Alternating Current Class 12 Physics Chapter 7 Notes

Alternating Current

Alternating Voltage (AC Voltage): AC voltage is characterized by its continuous variation over time. It follows a sinusoidal waveform, commonly represented by a sine function. In AC circuits, the voltage periodically changes its polarity, cycling between positive and negative values.
Alternating Current (AC Current): AC current is the flow of electric charge that corresponds to AC voltage. It also varies with time, oscillating back and forth, matching the voltage waveform.
Applications of AC Voltage: AC voltage is widely used because of its advantages. It is the standard form of electrical power supplied to homes and businesses. Many electrical devices, including household appliances, lighting, and industrial machinery, operate on AC voltage.
Efficient Transmission and Transformation: AC voltage can be easily transformed from one voltage level to another using devices called transformers. This property makes it suitable for long-distance power transmission, where high-voltage AC is transmitted over power lines and then transformed to lower voltages for local distribution.
Characteristics of AC Circuits: AC circuits exhibit several characteristics that differ from DC circuits. These include impedance (analogous to resistance in DC), reactance (related to the effect of capacitors and inductors), and phase differences between voltage and current.
Applications of AC Circuit Theory: AC circuit theory is applied in various fields, including electronics, telecommunications, power generation, and electric motor operation. Understanding AC circuit principles is essential for designing and analyzing electrical systems.
Signal Processing: AC signals are fundamental in signal processing and communication systems. Radio broadcasting, television, and data transmission rely on the modulation and demodulation of AC signals to carry information.
Frequency of AC: The frequency of AC voltage and current is measured in hertz (Hz) and represents the number of cycles per second. The standard frequency for AC mains power varies by region, with 50 Hz in many parts of the world and 60 Hz in others.
Harmonics: In addition to the fundamental frequency, AC waveforms may contain harmonics—integer multiples of the fundamental frequency. Harmonics can have significant effects on power quality and device operation.
AC Circuit Analysis: Analyzing AC circuits involves dealing with complex numbers, phasors, and impedance. AC circuit analysis techniques include nodal analysis, mesh analysis, and phasor diagrams.

Understanding AC circuits is crucial for electrical engineers, technicians, and anyone working with electrical systems. AC circuits form the basis for many electrical technologies and applications, from power generation and distribution to modern electronic devices.

AC Voltage applied to a Resistor

AC Voltage (Alternating Voltage): AC voltage is characterized by its continuous variation with time, following a sinusoidal waveform. It periodically changes direction, oscillating between positive and negative values. The amplitude of the voltage waveform is denoted as vm, and its angular frequency is represented by ω.
Ohm’s Law for AC Circuits: Ohm’s law, which relates voltage (V), current (I), and resistance (R), also holds for AC circuits. For a resistor in an AC circuit, the instantaneous current i(t) is given by:i(t) = (vm / R) * sin(ωt)This equation shows that the current varies sinusoidally with time, just like the applied voltage.
Phase Relationship: In a purely resistive AC circuit (like the one with a resistor), the voltage and current are in phase with each other. This means that they reach their maximum, minimum, and zero values at the same points in time.
Average Power Dissipation: Although the average current over one complete cycle is zero in an AC circuit, there is still power dissipation due to Joule heating. The instantaneous power dissipated in a resistor is given by:p(t) = (1/2) * (vm^2 / R) * sin^2(ωt)The average power over a cycle is calculated as:P = (1/2) * (vm^2 / R)This power represents the energy dissipated as heat in the resistor.
Root Mean Square (RMS) Current and Voltage: To express AC power in a way similar to DC power (P = I^2R), RMS values are introduced. The RMS current (Irms or I) and RMS voltage (Vrms or V) are defined as:Irms = 0.707 * ImaxVrms = 0.707 * VmaxThese values represent the effective or equivalent DC values of current and voltage that produce the same power dissipation.
Power in AC Circuits: Using RMS values, the power dissipation in an AC circuit can be calculated as:P = Vrms * IrmsThis formula allows for the direct calculation of power in AC circuits, similar to the DC case.
Advantages of RMS Values: RMS values are commonly used for AC measurements and specifications because they simplify calculations and allow for direct comparisons with DC circuits.

