Introduction

Introduction

The World Runs on Alternating Current

Every time you switch on a light, charge your phone, or turn on a fan, you are using alternating current (AC). The electric supply in our homes and offices is not a steady, unchanging current like the one from a battery. Instead, it is a voltage that varies sinusoidally with time — rising, falling, reversing direction, and repeating this cycle many times every second.

In the previous chapters, we studied direct current (DC) sources — batteries and cells that push charge in one direction continuously. While DC is essential for portable devices and electronic circuits, the electrical energy sold by power companies worldwide is transmitted and distributed almost exclusively as alternating current. Understanding why AC dominates modern electrical systems, and how it behaves in circuits, is the heart of this chapter.

{{VISUAL: diagram: comparison table showing DC voltage as a flat horizontal line and AC voltage as a sinusoidal wave plotted against time, both on the same axes}}

What is Alternating Voltage and Alternating Current?

An alternating voltage is a potential difference that varies periodically with time, typically as a sine function:

v = vₘ sin(ωt)

Here, vₘ is the amplitude or peak voltage, ω is the angular frequency, and t is time. The voltage oscillates between +vₘ and −vₘ, crossing zero twice in every cycle.

{{KEY: type=definition | title=Alternating Voltage | text=A voltage that varies sinusoidally with time, periodically reversing its polarity. Mathematically represented as v = vₘ sin(ωt), where vₘ is the peak voltage and ω is the angular frequency.}}

When such a voltage is applied across a circuit, it drives a current that also varies sinusoidally with time. This is called alternating current (AC). Unlike direct current, AC reverses direction periodically. In India, the standard AC supply has a frequency of 50 Hz, meaning the current completes 50 full cycles every second.

{{VISUAL: photo: realistic photo of a standard Indian household electrical socket and plug, with a digital multimeter displaying 230V AC}}

{{KEY: type=concept | title=AC vs DC | text=In direct current (DC), charge flows steadily in one direction. In alternating current (AC), charge oscillates back and forth, reversing direction periodically. AC voltage varies sinusoidally; DC voltage remains constant.}}

Why Do We Prefer AC Over DC?

If DC is simpler to understand and analyze, why does the entire electrical grid run on AC? The answer lies in three major practical advantages:

1. Easy Voltage Transformation

AC voltages can be easily and efficiently converted from one level to another using transformers. A transformer is a simple device with no moving parts, made of two coils wound around an iron core. It can step up (increase) or step down (decrease) AC voltage with minimal energy loss.

This is crucial for power transmission. Electrical energy is generated at moderate voltages (typically 11–25 kV) at power stations, then stepped up to very high voltages (132 kV, 220 kV, or even 765 kV) for transmission over long distances. Near homes and factories, it is stepped down again to safer, usable levels (230 V for homes, 400 V for three-phase industrial supply).

Why does high voltage matter? Power loss during transmission is proportional to I²R, where I is the current and R is the resistance of the transmission line. For a given power P = VI, increasing voltage V reduces current I. Lower current means drastically lower resistive losses, making transmission economical.

{{FORMULA: expr=P_loss = I² R | symbols=P_loss:power lost as heat (W), I:current in transmission line (A), R:resistance of transmission line (Ω)}}

{{KEY: type=points | title=Advantages of AC over DC | text=- AC voltage can be stepped up or down efficiently using transformers.

• High-voltage AC transmission reduces current, minimizing I²R power losses.
• AC generators are simpler, more robust, and cheaper than DC generators.
• AC motors are self-starting and require less maintenance than DC motors.}}

2. Economic Long-Distance Transmission

Because AC can be transmitted at high voltages with low currents, the energy loss over long-distance transmission lines is minimized. This makes the entire electrical grid economically viable. Imagine transmitting power at 230 V over hundreds of kilometers — the resistive losses would consume most of the energy before it reached consumers.

{{VISUAL: diagram: schematic diagram of AC power transmission from power station to home, showing step-up transformer at power station (11 kV to 220 kV), high-voltage transmission lines, step-down transformer at substation (220 kV to 11 kV), and final step-down transformer near homes (11 kV to 230 V)}}

3. Versatility in Everyday Devices

AC circuits exhibit special properties that are exploited in countless devices. When you tune a radio to a favourite station, you are adjusting an LC circuit (inductor-capacitor circuit) to resonate at the station's frequency — a phenomenon unique to AC circuits. Similarly, AC motors, used in fans, refrigerators, and washing machines, are simpler and more durable than their DC counterparts.

