}\right)\n\n# RL Series Circuit Analysis\n\n* **Circuit Description:**\n * A circuit consisting of a resistor (R$) and an inductor ($L) connected in series with an AC source.\n\n\n\n* **Series Circuit Characteristics:**\n * Same current I_{rms} flows through both components.\n * Voltages V_R$and$V_L are different across components.\n\n* **Component Phase Relations:**\n * Across Resistor (R$): Voltage$V_R$is in phase with$I_{rms}.\n * Across Inductor (L$): Voltage$V_L$leads current$I_{rms}$by$90^\circ\n\n* **Combined Phase Relationship:**\n * Total voltage V_{rms}$leads current$I_{rms}$by phase angle$\theta$, where$0^\circ < \theta < 90^\circ\n\n* **Derivation of Impedance (Z):**\n * **Impedance Definition:** Combined opposition offered by resistor and inductor to the flow of alternating current.\n * Applying Pythagorean theorem (H^2 = B^2 + P^2):\n * (V_{rms})^2 = (V_R)^2 + (V_L)^2\n * (I_{rms} Z)^2 = (I_{rms} R)^2 + (I_{rms} X_L)^2\n * I_{rms}^2 Z^2 = I_{rms}^2 R^2 + I_{rms}^2 X_L^2\n * Dividing both sides by I_{rms}^2$gives:$Z^2 = R^2 + X_L^2\n * Z = \sqrt{R^2 + X_L^2}\n * Substituting X_L = \omega L = 2 \pi f L.\n * Z = \sqrt{R^2 + (\omega L)^2} = \sqrt{R^2 + (2 \pi f L)^2}\n\n* **Impedance Diagram and Phase Angle (\theta):**\n * \tan(\theta) = \frac{X_L}{R} = \frac{V_L}{V_R}\n * Phase angle equation: \theta = \tan^{-1}\left(\frac{X_L}{R}\right) = \tan^{-1}\left(\frac{\omega L}{R}\right) = \tan^{-1}\left(\frac{2 \pi f L}{R}\right)$$