Understanding AC voltage and current in resistive circuits is fundamental in electrical engineering and plays a crucial role in the analysis and design of AC circuits and systems.

AC circuits and the representation of AC current and voltage using rotating vectors (phasors)

Phasors: Phasors are vectors used to represent sinusoidally varying quantities in AC circuits. These vectors rotate about the origin with an angular speed ω, which corresponds to the angular frequency of the AC source. Phasors help visualize the relationship between voltage and current in AC circuits.
Phasor Diagram: Phasors are typically represented in a phasor diagram, which is a graphical tool used to analyze AC circuits. In the diagram, the vertical components of the phasors represent the instantaneous values of sinusoidally varying quantities (such as voltage or current), and the magnitudes of the phasors represent the amplitudes or peak values of these quantities.
Phase Relationship: The phase relationship between voltage and current in an AC circuit can be determined by examining the angle between the voltage and current phasors at a specific time. This angle represents the phase difference between the two quantities.
Resistor (Zero Phase Angle): In a purely resistive AC circuit, such as the one with a resistor, the voltage and current phasors are in the same direction at all times. This indicates that the phase angle between voltage and current is zero degrees. In other words, voltage and current are in phase.
Projection on Vertical Axis: The projection of a phasor onto the vertical axis of the phasor diagram represents the instantaneous value of the corresponding sinusoidally varying quantity. For example, the projection of the voltage phasor represents the instantaneous voltage, and the projection of the current phasor represents the instantaneous current.
Frequency of Rotation: The angular speed of phasors in the phasor diagram corresponds to the angular frequency (ω) of the AC source. As the phasors rotate, they generate sinusoidal curves that represent the variations of voltage and current with time.
Phase Angle: The phase angle between voltage and current is measured in degrees and indicates how much one lags or leads the other in time within an AC cycle. In the case of a resistor, this phase angle is zero, indicating that voltage and current are in phase.

Understanding phasors and phasor diagrams is essential for analyzing more complex AC circuits involving inductors, capacitors, and combinations of circuit elements. Phasor diagrams simplify the analysis of AC circuits by visually representing the phase relationships between voltage and current.

AC voltage applied to an inductor

Purely Inductive AC Circuit: The circuit consists of an AC voltage source connected to an inductor. In this analysis, it is assumed that the inductor has negligible resistance, making it a purely inductive AC circuit.
Induced EMF in the Inductor: When AC voltage is applied to the inductor, it generates an induced electromotive force (emf) due to the changing magnetic flux within the inductor. This induced emf follows Faraday’s law.
Inductive Reactance (XL): The opposition offered by the inductor to the flow of AC current is quantified by inductive reactance (XL), which is analogous to resistance in a DC circuit. XL depends on the inductance (L) of the inductor and the frequency (ω) of the AC source and is given by XL = ωL.
Phase Relationship: In a purely inductive circuit, the current lags behind the voltage by one-quarter (1/4) cycle or a phase angle of π/2 radians. This phase relationship is due to the behavior of the inductor in responding to changes in voltage.
Amplitude of Current: The amplitude of the current (im) in a purely inductive circuit is determined by the amplitude of the voltage (vm) and the inductive reactance (XL) and is given by im = vm / XL.
Instantaneous Power: The instantaneous power supplied to the inductor varies with time and depends on the phase difference between voltage and current. The power is calculated as the product of voltage and current at each instant.
Average Power: When averaged over a complete cycle, the average power supplied to an inductor in a purely inductive circuit is zero. This means that, on average, no net electrical energy is consumed or dissipated as heat in the inductor.
No Real Power Dissipation: Unlike resistors, which dissipate electrical energy as heat (Joule heating), inductors in purely inductive AC circuits do not consume real power on average. The power alternates between positive and negative values but integrates to zero over a cycle.