{{KEY: type=exam | title=NCERT Emphasis | text=NCERT highlights the economic transmission of electrical energy and the role of transformers as the primary reasons for preferring AC over DC. Expect 2-3 mark questions asking you to justify this preference.}}

What Will We Study in This Chapter?

In this chapter, we will explore how AC voltages and currents behave when applied to different circuit elements:

1. Pure resistors — where voltage and current remain in phase.
2. Pure inductors — where current lags voltage by 90°.
3. Pure capacitors — where current leads voltage by 90°.
4. LCR circuits — combining all three elements, leading to the phenomenon of resonance.
5. Power in AC circuits — understanding the difference between peak, RMS, and average power.
6. Transformers — the device that makes AC transmission practical.

Each element responds to AC in a unique way, introducing the concept of reactance and impedance. We will also learn why the average current in AC is zero, but the average power is not.

{{VISUAL: chart: three separate graphs stacked vertically, each showing voltage (dashed line) and current (solid line) vs time for (a) resistor - in phase, (b) inductor - current lagging, (c) capacitor - current leading}}

AC circuits are the backbone of modern electrical engineering. Mastering them opens the door to understanding everything from household wiring to radio communication.

A Note on Terminology

You will often encounter phrases like "AC voltage" and "AC current". Strictly speaking, these are redundant or contradictory — "AC" already stands for alternating current, so "AC current" literally means "alternating current current." Similarly, "AC voltage" means "alternating current voltage."

Despite this, these phrases have become so universally accepted in engineering and physics that we use them throughout this chapter. Just remember: AC refers to any electrical quantity that varies sinusoidally with time.

{{ZOOM: title=Historical Context | text=The AC vs DC debate — known as the "War of Currents" — was famously fought between Nikola Tesla (AC advocate) and Thomas Edison (DC advocate) in the late 1800s. Tesla's AC system, championed by George Westinghouse, ultimately won due to transformers and long-distance transmission efficiency.}}

By the end of this chapter, you will not only understand why the world runs on AC, but also how to analyze AC circuits mathematically and apply these principles to real-world scenarios — from power transmission to radio tuning. Let's begin by studying the simplest case: AC voltage applied to a resistor.

AC Voltage Applied to a Resistor

Page 2: AC Voltage Applied to a Resistor

When we connect a purely resistive circuit to an alternating voltage source, the behavior differs dramatically from DC circuits — not in how Ohm's law applies, but in how the voltage and current vary continuously with time. Understanding this relationship is the foundation for analyzing all AC circuits.

The Circuit and Applied Voltage

Consider a resistor of resistance R connected directly to an AC voltage source. The source produces a sinusoidally varying potential difference across its terminals.

{{VISUAL: diagram: circuit diagram showing an AC source (represented by ~ symbol) connected to a single resistor R, with voltage v and current i labeled}}

The instantaneous voltage at any time t is given by:

{{FORMULA: expr=v = v_m sin ωt | symbols=v:instantaneous voltage (V), v_m:peak or maximum voltage (V), ω:angular frequency (rad/s), t:time (s)}}

Here, v_m is the amplitude — the maximum value the voltage reaches during one complete cycle. The angular frequency ω determines how rapidly the voltage oscillates and is related to the regular frequency by ω = 2πf.

{{KEY: type=definition | title=Instantaneous Voltage in AC | text=The voltage across an AC source at any instant t is given by v = v_m sin ωt, where v_m is the peak voltage and ω is the angular frequency in radians per second.}}

Current Through the Resistor

To find the current, we apply Kirchhoff's loop rule (Σε(t) = 0) around the circuit. Since there's only the source and resistor:

v_m sin ωt = i R

Rearranging for current:

i = (v_m / R) sin ωt

Because R is constant, we can write:

i = i_m sin ωt

where the current amplitude is:

i_m = v_m / R

{{KEY: type=concept | title=Ohm's Law in AC Circuits | text=For a pure resistor, Ohm's law holds in the same form as DC. The peak current i_m equals the peak voltage v_m divided by resistance R. Both voltage and current vary sinusoidally with time.}}

{{VISUAL: chart: graph showing sinusoidal voltage v and current i plotted against time t, both curves overlapping perfectly with same phase, reaching maxima and minima simultaneously}}

Notice something crucial: the voltage and current are in phase. They reach their maximum values, zero values, and minimum values at exactly the same instants. There is no time lag between voltage and current in a resistive circuit.

