Alternating Current — Class 12 Physics Notes

Topics covered in Alternating Current

1. 1.Introduction Of Alternating Current

   Short Answer

   Alternating current (AC) is a type of electrical current where the flow of electric charge periodically reverses direction. Unlike direct current (DC) where the flow of electric charge is in a single direction, AC changes its direction and magnitude over time in a sinusoidal manner. AC is commonly used in homes and offices because it can be easily transformed to different voltages, is more efficient for long-distance transmission, and is utilized by most electrical devices.

   Long Answer

   Introduction to Alternating Current (AC)

   1. What is Alternating Current?

      • Alternating current (AC) refers to the flow of electric charge that changes direction periodically. This is in contrast to direct current (DC), where the electric charge flows in one direction continuously.

   2. Characteristics of AC

      • The voltage and current in an AC circuit vary with time, typically in a sinusoidal pattern, meaning they rise and fall in a wave-like manner.

   3. Common Use

      • AC is the standard form of electricity supplied to homes and businesses. It powers most of the electrical devices we use daily, such as lights, refrigerators, and televisions.

   4. Advantages of AC Over DC

      • Efficient Transmission: AC can be transmitted over long distances more efficiently than DC.
      • Voltage Transformation: AC voltages can be easily changed using transformers. This allows for the electricity to be generated at high voltages (for efficient transmission) and then stepped down to lower voltages for safe use in homes and businesses.
      • Economic Benefits: Due to the ease of changing voltages and efficient long-distance transmission, AC is more economical to distribute than DC.

   5. Applications in Daily Life

      • AC is utilized in various daily devices and also in systems requiring varying voltage levels. Tuning a radio to a specific station is an example of exploiting the unique characteristics of AC circuits.

   6. Significance in Electrical Engineering

      • The study of AC circuits opens up a realm of possibilities in electrical engineering, including the design and operation of many types of electrical devices, power distribution systems, and much more.

   Real-life Applications and Careers

   • Home and Office Appliances: From lighting to heating and cooling systems, AC powers nearly all electrical appliances in domestic and commercial settings.
   • Entertainment and Media: Devices like radios, televisions, and sound systems all operate on AC.
   • Industrial Applications: Heavy machinery and industrial equipment often run on AC because of its ability to be easily transformed and transmitted.
   • Careers: Understanding AC is fundamental for careers in electrical engineering, power distribution, renewable energy, and appliance design, among others.

2. 2.AC Voltage Applied To A Resistor

   Short Answer

   When AC voltage is applied to a resistor, the current that flows through it also alternates. This results in power dissipation known as Joule heating, calculated by 2i2R. Even though the current alternates, resulting in a zero average current over time, the power is not zero because the square of the current (2i2) is always positive. The average power dissipation is half the product of the square of the maximum current and the resistance (12221​Im2​R).

   Long Answer

   Derivation of AC Voltage Applied to a Resistor

   1. AC Circuit Basics:

      • An alternating current (AC) circuit with a resistor involves a voltage source e that produces a voltage that varies sinusoidally over time, =sin⁡v=Vm​sin(ωt), where Vm​ is the maximum voltage and ω is the angular frequency.

   2. Joule Heating in AC Circuits:

      • As the AC current flows through the resistor, power is dissipated in the form of heat. This is known as Joule heating, described by the formula 2i2R, where i is the instantaneous current and R is the resistance.

   3. Instantaneous Power:

      • The instantaneous power dissipated can be represented as =2=2sin⁡2p=i2R=Im2​sin2(ωt)R, where Im​ is the maximum current.

   4. Average Power:

      • To find the average power over one cycle, we use the fact that the average value of sin⁡2sin2(ωt) over one cycle is 1221​, leading to the average power formula =122Pavg​=21​Im2​R.

   5. RMS Value:

      • The root mean square (rms) value of current, Irms​, is used to express the effective current that would produce the same power dissipation as a DC current. It is calculated as =2Irms​=2​Im​​.

   6. AC Power Equation:

      • The average power in an AC circuit can thus be expressed in a similar form to DC power: =2P=Irms2​R.

   Derivation Steps

   1. AC Voltage Source:

      • The AC voltage across the resistor is =sin⁡v=Vm​sin(ωt).

   2. Ohm's Law Application:

      • By applying Ohm's Law, =V=IR, the instantaneous current can be calculated as =sin⁡i=RVm​​sin(ωt).