AC voltage applied to a capacitor

Purely Capacitive AC Circuit: The circuit consists of an AC voltage source connected to a capacitor, resulting in a purely capacitive AC circuit.
Capacitive Reactance (Xc): The opposition offered by the capacitor to the flow of AC current is quantified by capacitive reactance (Xc), which is analogous to resistance in a DC circuit. Xc depends on the capacitance (C) of the capacitor and the frequency (ω) of the AC source and is given by Xc = 1 / (ωC).
Phase Relationship: In a purely capacitive circuit, the current leads the voltage by one-quarter (1/4) cycle or a phase angle of π/2 radians. This phase relationship is due to the behavior of the capacitor in responding to changes in voltage.
Amplitude of Current: The amplitude of the current (im) in a purely capacitive circuit is determined by the amplitude of the voltage (vm) and the capacitive reactance (Xc) and is given by im = vm / Xc.
Instantaneous Power: The instantaneous power supplied to the capacitor varies with time and depends on the phase difference between voltage and current. The power is calculated as the product of voltage and current at each instant.
Average Power: When averaged over a complete cycle, the average power supplied to a capacitor in a purely capacitive circuit is zero. This means that, on average, no net electrical energy is consumed or dissipated as heat in the capacitor.
No Real Power Dissipation: Similar to inductors in purely inductive circuits, capacitors in purely capacitive AC circuits do not consume real power on average. The power alternates between positive and negative values but integrates to zero over a cycle.

LCR circuit connected to an AC source

LCR Circuit: The circuit consists of a series combination of three components – an inductor (L), a capacitor (C), and a resistor (R) – connected to an AC voltage source (e).

AC Voltage Source: The AC voltage source is represented by v(t) = vm * sin(ωt), where vm is the amplitude of the AC voltage, ω is the angular frequency, and t represents time.

Kirchhoff’s Loop Rule: The Kirchhoff’s loop rule is applied to the circuit, which results in a second-order differential equation representing the behavior of the circuit components:

L * (di/dt) + R * i + (1/C) * ∫(i dt) = vm * sin(ωt)

This equation represents the interplay between the inductor’s self-inductance (L), the resistor’s resistance (R), and the capacitor’s capacitance (C) concerning the current (i) and the applied AC voltage (vm * sin(ωt)).

Analysis Methods:

Phasor Analysis: One method of analysis involves using phasors, which are rotating vectors that represent the amplitude and phase of AC quantities. Phasors simplify the analysis of AC circuits by converting sinusoidal voltages and currents into vectors. The phase relationship between voltage and current can be visually represented using phasor diagrams.
Analytical Solution: The second method involves solving the differential equation (Eq. 7.20) analytically to obtain the time-dependent current (i(t)). This approach allows us to derive the explicit mathematical expression for the current as a function of time.

Phase Relationship: The phase relationship between the current and the applied AC voltage in an LCR circuit depends on the values of L, C, and R. By analyzing the circuit, one can determine whether the current leads or lags the voltage and by how much.

Resonance: Special attention is often given to the phenomenon of resonance in LCR circuits. At resonance, the impedance of the inductor and the capacitor cancels each other out, resulting in a maximum current response to the applied voltage.

The analysis of the LCR circuit connected to an AC source using phasor diagrams

Components of the LCR Circuit:

The LCR circuit consists of a series combination of three components: an inductor (L), a capacitor (C), and a resistor (R).
These components are connected to an AC voltage source represented by v(t) = vm * sin(ωt), where vm is the amplitude of the AC voltage, ω is the angular frequency, and t represents time.

Phasor Analysis:

Phasors are used to represent the amplitude and phase of AC quantities in a graphical manner. They simplify the analysis of AC circuits.
The phasors for voltage across the inductor (VL), resistor (VR), capacitor (VC), and source voltage (V) are determined based on their phase relationships with the current (I).
VR is always parallel to I, VC is p/2 behind I, and VL is p/2 ahead of I.