{{ZOOM: title=Why "in phase" matters | text=In AC circuits containing inductors or capacitors, voltage and current do NOT peak at the same time — they become "out of phase." The purely resistive case is special because phase difference is zero, making analysis simpler and serving as the reference for more complex circuits.}}

Average Current and the Zero Problem

Because the current varies sinusoidally — spending equal time in positive and negative directions — the average current over one complete cycle is zero:

I_avg = 0

Does this mean no energy is consumed? Absolutely not.

Power dissipation depends on i²R, and since i² is always positive (whether i itself is positive or negative), energy is continuously dissipated as heat.

{{KEY: type=points | title=Why Average Current is Zero but Power is Not | text=- The current alternates direction each half-cycle, so its algebraic average is zero.

• Power dissipation depends on i² (Joule heating), which is always positive.
• Energy is continuously converted to heat even though net charge displacement per cycle is zero.
• We need a new measure of "effective" current to quantify power properly.}}

Instantaneous and Average Power

The instantaneous power dissipated at any moment is:

p = i² R = i_m² R sin² ωt

To find meaningful power consumption, we take the time-average over one cycle:

P = <i² R> = i_m² R <sin² ωt>

Using the trigonometric identity sin² ωt = (1/2)(1 − cos 2ωt) and noting that the average of cos 2ωt over a full cycle is zero:

<sin² ωt> = 1/2

Therefore, the average power is:

P = (1/2) i_m² R

{{VISUAL: chart: plot showing instantaneous power p = i_m² R sin² ωt oscillating between 0 and i_m² R, with horizontal dashed line at P = (1/2) i_m² R indicating average power}}

Root Mean Square (RMS) Values

To express AC power in the same form as DC power (P = I²R), we define a special equivalent current called the root mean square or effective current.

{{KEY: type=definition | title=RMS or Effective Current | text=The RMS current I (or I_rms) is defined as the square root of the mean (average) of the square of the instantaneous current. It represents the equivalent DC current that would dissipate the same average power in a resistor.}}

Mathematically:

I = √<i²> = √(i_m² <sin² ωt>) = √(i_m² × 1/2)

I = i_m / √2 = 0.707 i_m

Similarly, the RMS voltage is:

V = v_m / √2 = 0.707 v_m

Now the average power becomes beautifully simple:

P = I² R = V² / R = I V

These equations mirror the DC forms exactly, which is why RMS values are universally used in AC measurements.

{{KEY: type=exam | title=RMS Values in Numerical Problems | text=CBSE numerical problems almost always give RMS values (household supply is 220 V RMS, not peak). To find peak values, multiply by √2 ≈ 1.414. To convert peak to RMS, divide by √2. Always check what the question provides and what it asks for.}}

{{VISUAL: diagram: comparison table showing relationship between peak and RMS values, with formulas I = i_m/√2 and V = v_m/√2, and numerical example showing 220 V RMS corresponds to 311 V peak}}

Practical Significance

When we say household voltage is 220 V, we mean 220 V RMS. The actual peak voltage swinging across the wires is:

v_m = √2 × V = 1.414 × 220 V = 311 V

All AC ammeters and voltmeters are calibrated to read RMS values, because these directly relate to power consumption. A "100 W" bulb rated at "220 V" uses those RMS figures — the resistor dissipates 100 watts of average power when connected to 220 V RMS supply.

Key Takeaway: RMS values allow us to treat AC circuits using the same power and voltage-current relations as DC circuits, making calculations intuitive and measurements practical.

{{KEY: type=concept | title=Phase Relationship in Resistive Circuits | text=In a purely resistive AC circuit, voltage and current are in phase — they reach their peak, zero, and minimum values at the same instants. This is unique to resistors; inductors and capacitors introduce phase differences that we will explore in subsequent sections.}}

Representation of AC Current and Voltage by Rotating Vectors — Phasors

Representation of AC Current and Voltage by Rotating Vectors — Phasors

In our study of resistive AC circuits, we saw that current and voltage remain in phase — they rise and fall together. However, when we introduce inductors and capacitors into AC circuits, this simple relationship breaks down. Current and voltage begin to exhibit a phase difference, meaning one reaches its peak before or after the other.

To visualize and analyze these phase relationships effectively, physicists and engineers use a powerful graphical tool called phasors. Understanding phasors is essential not only for solving AC circuit problems but also for grasping how real-world electrical systems — from power grids to radio transmitters — actually work.

