📘 Alternating Current
Physics • Class XII
Introduction to Alternating Current
Direct current (dc) sources have been previously discussed, but alternating current (ac) sources are more common in everyday applications. AC voltage varies with time, typically in a sine wave pattern, and is essential for most electrical devices. The preference for ac voltage over dc voltage is due to its efficient conversion between voltages using transformers and its economical transmission over long distances.
Importance of AC Voltage
AC voltage is crucial because: - Most electrical energy sold is transmitted as alternating current. - AC circuits exhibit characteristics that are utilized in various devices, such as radios. - The abbreviation 'ac' is widely accepted despite being somewhat redundant.
Nicola Tesla and AC Technology
Nicola Tesla (1856-1943) was a Serbian-American scientist and inventor who significantly contributed to the development of alternating current technology. His inventions include: - The rotating magnetic field, foundational for ac machinery. - The induction motor and polyphase system of ac power. - The Tesla coil, used in radio and television equipment.
AC Voltage Applied to a Resistor
When an ac voltage is applied to a resistor, the voltage can be expressed as: v(t) = v_m sin(ωt) where: - v_m = amplitude of the oscillating potential difference - ω = angular frequency. Using Kirchhoff’s loop rule, the current through the resistor can be derived as: i(t) = (v_m/R) sin(ωt) This shows that the current also varies sinusoidally and is in phase with the voltage.
Key Equations
Important equations related to ac voltage and current through a resistor include: 1. v(t) = v_m sin(ωt) 2. i(t) = (v_m/R) sin(ωt) 3. i_m = v_m/R (Ohm's law for ac). Both voltage and current reach their maximum and minimum values simultaneously, indicating they are in phase.
Average Current in AC Circuits
In a pure resistor, the average current over one complete cycle is zero, as the instantaneous current values sum to zero. This characteristic is important for understanding the behavior of ac circuits.
George Westinghouse and Alternating Current
George Westinghouse (1846 – 1914) was a key proponent of alternating current (AC) technology, opposing Thomas Edison, who advocated for direct current (DC). Westinghouse believed AC was essential for the future of electricity. He founded a company named after him and collaborated with inventors like Nicola Tesla to develop AC motors and high-tension current transmission systems, significantly advancing large-scale lighting.
Joule Heating and Power Dissipation
Joule heating occurs when an AC current passes through a resistor, leading to energy dissipation. The instantaneous power dissipated in a resistor is given by the formula: p = i²R The average power over a cycle can be calculated using the average value of current squared and resistance.
Root Mean Square (RMS) Current
The root mean square (rms) current, denoted as I_rms or I, is defined to express AC power in a form similar to DC power (P = I²R). The rms current is calculated as: I = i_m / √2 = 0.707 i_m The average power in terms of rms current is given by: P = I²R.
RMS Voltage and Power Relations
The rms voltage (V) is similarly defined as: V = v_m / √2 = 0.707 v_m The relationship between AC current and voltage is analogous to DC, expressed as: V = IR.
Example Calculation of AC Parameters
For a light bulb rated at 100W for a 220V supply: (a) Resistance (R) can be calculated using: R = V² / P = (220V)² / 100W = 484Ω. (b) The peak voltage (v_m) is calculated as: v_m = √2 * V = 311V. (c) The rms current (I) through the bulb is calculated using: I = P / V = 100W / 220V = 0.454A.
Phasors in AC Circuits
Phasors are vectors that represent sinusoidally varying quantities in AC circuits. They rotate about the origin with angular speed ω. The vertical components of phasors V and I represent the instantaneous voltage (v) and current (i). The magnitudes of phasors represent the peak values (vm and im) of these quantities.
AC Voltage Applied to an Inductor
When an AC source is connected to an inductor, the circuit is considered purely inductive if the inductor has negligible resistance. The voltage across the source is given by v = vm sin(ωt). Using Kirchhoff’s loop rule, the relationship between voltage and current in the inductor is described by the equation: d(i)/dt = (vm/L) sin(ωt).
Current in an Inductive Circuit
The current i(t) can be expressed as: i(t) = (vm/XL) sin(ωt - π/2). The amplitude of the current is given by im = vm/XL.
Inductive Reactance in AC Circuits
Inductive reactance limits current in a purely inductive circuit, similar to resistance in a purely resistive circuit. It is directly proportional to inductance and frequency of the current. In an inductor, the current lags the voltage by π/2 or one-quarter cycle. The average power supplied to an inductor over one complete cycle is zero.
Calculating Inductive Reactance and RMS Current
To find the inductive reactance (X_L) and RMS current (I_L) in a circuit, use the following formulas: - Inductive Reactance: X_L = ωL = 2πfL - RMS Current: I_L = V/X_L.
AC Voltage Applied to a Capacitor
In a purely capacitive AC circuit, the capacitor limits or regulates current but does not completely prevent charge flow. The instantaneous voltage (v) across the capacitor is given by v = q/C. The current (i) can be derived from the charge (q) as i = dq/dt.