Impedance Diagram:

An impedance diagram is constructed, representing the impedance (Z) as the hypotenuse of a right triangle. VR, VC, and VL are involved in this construction.
The impedance Z is defined as Z = vm / Im, where Im is the amplitude of the current.
The impedance Z is related to the resistance (R), capacitive reactance (XC), and inductive reactance (XL) by the equation Z = √(R^2 + (XC – XL)^2).

Phase Angle (φ):

The phase angle φ is the angle between VR and V and can be determined using the tangent function.
tan(φ) = (XC – XL) / R

Predominantly Capacitive or Inductive Circuit:

Depending on the values of XC and XL, the circuit can be predominantly capacitive or inductive.
If XC > XL, the circuit is predominantly capacitive, and the current leads the source voltage (φ is positive).
If XC < XL, the circuit is predominantly inductive, and the current lags the source voltage (φ is negative).

Steady-State vs. Transient Solution:

The phasor diagram provides the steady-state solution, which describes the circuit’s behavior after a sufficiently long time when transient effects have diminished.
Transient solutions exist for non-zero initial conditions and are part of the general solution.

Resonance

Resonance is a fascinating phenomenon observed in series RLC circuits, and it occurs when the impedance of the circuit is minimized at a specific frequency. Here are the key points about resonance in series RLC circuits:

Natural Frequency:

Resonance is a characteristic behavior exhibited by systems, including series RLC circuits, that tend to oscillate at a specific natural frequency.
The natural frequency is an inherent property of the system and is determined by the values of the inductance (L) and capacitance (C) in the circuit.

Driving Frequency and Amplitude:

When an RLC circuit is driven by an external voltage source with a frequency (ω) that is close to the circuit’s natural frequency (ω₀), the amplitude of the current response becomes significantly large.
The amplitude of the driving voltage source is represented as vm.

Current Amplitude at Resonance:

At the resonant frequency (ω = ω₀), the impedance of the series RLC circuit is minimized, meaning the circuit offers minimal opposition to the flow of current.
The impedance (Z) of the circuit is determined by the values of resistance (R), inductive reactance (XL = ωL), and capacitive reactance (XC = 1/ωC).
The resonant frequency (ω₀) is given by the formula ω₀ = 1 / √(LC).

Resonance Curve:

The current amplitude (im) at resonance is maximum and is equal to the driving voltage amplitude (vm) divided by the resistance (R) of the circuit, i.e., im = vm / R.

Applications of Resonance:

Resonant circuits find practical applications in tuning mechanisms, such as those used in radios and TV sets.
In radios, the antenna picks up signals from multiple broadcasting stations. By adjusting the capacitance in the tuning circuit, the circuit’s resonant frequency can be made nearly equal to the frequency of the desired radio station’s signal. When resonance occurs, the current amplitude for that specific frequency is maximized, allowing the station to be heard clearly.

Resonance Exclusive to RLC Circuits:

It’s crucial to note that the phenomenon of resonance is exhibited only in series RLC circuits, where both inductance (L) and capacitance (C) are present in the circuit.
Resonance cannot occur in RL (inductor and resistor) or RC (resistor and capacitor) circuits, as the voltages across L and C do not cancel each other out in these configurations.

Power in AC Circuits, the Power Factor

The concept of power in AC circuits, especially the power factor, is crucial for understanding how electrical energy is used and distributed in various types of circuits. Here are the key points about power in AC circuits and the power factor:

Instantaneous Power (p):

In an AC circuit, when a voltage v = vm sin(wt) is applied, it drives a current i = im sin(wt + φ), where φ is the phase angle between the voltage and current.
The instantaneous power supplied by the source at any time t is given by the product of voltage and current at that moment: p(t) = v(t) * i(t).

Average Power:

To find the average power over a complete cycle, we consider the time-dependent term in the expression for instantaneous power.
The average power (P) is given by the average of the time-dependent term: P = V * I * cos(φ), where V is the rms voltage, I is the rms current, and φ is the phase angle between them.
The power factor (cos(φ)) is a dimensionless quantity that represents the cosine of the phase angle between voltage and current.