What is a Phasor?

{{KEY: type=definition | title=Phasor | text=A phasor is a rotating vector that represents the amplitude and phase of a sinusoidally varying quantity (voltage or current). The phasor rotates counterclockwise about the origin with angular speed ω, and its vertical projection at any instant gives the instantaneous value of the oscillating quantity.}}

Let's break this down carefully. Imagine a vector of length v_m (the peak voltage) rotating in the anticlockwise direction about the origin with angular frequency ω. At any time t, the vertical component of this rotating vector equals v_m sin(ωt) — exactly the expression for our AC voltage!

Similarly, a current phasor of length i_m rotating at the same angular speed ω has a vertical projection of i_m sin(ωt − φ), where φ is the phase difference between voltage and current.

{{VISUAL: diagram: a phasor vector V rotating counterclockwise with angular velocity ω, showing its vertical projection v = v_m sin(ωt) on the y-axis and the angle ωt from the horizontal axis}}

Why phasors are not true vectors: Although we represent phasors as rotating arrows, voltage and current are scalar quantities, not vectors. They have no direction in physical space. The phasor representation is a mathematical convenience — it turns out that the amplitudes and phases of harmonically varying scalars combine using the same rules as vector projections. This coincidence makes phasors an incredibly useful tool.

Phasor Diagram for a Resistive Circuit

Let's revisit the purely resistive circuit from earlier. An AC source with voltage v = v_m sin(ωt) is connected to a resistor R. We found that the current is:

i = i_m sin(ωt)

where i_m = v_m / R.

Notice that both voltage and current are sine functions of the same argument (ωt) — they have zero phase difference. In the phasor diagram, this means the voltage phasor V and current phasor I point in the same direction at all times.

{{VISUAL: diagram: phasor diagram showing voltage phasor V and current phasor I aligned in the same direction, both rotating counterclockwise, representing zero phase difference for a resistive circuit}}

{{KEY: type=concept | title=Phase Relationship in Resistive Circuits | text=In a purely resistive AC circuit, the voltage and current phasors are always aligned. The current is in phase with the voltage, meaning they reach their maximum, zero, and minimum values simultaneously. The phase angle φ between them is zero.}}

As both phasors rotate together at angular speed ω, their vertical projections trace out the sinusoidal graphs of v and i versus time. The fact that they stay aligned confirms that voltage and current oscillate in step.

Constructing and Interpreting Phasor Diagrams

Step-by-Step Construction

1. Choose a reference: Typically, we take the voltage phasor V as the reference, drawn along the positive x-axis at t = 0.
2. Draw the current phasor I: Position it at an angle φ relative to V, where φ is the phase difference. If current lags voltage, draw I at angle −φ (clockwise from V). If current leads, draw I at angle +φ (counterclockwise from V).
3. Rotate the entire diagram: Imagine both phasors rotating counterclockwise at angular frequency ω. The vertical projections give instantaneous values.

{{KEY: type=points | title=Phasor Diagram Conventions | text=- Phasor length represents the amplitude (peak value) of voltage or current.

• The angle between phasors represents the phase difference φ.
• Counterclockwise rotation corresponds to increasing time.
• Vertical projection at any instant gives the instantaneous value of the quantity.}}

Reading Phase Information

The angle φ between the voltage and current phasors tells us:

• φ = 0: Current and voltage are in phase (pure resistor).
• φ = +π/2 (or +90°): Current leads voltage by a quarter cycle (pure capacitor, as we'll see later).
• φ = −π/2 (or −90°): Current lags voltage by a quarter cycle (pure inductor).

{{VISUAL: chart: graph showing sinusoidal voltage v and current i plotted against ωt, with current lagging voltage by π/2, demonstrating the phase difference visually in the time domain}}

Why Phasors Matter: A Practical Perspective

Simplifying AC Circuit Analysis

In DC circuits, we use Kirchhoff's laws with simple algebra. In AC circuits, voltages and currents are time-varying sinusoids, and adding them algebraically is cumbersome. Phasors convert these sinusoids into rotating vectors, allowing us to:

• Add and subtract AC quantities using vector addition instead of trigonometric identities.
• Visualize phase relationships at a glance.
• Calculate impedance (the AC equivalent of resistance) using phasor algebra.