   3. Current Amplitude:

      • The amplitude of the current (Im​) is thus /Vm​/R.

   4. Power Dissipation Formula:

      • The instantaneous power dissipated in the resistor is =2=(sin⁡)2p=i2R=(RVm​​sin(ωt))2R.

   5. Average Power Over a Cycle:

      • Using the trigonometric identity, sin⁡2=12(1−cos⁡(2))sin2(ωt)=21​(1−cos(2ωt)), and knowing that the average of cos⁡(2)cos(2ωt) over one cycle is zero, the average power becomes =122Pavg​=21​Im2​R.

   Real-life Applications and Careers

   • Electronics and Appliance Design: Understanding how AC voltage interacts with resistors helps in designing safe and efficient electronic circuits for appliances.
   • Electrical Engineering: Electrical engineers must be familiar with these concepts to work in areas like power generation, transmission, and distribution.
   • Energy Sector: Knowledge of AC power is vital in the energy sector, especially for roles focusing on improving the efficiency of power transmission and reducing energy losses.

   Activity to Understand AC in Resistors

   • Simulation Activity: Use an online circuit simulation tool to build a simple AC circuit with a resistor. Observe how the current and voltage change over time and how the power dissipation is calculated.
   
   

3. 3.Representation of AC Current And Voltage By Rotating Vectors — Phasors

   Short Answer: The image is showing how alternating current (AC) and voltage can be represented using rotating vectors called phasors. In simple terms, it's a way to visualize how AC changes over time using a circular motion, which helps us understand it better.

   Long Answer: Let's understand this step by step:

   1. Phasors: These are rotating vectors that represent the magnitude and phase of sinusoidal functions like voltage and current in AC circuits. They rotate because AC signals vary with time in a cyclical fashion, just like a point moving in a circle.

   2. Graph (a) – Static Representation:

      • The vertical line represents the maximum value of current (I) or voltage (V) at a particular instant.
      • The horizontal line represents the time axis.
      • The dashed line shows the projection of the rotating vector on the vertical axis, giving us the value of the current or voltage at that instant.

   3. Graph (b) – Dynamic Representation:

      • The sine wave shows how the value of current (i) or voltage (v) changes over time.
      • The dots on the sine wave correspond to the position of the phasor at different times.
      • The phasor rotates counterclockwise, and as it does, the projection on the vertical axis goes up and down, creating the sine wave.

   4. Real-Life Applications: This concept is used everywhere AC power is used, which is in our homes, schools, and workplaces. It's also vital in designing and understanding electrical circuits and systems in many industries like power distribution, consumer electronics, and telecommunications.

   5. Careers: Knowledge of AC and phasors is essential for electrical engineers, electronic engineers, and anyone working in power systems or any industry that uses AC power.

4. 4.AC Voltage Applied To An Inductor

   Short Answer: When AC voltage is applied to an inductor, it creates an alternating current that lags the voltage by 90 degrees due to the inductor's property of self-inductance.

   Long Answer with Derivation:

   1. 1. Inductor in an AC Circuit: An inductor is a passive component that resists changes in electric current. When an AC voltage is applied, the inductor stores energy in its magnetic field. The inductor can either charge up by storing the energy or release the stored energy, which affects the current in the circuit.

   2. 2. Faraday's Law of Induction: Faraday's law tells us that an induced electromotive force (EMF) in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. For an inductor, this EMF (self-induced) is given by =−e=−LdtdI​, where L is the inductance.

   3. 3. AC Voltage Source: The applied AC voltage can be described by the equation =sin⁡V(t)=Vm​sin(ωt), where Vm​ is the maximum voltage, ω is the angular frequency, and t is the time.

   4. 4. Applying Kirchhoff's Voltage Law: According to Kirchhoff's voltage law, the sum of all electrical potential differences (voltage) around a loop must equal zero. For our circuit with an inductor and AC source, we get −=0V(t)−LdtdI​=0.

   5. 5. Solving the Differential Equation: To find the current I(t) as a function of time, we need to solve the differential equation =sin⁡LdtdI​=Vm​sin(ωt).

   6. 6. Integration of Current: Integrating both sides of the equation with respect to time, we get =∫sin⁡I(t)=ωLVm​​∫sin(ωt)dt, which leads to =(−cos⁡)+I(t)=ωLVm​​(−cos(ωt))+C, where C is the integration constant.

   7. 7. Determining the Integration Constant: Because the current in the circuit has a sinusoidal oscillation that is symmetric about zero, the integration constant C will be zero (no DC component).