Power Factor in Different Types of Circuits:

Resistive Circuit (Case i):
- In a purely resistive circuit, the voltage and current are in phase (φ = 0), leading to a power factor of 1.
- Maximum power is dissipated in the resistor.
Purely Inductive or Capacitive Circuit (Case ii):
- In circuits with only inductors or capacitors, the phase difference between voltage and current is π/2 (φ = π/2 or -π/2).
- In such cases, the power factor is 0 (cos(φ) = 0), indicating no power dissipation, even though current flows.
LCR Series Circuit (Case iii):
- In an LCR series circuit, the power factor depends on the phase difference between voltage and current.
- Power is dissipated only in the resistor (R), and the power factor can be calculated as cos(φ) = (Xc – XL) / R, where Xc is capacitive reactance, XL is inductive reactance, and R is resistance.
Power Dissipation at Resonance in LCR Circuit (Case iv):
- At the resonant frequency, Xc – XL = 0, and the phase angle φ is 0.
- Therefore, the power factor is 1 (cos(φ) = 1), indicating maximum power dissipation through the resistor (R) at resonance.

Transformers

Transformers are fundamental devices in electrical engineering used for the conversion of alternating current (AC) voltages from one level to another, either increasing (step-up) or decreasing (step-down) the voltage. They operate on the principle of mutual induction and are critical components in electrical power generation, transmission, and distribution systems. Here are the key points about transformers:

Basic Structure of a Transformer:
- A transformer consists of two sets of coils, known as windings, which are insulated from each other.
- These windings are wound on a soft-iron core. The core provides a path for the magnetic flux generated by the primary winding to link with the secondary winding.
- The primary winding has Np turns, while the secondary winding has Ns turns. The primary coil is typically connected to the input voltage source, while the secondary coil is connected to the load.
Principle of Mutual Induction:
- When an alternating voltage is applied to the primary coil, it produces an alternating current that generates an alternating magnetic flux in the transformer core.
- This changing magnetic flux links with both the primary and secondary windings, inducing an electromotive force (emf) or voltage in the secondary winding.
- The magnitude of the induced voltage in the secondary depends on the turns ratio between Ns and Np.
Ideal Transformer:
- In an ideal transformer, the primary winding has negligible resistance, and all the magnetic flux generated links both windings.
- The ideal transformer operates with perfect efficiency, meaning that the power input equals the power output (ignoring losses).
Turns Ratio and Voltage Transformation:
- The turns ratio, denoted as Ns/Np, determines the voltage transformation between the primary and secondary sides of the transformer.
- If Ns > Np, it’s a step-up transformer, which increases the voltage from primary to secondary.
- If Ns < Np, it’s a step-down transformer, which decreases the voltage from primary to secondary.
Current Transformation:
- The current in the secondary winding is inversely proportional to the turns ratio. In a step-up transformer, secondary current is smaller than primary current (Is < Ip), and in a step-down transformer, secondary current is larger (Is > Ip).
Efficiency and Losses:
- Real transformers have some energy losses due to factors like flux leakage, resistance of windings, eddy currents in the core, and hysteresis losses.
- These losses result in reduced efficiency compared to ideal transformers.
Applications of Transformers:
- Transformers are widely used in electrical power generation, transmission, and distribution systems to change voltage levels efficiently.
- They are essential for long-distance transmission of electrical energy, reducing power losses.
- Transformers are used in electronic devices, power supplies, and various industrial applications to provide different voltage levels as needed.
Power Transmission and Distribution:
- Transformers play a critical role in power transmission and distribution by stepping up the voltage at power plants for long-distance transmission and stepping it down for safe distribution to homes and businesses.
Improving Efficiency:
- Techniques such as using laminated cores, minimizing resistance in windings, and selecting appropriate materials help reduce energy losses in transformers.

Transformers are essential components in modern electrical systems, allowing for efficient and safe electrical power distribution over vast distances while providing the flexibility to adapt voltage levels to suit various applications.