{{ZOOM: title=Phasors and Complex Numbers | text=Advanced AC circuit analysis often represents phasors as complex numbers, where the real part is the horizontal projection and the imaginary part is the vertical projection. The magnitude gives amplitude, and the argument gives phase. This formalism is central to electrical engineering.}}

Real-World Applications

• Power transmission: Engineers use phasor diagrams to analyze three-phase power systems, ensuring voltage and current remain balanced.
• Radio frequency circuits: Phasors help design filters, amplifiers, and oscillators by tracking how signals shift in phase.
• Impedance matching: In audio systems and antennas, matching impedances (magnitudes and phases) maximizes power transfer.

{{KEY: type=exam | title=CBSE Exam Focus | text=CBSE frequently asks you to draw phasor diagrams for R, L, C circuits and to explain the phase relationship between v and i. Practice sketching phasors with correct angles and labeling peak values. 3-mark questions often test your understanding of what "leading" or "lagging" means graphically.}}

Connecting Phasors to the Time Domain

It's crucial to remember: phasors are a snapshot tool. At any instant t, the actual voltage and current are the vertical projections of the rotating phasors. The full time-domain behavior emerges as these phasors rotate.

For example, consider two phasors V and I separated by angle φ. At t = 0:

• V has projection v_m sin(0) = 0
• I has projection i_m sin(−φ) (non-zero if φ ≠ 0)

As time progresses, both rotate, and their projections oscillate sinusoidally. The phasor diagram encodes this entire oscillation in a single, elegant picture.

{{VISUAL: diagram: combined view showing a phasor diagram on the left with V and I separated by phase angle φ, and on the right the corresponding sinusoidal graphs of v and i versus ωt, illustrating how projections create the time-domain waveforms}}

Phasors transform the complexity of time-varying AC signals into the simplicity of rotating geometry — making invisible phase relationships visible.

AC Voltage Applied to an Inductor

AC Voltage Applied to an Inductor

When an inductor is connected to an alternating current (AC) source, it behaves very differently from a resistor. While a resistor opposes current uniformly, an inductor opposes changes in current through electromagnetic induction. This creates a fascinating phase relationship between voltage and current that is fundamental to understanding AC circuits.

Let us derive the behaviour of a purely inductive AC circuit — one where the inductor has negligible resistance in its windings.

The Circuit Setup

Consider an AC source with instantaneous voltage v = vₘ sin ωt connected across an inductor of self-inductance L. The circuit is shown below.

{{VISUAL: diagram: labeled circuit diagram showing an AC source connected to a pure inductor L with voltage v and current i marked}}

Since the circuit is purely inductive (no resistor), applying Kirchhoff's loop rule gives:

v - L (di/dt) = 0

The term L (di/dt) represents the self-induced emf in the inductor. The negative sign comes from Lenz's law — the induced emf opposes the change in current.

{{KEY: type=concept | title=Self-Induced EMF in an Inductor | text=When current through an inductor changes, a back emf is induced that opposes the change. This is quantified as ε = -L(di/dt), where L is the self-inductance. This opposition is the basis of inductive reactance in AC circuits.}}

Deriving the Current Expression

Rearranging the equation:

di/dt = v/L = (vₘ/L) sin ωt

This tells us that the rate of change of current varies sinusoidally with time, in phase with the applied voltage.

To find the current i(t), we integrate both sides with respect to time:

1. Integrate di/dt:

   ∫ di = (vₘ/L) ∫ sin(ωt) dt
   

2. Evaluate the integral:

   i = -(vₘ/ωL) cos(ωt) + constant
   

3. Determine the constant: Since the AC source oscillates symmetrically about zero, the current must also oscillate symmetrically about zero with no DC component. Hence, the integration constant is zero.

4. Use trigonometric identity: Using -cos(ωt) = sin(ωt - π/2), we get:

{{FORMULA: expr=i = iₘ sin(ωt - π/2) | symbols=i:instantaneous current (A), iₘ:peak current (A), ω:angular frequency (rad/s), t:time (s)}}

Where the peak current is:

iₘ = vₘ/(ωL)

{{KEY: type=definition | title=Inductive Reactance | text=The quantity ωL is called inductive reactance, denoted by Xₗ. It is measured in ohms (Ω) and plays the same role in an AC inductive circuit as resistance does in a DC circuit. Mathematically, Xₗ = ωL = 2πνL.}}

Understanding Inductive Reactance

The inductive reactance Xₗ limits the current in a purely inductive circuit:

Xₗ = ωL = 2πνL

Where ν is the frequency of the AC source in hertz (Hz).