   8. 8.Final Current Equation: The final equation for the current in the circuit becomes =cos⁡(+2)I(t)=ωLVm​​cos(ωt+2π​), which can also be written as =sin⁡(−2)I(t)=Im​sin(ωt−2π​), indicating that the current lags the voltage by 90 degrees.

   9. 9.Inductive Reactance: The ratio Im​Vm​​ is defined as the inductive reactance =XL​=ωL, which is the opposition that an inductor offers to AC.

   10. 10. Amplitude of Current: The amplitude of the current is given by =Im​=ωLVm​​.

   In real-life applications, this behavior is significant in AC power systems, radio transmitters, and other electronics where inductors are used to control the flow of alternating current.

   In careers like electrical and electronic engineering, especially in the fields of power distribution and RF design, understanding the interaction of AC voltage with inductors is essential for designing efficient circuits and systems.

5. 5.AC Voltage Applied To A Capacitor

   Short Answer: In the image, you have an AC source connected to a capacitor. When AC voltage is applied to a capacitor, it causes the capacitor to charge and discharge cyclically at the frequency of the AC source, leading to an alternating current through the circuit.

   Long Answer with Derivation: Let's go through the steps to derive the behavior of an AC voltage applied to a capacitor:

   1. Capacitor in an AC Circuit: A capacitor is a device that stores energy in an electric field between its plates. When connected to an AC source, it charges and discharges repeatedly with the alternating voltage.

   2. AC Voltage Source: The applied AC voltage can be described by =sin⁡V(t)=Vm​sin(ωt), where Vm​ is the peak voltage, ω is the angular frequency, and t is the time.

   3. Capacitor's Charge and Voltage: The relationship between the charge Q on a capacitor and the voltage V across it is given by =Q=CV, where C is the capacitance.

   4. Current in terms of Charge: The current I flowing through the capacitor is the time rate of change of charge, given by =I=dtdQ​.

   5. Substituting Charge with Voltage: Using =Q=CV, we can express the current as =I=CdtdV​.

   6. Applying the AC Voltage to the Capacitor: Substituting the AC voltage expression into the current equation gives us =(sin⁡)I=Cdtd​(Vm​sin(ωt)).

   7. Differentiating Voltage: Differentiating sin⁡Vm​sin(ωt) with respect to time t yields =cos⁡I=CωVm​cos(ωt).

   8. Resulting Current: The resulting current through the capacitor is =cos⁡I(t)=CωVm​cos(ωt), which can also be written as =sin⁡(+2)I(t)=Im​sin(ωt+2π​), showing that the current leads the voltage by 90 degrees.

   9. Capacitive Reactance: The ratio of the peak voltage to the peak current gives us the capacitive reactance =1XC​=ωC1​, which is the opposition that a capacitor presents to AC.

   10. Amplitude of Current: The amplitude of the current is given by =Im​=ωCVm​.

   In real life, capacitors in AC circuits are used for filtering, tuning, and power factor correction. Understanding the interaction of AC voltage with capacitors is vital for careers in electronics and electrical engineering, especially in designing circuits for power management and signal processing.

6. 6.Power In AC Circuit: The Power Factor

   Short Answer: Power in an AC (Alternating Current) circuit is the measure of how much work can be done by the electricity as it flows. The power factor is a number between 0 and 1 that tells us how efficiently the electricity is being used to do work. The derivations help us understand how to calculate the power in AC circuits depending on whether they have just resistance (like heaters), just inductance (like coils), just capacitance (like in tuning circuits), or a mix of these.

   Long Answer:

   To provide a detailed explanation of the derivation of the power factor in an AC circuit, let's break it down into understandable parts. The power factor (PF) represents the cosine of the angle between the current and voltage waveforms in an AC circuit. This angle is known as the phase angle (ϕ). The power factor is crucial because it gives us insight into how efficiently the power is being used in the circuit.

   Understanding the Components:

   1. Real Power (P): This is the power that actually does work in the circuit, measured in watts (W). It is the product of the voltage (V), current (I), and the cosine of the phase angle (cos⁡cos(ϕ)). It represents the average power over a cycle and is given by: =cos⁡P=VIcos(ϕ).

   2. Reactive Power (Q): This power does not do any real work but is necessary for the functioning of reactive components like inductors and capacitors. It is measured in volt-amperes reactive (VAR) and is given by: =sin⁡Q=VIsin(ϕ).

