Understanding Purely Resistive, Inductive, and Capacitive Circuits

Purely resistive, purely inductive, and purely capacitive circuits represent three fundamental types of AC circuits in which only resistance, inductance, or capacitance is present, respectively. Their behavior is essential for understanding alternating current systems, as each circuit exhibits a unique relationship between current and voltage, as well as distinctive energy and power properties.

Definitions and Basic Characteristics

A purely resistive circuit contains only resistors and exhibits no phase difference between the current and the voltage. The voltage and current waveforms reach their maximum and zero values simultaneously. Power is consumed continuously as heat.

A purely inductive circuit contains only an inductor, leading to the current lagging the voltage by $90^\circ$. The energy is alternately stored in the magnetic field and returned to the source, resulting in zero average power consumption.

A purely capacitive circuit comprises only capacitors, and the current leads the voltage by $90^\circ$. Energy is alternately stored and released from the electric field of the capacitor, with the average power consumed over a cycle being zero.

The phase relationships and energy characteristics of these circuits are tested frequently in JEE and other competitive exams. Additional details can be found in related topics such as Current Electricity.

Mathematical Representation and Standard Formulas

For each type of circuit, the mathematical relationship between the alternating voltage and the resulting current is distinct and fundamental for solving associated physics problems.

In a purely resistive circuit, if $V = V_0 \sin(\omega t)$, the current is $I = I_0 \sin(\omega t)$, where $I_0 = \dfrac{V_0}{R}$. The impedance is equal to resistance, $Z = R$, and phase angle $\phi = 0^\circ$.

In a purely inductive circuit, $V = V_0 \sin(\omega t)$ and $I = I_0 \sin(\omega t - 90^\circ)$, with $I_0 = \dfrac{V_0}{\omega L}$, $Z = \omega L$, and phase angle $\phi = +90^\circ$.

For a purely capacitive circuit, $V = V_0 \sin(\omega t)$ and $I = I_0 \sin(\omega t + 90^\circ)$, with $I_0 = V_0 \omega C$, $Z = \dfrac{1}{\omega C}$, and phase angle $\phi = -90^\circ$.

Reactance and impedance calculation methods for each circuit are frequently applied in topics like RC Circuit.

Phasor Diagrams and Phase Relationships

Phasor diagrams are used to represent the phase difference and magnitude of alternating voltage and current in these circuits. They provide a visual comparison of current and voltage orientation for each circuit type.

In a purely resistive circuit, the voltage and current phasors are aligned, indicating no phase difference. In a purely inductive circuit, the voltage phasor leads the current phasor by $90^\circ$. In a purely capacitive circuit, the current phasor leads the voltage phasor by $90^\circ$.

These phase differences are critical for graphical analysis and are foundational for understanding advanced alternating current topics, which can also be reviewed in the Electromagnetic Induction and AC Notes.

Comparison Table: Resistive, Inductive, and Capacitive Circuits

Equations for Current and Power in Each Circuit

The current in a purely resistive circuit is $I = \dfrac{V}{R}$. True power consumed is $P = VI$ as voltage and current are in phase, resulting in a power factor of 1.

In a purely inductive circuit, current is $I = \dfrac{V}{\omega L}$. The average power consumed is zero, as the power alternates between source and the magnetic field of the inductor, with phase difference $90^\circ$ making the power factor zero.

For a purely capacitive circuit, $I = V \omega C$. Here, the alternating power is stored and returned by the capacitor’s electric field, leading to a zero average value and a power factor of zero.

For fundamental principles on electric fields, see the basics in Electric Field Lines.

Examples of Purely Resistive, Inductive, and Capacitive Circuits

• Resistive: An incandescent bulb in AC supply
• Inductive: An ideal air core inductor in AC
• Capacitive: Quality air dielectric capacitor in AC

These idealized examples are widely referenced for conceptual and numerical questions in entrance and board examinations.

Sample Calculation: Connected to AC Source

Consider a $20\,\mathrm{V}$, $50\,\mathrm{Hz}$ AC source connected independently across a $10\,\Omega$ resistor, a $200\,\mathrm{mH}$ inductor, and a $50\,\mu\mathrm{F}$ capacitor. The angular frequency is $\omega = 2\pi f = 314\,\mathrm{rad/s}$.

For the resistor: $I_R = \dfrac{20}{10} = 2\,\mathrm{A}$. For the inductor: $I_L = \dfrac{20}{314 \times 0.2} \approx 0.32\,\mathrm{A}$. For the capacitor: $I_C = 20 \times 314 \times 50 \times 10^{-6} \approx 0.314\,\mathrm{A}$.

Such stepwise calculations are fundamental to AC circuit numericals and are supplemented in detailed topics like Purely Resistive, Inductive, and Capacitive Circuit.

JEE Main Exam Tips and Common Errors

• Remember: in inductive circuits, current lags voltage by $90^\circ$
• In capacitive circuits, current leads voltage by $90^\circ$
• Average power is zero for purely inductive or capacitive circuits
• Use rad/s for $\omega$ not Hz
• Real elements are never perfectly ideal

Conceptual clarity on phase relationships and impedance values is critical for scoring in questions related to AC circuits.

Summary of Key Points

• Resistive: power consumed, current and voltage in phase, $Z=R$
• Inductive: zero average power, current lags voltage, $Z=\omega L$
• Capacitive: zero average power, current leads voltage, $Z=1/\omega C$
• Phase difference is a core distinguishing feature

For advanced understanding, topics such as impedance, reactance, and power factor are interconnected with AC circuit analysis and are foundational for Electronic Devices studies in the curriculum.