Key Properties of Inductive Reactance

{{KEY: type=points | title=Characteristics of Xₗ | text=- Xₗ is directly proportional to frequency: higher frequency means higher opposition to current.

• Xₗ is directly proportional to inductance L: larger inductors oppose current changes more strongly.
• Xₗ has SI unit ohm (Ω), same as resistance.
• Unlike resistance, Xₗ does not dissipate energy as heat.}}

The current amplitude can now be written as:

iₘ = vₘ/Xₗ

This is analogous to Ohm's law for resistive circuits, but with reactance replacing resistance.

{{VISUAL: chart: graph showing how inductive reactance Xₗ increases linearly with frequency ν, with frequency on x-axis and Xₗ on y-axis}}

Phase Relationship: Current Lags Voltage

Comparing the voltage equation v = vₘ sin ωt with the current equation i = iₘ sin(ωt - π/2), we observe a critical phase difference.

The term π/2 (or 90°) in the current expression indicates that current lags voltage by one-quarter cycle.

What Does "Lagging" Mean?

At any instant, if we compare the phase of current with voltage:

• The current reaches its maximum value π/2 radians (or T/4 seconds) after the voltage reaches its maximum.
• The current crosses zero π/2 radians after the voltage crosses zero.

{{VISUAL: diagram: phasor diagram showing voltage phasor V along positive x-axis and current phasor I lagging by 90 degrees (pointing downward along negative y-axis)}}

In the phasor diagram, the current phasor I is rotated 90° clockwise (or π/2 radians behind) relative to the voltage phasor V. When both phasors rotate counterclockwise with angular frequency ω, they generate the sinusoidal voltage and current waveforms.

{{VISUAL: chart: graph of voltage v and current i versus ωt, showing voltage as a sine wave and current as a cosine wave shifted right by π/2, with one complete cycle marked}}

{{KEY: type=exam | title=Phase Difference in Inductive Circuits | text=In purely inductive AC circuits, current always lags voltage by exactly 90°. This is a standard exam question. Remember the mnemonic: ELI — in an inductor (L), emf (E) leads current (I).}}

Power Consumption in an Inductor

Does an inductor consume electrical energy like a resistor? Let's calculate the instantaneous power:

p = v × i = vₘ sin(ωt) × iₘ sin(ωt - π/2)

Using the identity sin(ωt - π/2) = -cos(ωt):

p = -vₘ iₘ sin(ωt) cos(ωt) = -(vₘ iₘ/2) sin(2ωt)

The average power over one complete cycle is:

Pₐᵥ = -(vₘ iₘ/2) × (average of sin(2ωt))

Since the average value of sin(2ωt) over a full cycle is zero:

The average power consumed by a pure inductor over a complete AC cycle is zero.

Energy Oscillation, Not Dissipation

Although the inductor does not consume energy on average, energy is alternately stored in the magnetic field when current builds up, and returned to the circuit when current decreases. This is fundamentally different from a resistor, which dissipates energy irreversibly as heat.

{{ZOOM: title=Why Inductors Are Called "Wattless" Components | text=An ideal inductor consumes zero average power (zero watts) even though current flows through it. The energy oscillates between the inductor's magnetic field and the source. In practical inductors, a small amount of power is dissipated due to the resistance of the wire windings.}}

{{KEY: type=concept | title=Power in a Pure Inductor | text=In a purely inductive AC circuit, instantaneous power oscillates sinusoidally but its average over a complete cycle is zero. The inductor stores energy in its magnetic field during one half-cycle and returns it to the circuit during the next half-cycle, resulting in no net energy consumption.}}

Worked Example

Example 7.2: A pure inductor of 25.0 mH is connected to a 220 V, 50 Hz AC source. Find (a) the inductive reactance, and (b) the rms current in the circuit.

Solution

(a) Inductive Reactance:

Xₗ = 2πνL = 2 × 3.14 × 50 × 25 × 10⁻³ Ω
   = 7.85 Ω

(b) RMS Current:

Using I = V/Xₗ:

I = 220 V / 7.85 Ω = 28.0 A

Notice how the inductor limits current even though it consumes no net power!

Key Takeaway: In a purely inductive AC circuit, the inductive reactance Xₗ = ωL limits current, and the current lags the voltage by exactly 90°. No average power is consumed — energy merely oscillates between the source and the inductor's magnetic field.