   3. Apparent Power (S): This is the product of the RMS (Root Mean Square) values of voltage and current, measured in volt-amperes (VA). It represents the total power flowing through the circuit and is given by: =S=VI.

   Derivation:

   The power factor is the ratio of real power to apparent power, which can be mathematically expressed as:

   =PF=SP​

   Since =cos⁡P=VIcos(ϕ) and =S=VI, we can substitute these values into the power factor equation:

   =cos⁡PF=VIVIcos(ϕ)​

   =cos⁡PF=cos(ϕ)

   This equation shows that the power factor is indeed the cosine of the phase angle between the current and voltage, highlighting the efficiency of power usage in the circuit. A power factor of 1 (or 100%) means that all the power is effectively used for work (perfect efficiency), while a power factor less than 1 indicates the presence of reactive power, which does not contribute to work but is necessary for the functioning of the circuit.

   Understanding Through an Activity:

   • Visual Activity: Imagine a clock with two hands, one representing voltage and the other current. If both hands point in the same direction (indicating a phase angle of 0 degrees), the power factor is 1, as voltage and current are perfectly in phase. If the hands form an angle, the cosine of this angle gives the power factor, demonstrating how much out of phase the voltage and current are.

   • Practical Implication: In real-world electrical systems, achieving a power factor close to 1 is desirable because it means the electrical power is being used efficiently. This is often achieved through the use of capacitors or inductors to adjust the phase angle, thus improving the power factor.

   Real-Life Application:

   The concept of power factor is widely used in the design and operation of AC power systems, including residential, commercial, and industrial sectors. Engineers strive to improve the power factor in electrical installations to reduce energy losses, improve voltage levels, and lower electricity costs. This is particularly important in industries with large motors or inductive loads, where a poor power factor leads to higher charges and inefficient energy use.

   Understanding and managing the power factor is a key aspect of electrical engineering and energy management careers, highlighting its importance in both practical and environmental aspects of electrical power usage.

7. 7.Transformer

   Short Answer:

   A transformer is a device that transfers electrical energy from one circuit to another through coils, typically to increase or decrease voltage. It works on the principle of electromagnetic induction, with no moving parts. Transformers are used in power distribution to adjust voltages, in electronics to adapt levels for devices, and in various equipment like chargers and adapters.

   Long Answer:

   What is a Transformer? A transformer is a static device that transfers electrical energy from one circuit to another through the process of electromagnetic induction. It can either increase (step-up) or decrease (step-down) the voltage level while keeping the power (the product of voltage and current) the same.

   How Does a Transformer Work?

   1. Primary Coil: When an alternating current (AC) flows through the primary coil, it generates an alternating magnetic field around it.
   2. Magnetic Core: The magnetic field lines pass through a soft iron core, which serves to concentrate the magnetic field and reduce losses.
   3. Secondary Coil: The alternating magnetic field induces an alternating voltage in the secondary coil by Faraday's Law of Electromagnetic Induction.
   4. Energy Transfer: The energy is thus transferred from the primary coil to the secondary coil, without any physical connection between them.

   The Principle of Operation: The operation of a transformer is based on Faraday's law, which states that the induced electromotive force (EMF) in any closed circuit is equal to the rate of change of the magnetic flux through the circuit.

   Mathematical Derivation: Let's derive the formula relating the primary and secondary voltages with the number of turns in the coils.

   Assuming an ideal transformer:

   • Np​ = Number of turns in the primary coil
   • Ns​ = Number of turns in the secondary coil
   • Vp​ = Voltage across the primary coil
   • Vs​ = Voltage across the secondary coil
   • ϕ = Magnetic flux through the core

   Faraday's Law states: =−EMF=−Ndtdϕ​

   For the primary coil: =−Vp​=−Np​dtdϕ​

   For the secondary coil: =−Vs​=−Ns​dtdϕ​

   By eliminating dtdϕ​ from both equations, we get the transformer's basic equation: =Vp​Vs​​=Np​Ns​​

   This equation shows that the ratio of the voltages is equal to the ratio of the turns on the coils.

   Applications in Real Life and Careers: Transformers are fundamental in the electrical power industry. They are used in power plants, substations, and distribution lines to ensure that electricity is transmitted efficiently over long distances. They also have applications in electronics, audio systems for impedance matching, and in various industries for voltage control and conversion. Careers related to transformers include electrical engineering, power systems engineering, and electronics design.