AC Voltage Applied to a Capacitor

AC Voltage Applied to a Capacitor

When a capacitor is connected to an alternating current (AC) source, it exhibits behaviour fundamentally different from its response in a DC circuit. Unlike a resistor or inductor, the capacitor does not dissipate energy — instead, it stores and releases energy alternately as the AC voltage changes polarity. Understanding this behaviour is crucial for analyzing AC circuits and applications like filters, tuning circuits, and power factor correction.

The Purely Capacitive AC Circuit

Consider a purely capacitive circuit where a capacitor of capacitance C is connected across an AC source with voltage v = vₘ sin(ωt), as shown in the circuit diagram. Here, vₘ is the peak voltage and ω = 2πν is the angular frequency of the source.

{{VISUAL: diagram: circuit showing an AC voltage source connected to a capacitor C with voltage v across it and current i flowing through it}}

In a DC circuit, once the capacitor is fully charged, current stops flowing. However, in an AC circuit, the voltage continuously changes direction and magnitude. This means the capacitor is constantly charging and discharging, allowing a continuous alternating current to flow through the circuit even though no actual charge crosses the gap between the capacitor plates.

Deriving the Current in a Capacitive Circuit

At any instant t, let the charge on the capacitor be q. The voltage across the capacitor is related to the charge by:

v = q/C

Since the capacitor is connected directly to the AC source, by Kirchhoff's voltage law:

vₘ sin(ωt) = q/C

Therefore, the charge on the capacitor at time t is:

q = C vₘ sin(ωt)

The instantaneous current i is the rate of change of charge:

i = dq/dt = d/dt [C vₘ sin(ωt)] = ωC vₘ cos(ωt)

Using the trigonometric identity cos(ωt) = sin(ωt + π/2), we can rewrite this as:

i = iₘ sin(ωt + π/2)

where the peak current (or current amplitude) is:

iₘ = ωC vₘ

{{FORMULA: expr=i = iₘ sin(ωt + π/2), iₘ = ωC vₘ | symbols=i:instantaneous current (A), iₘ:peak current (A), ω:angular frequency (rad/s), C:capacitance (F), vₘ:peak voltage (V), t:time (s)}}

{{KEY: type=concept | title=Current Leads Voltage by π/2 | text=In a purely capacitive AC circuit, the current leads the voltage by a phase angle of π/2 radians (or 90°). This means the current reaches its maximum value one-quarter cycle earlier than the voltage.}}

Capacitive Reactance

Looking at the expression for peak current iₘ = ωC vₘ, we can rearrange it as:

iₘ = vₘ / (1/ωC)

This form is analogous to Ohm's law I = V/R for a resistive circuit. The quantity 1/ωC plays the role of resistance in limiting the current amplitude. We call this the capacitive reactance, denoted by Xc:

{{KEY: type=definition | title=Capacitive Reactance | text=Capacitive reactance Xc is the opposition offered by a capacitor to the flow of alternating current, defined as Xc = 1/ωC = 1/(2πνC). Its SI unit is ohm (Ω).}}

Therefore, the amplitude of current can be written as:

iₘ = vₘ / Xc

Properties of Capacitive Reactance

The capacitive reactance Xc has the following important characteristics:

• Inversely proportional to frequency: As frequency ν increases, Xc decreases. At very high frequencies, a capacitor offers almost no opposition to current.
• Inversely proportional to capacitance: Larger capacitors have lower reactance and allow more current to flow.
• Frequency dependence: Unlike resistance R, which is independent of frequency, Xc varies with frequency. At ν = 0 (DC), Xc → ∞, meaning the capacitor blocks DC current completely.

{{KEY: type=points | title=Key Features of Capacitive Reactance | text=- Dimension same as resistance, unit is ohm (Ω).

• Inversely proportional to both frequency and capacitance: Xc = 1/(2πνC).
• At DC (ν = 0), Xc → ∞, so capacitor blocks DC current.
• At very high frequencies, Xc → 0, so capacitor acts like a short circuit.}}

{{ZOOM: title=Why does a capacitor block DC but pass AC? | text=In DC, after initial charging, the voltage across the capacitor equals the source voltage, so no further charge flows — effectively infinite reactance. In AC, the voltage continuously reverses, forcing the capacitor to charge and discharge repeatedly, sustaining a continuous current flow even though no charge physically crosses the dielectric gap.}}

Phasor Diagram and Phase Relationship

The equation i = iₘ sin(ωt + π/2) tells us that the current waveform is π/2 radians ahead of the voltage waveform v = vₘ sin(ωt). We can visualize this using a phasor diagram.

{{VISUAL: diagram: phasor diagram showing voltage phasor V along the horizontal axis and current phasor I leading it by 90 degrees counterclockwise, with both rotating counterclockwise}}

In the phasor diagram:

• The voltage phasor V is taken along the horizontal reference direction.
• The current phasor I is π/2 ahead (leading) in the counterclockwise direction.
• Both phasors rotate counterclockwise with angular velocity ω.

At any instant t, the projection of these phasors on the vertical axis gives the instantaneous values of voltage and current.

{{VISUAL: chart: graph showing sinusoidal waveforms of voltage v and current i versus ωt, with current waveform shifted left by π/2 showing it reaches maximum before voltage}}

The graph clearly shows that the current reaches its maximum one-quarter cycle before the voltage does — hence we say the current leads the voltage.

In a capacitive circuit, current is the "eager one" — it reaches its peak before voltage does.

Power in a Purely Capacitive Circuit

The instantaneous power supplied to the capacitor is:

p = v × i = vₘ sin(ωt) × iₘ cos(ωt) = vₘ iₘ sin(ωt) cos(ωt)

Using the trigonometric identity sin(ωt) cos(ωt) = ½ sin(2ωt), we get:

p = (vₘ iₘ / 2) sin(2ωt)

This expression shows that the instantaneous power oscillates at twice the source frequency (2ω), being alternately positive and negative.

To find the average power consumed over a complete cycle, we calculate:

Pavg = <(vₘ iₘ / 2) sin(2ωt)> = (vₘ iₘ / 2) <sin(2ωt)>

Since the average value of sin(2ωt) over a complete cycle is zero:

Pavg = 0

{{KEY: type=concept | title=Zero Average Power in Capacitive Circuit | text=A pure capacitor does not dissipate energy. Over a complete AC cycle, the average power consumed is zero. Energy is alternately stored in the electric field when the capacitor charges, and returned to the source when it discharges.}}

This is a key difference from a resistor, which continuously dissipates energy as heat. The capacitor is a reactive element — it stores energy temporarily in its electric field and releases it back to the circuit.

{{KEY: type=exam | title=Common Exam Question | text=Questions often ask to compare current-voltage phase relationships in R, L, and C circuits, or to calculate reactance and current for given values of C, ν, and Vrms. Remember: in C, current leads voltage by 90°; in L, current lags voltage by 90°.}}

Worked Example: Capacitive Circuit Calculations

Example: A 15.0 μF capacitor is connected to a 220 V, 50 Hz AC source. Find: (a) the capacitive reactance (b) the rms and peak current in the circuit (c) what happens if the frequency is doubled

Solution:

(a) The capacitive reactance is:

Xc = 1/(2πνC) = 1/(2π × 50 Hz × 15.0 × 10⁻⁶ F) = 212 Ω

(b) The rms current is:

Irms = Vrms / Xc = 220 V / 212 Ω = 1.04 A

The peak current is:

iₘ = √2 × Irms = 1.414 × 1.04 A = 1.47 A

This current oscillates between +1.47 A and –1.47 A, and is π/2 ahead of the voltage.

(c) If the frequency is doubled to 100 Hz:

Xc(new) = 1/(2π × 100 Hz × 15.0 × 10⁻⁶ F) = 106 Ω

The capacitive reactance is halved, and consequently the current is doubled to approximately 2.08 A (rms).

This example illustrates the strong frequency dependence of capacitive circuits — higher frequencies result in lower reactance and higher currents.

Real-World Application: Lamp and Capacitor in Series

Scenario: A lamp is connected in series with a capacitor. Predict what happens when: (a) a DC source is connected (b) an AC source is connected (c) the capacitance is reduced in the AC case

Analysis:

(a) With DC, the capacitor charges up initially (lamp glows briefly), then once fully charged, current stops and the lamp goes off. Reducing C makes no difference — the lamp still remains off.

(b) With AC, the capacitor offers capacitive reactance Xc = 1/(2πνC), allowing continuous current to flow. The lamp glows continuously.

(c) If C is reduced, Xc increases, reducing the current through the circuit. The lamp will shine less brightly.

This simple demonstration highlights the AC-passing, DC-blocking nature of capacitors — a property exploited in coupling circuits, audio filters, and signal processing.

{{VISUAL: diagram: two circuits side-by-side showing lamp and capacitor in series, one connected to DC battery (lamp off) and one to AC source (lamp glowing)}}