Key Formulas for Capacitive Circuits
Important formulas related to capacitive circuits include: - Capacitive Reactance: X_C = 1/(ωC) - Amplitude of Current: i_m = v_m/X_C.
Capacitive Reactance
Capacitive reactance limits the amplitude of current in a purely capacitive circuit, similar to resistance in a resistive circuit. It is inversely proportional to frequency and capacitance.
Current and Voltage Relationship
In a capacitive circuit, the current is π/2 ahead of the voltage. This is illustrated in phasor diagrams, where the current phasor I leads the voltage phasor V.
Instantaneous and Average Power
The instantaneous power supplied to the capacitor is given by the formula: p_c = i v. The average power over a complete cycle is zero.
Capacitor Behavior with DC and AC Sources
When a DC source is connected to a capacitor, it charges and no current flows. In contrast, with an AC source, the capacitor offers capacitive reactance, allowing current to flow.
Capacitive Reactance Calculation
For a 15.0 μF capacitor connected to a 220 V, 50 Hz source, the capacitive reactance (X_C) can be calculated. If the frequency is doubled, the capacitive reactance is halved, and the current is doubled.
Inductor and Light Bulb Behavior
When an iron rod is inserted into an inductor connected to an AC source, the inductance increases, leading to increased inductive reactance.
Series LCR Circuit with AC Voltage
In a series LCR circuit connected to an AC source, Kirchhoff’s loop rule is applied to determine the relationship between charge on the capacitor (q) and current (i).
Alternating Current in LCR Circuits
This section discusses the analysis of alternating current (AC) in a series LCR circuit using phasors and analytical methods.
Phasor-Diagram Solution
In a series LCR circuit, the resistor (R), inductor (L), and capacitor (C) have the same current at any time. The phasor diagram illustrates the relationship between the phasors for current (I) and voltages across the components.
Phasor Relationships
The phasor relation can be expressed as VL + VR + VC = V. VC and VL can be combined into a single phasor due to their opposite directions.
Phase Angle Determination
The phase angle φ is determined from the relationship: tan(φ) = (VC - VL) / VR.
Circuit Behavior Based on Reactance
The behavior of the circuit is influenced by the reactance: - If XC > XL, φ is positive, and the circuit is predominantly capacitive. - If XC < XL, φ is negative, and the circuit is predominantly inductive.
Limitations of Phasor Analysis
While the phasor method provides the amplitude and phase of current in an LCR circuit, it has limitations. It does not account for initial conditions.
General Solution in Circuits
The general solution of a circuit is the sum of the transient solution and the steady-state solution.
Resonance in RLC Circuits
Resonance is a phenomenon in series RLC circuits where the system oscillates at a natural frequency. When driven by an energy source near this frequency, the amplitude of oscillation increases significantly.
Applications of Resonant Circuits
Resonant circuits are used in tuning mechanisms of radios and TVs.
Example Problem: RLC Circuit Analysis
Consider a resistor of 200 Ω and a capacitor of 15.0 mF connected in series to a 220 V, 50 Hz AC source. To find the current in the circuit, calculate the impedance and then use it to determine the current and the voltage across the resistor and capacitor.
Circuit Analysis and Current Calculation
To calculate the current in the circuit, use the formula: I = V / Z.
Voltage in Resistor and Capacitor
In a circuit, the voltage across the resistor (V_R) and the capacitor (V_C) can be calculated using Ohm's law.
Phase Difference and Total Voltage
The voltages V_R and V_C are out of phase by ninety degrees. Therefore, the total voltage across the resistor and capacitor is calculated using the Pythagorean theorem.
Power in AC Circuit: The Power Factor
The instantaneous power (p) supplied by the source in an AC circuit is given by p = v * i. The average power over a cycle is given by P = (1/2) * v_m * i_m * cos(φ).
Cases of Power Dissipation
1. **Resistive Circuit**: φ = 0, cos φ = 1, maximum power dissipation. 2. **Inductive or Capacitive Circuit**: φ = π/2, cos φ = 0, no power dissipation. 3. **LCR Series Circuit**: φ = tan⁻¹((X_C - X_L) / R), power is dissipated only in the resistor.
Power Factor Improvement
A low power factor indicates large power loss during transmission. The power factor can be improved by adding a capacitor of appropriate capacitance in the circuit.
Phasor Diagram and Current Components
In a phasor diagram, current (I) can be resolved into two components: I_p (power component) and I_q (wattless component).
Example 7.8: LCR Circuit Analysis
A sinusoidal voltage of peak value 283 V and frequency 50 Hz is applied to a series LCR circuit with R = 3 Ω, L = 25.48 mH, and C = 796 μF.
Resonance in LCR Circuits
At resonance, the frequency of the source is such that X_L = X_C, leading to maximum current and power dissipation.
Metal Detector Principle
A metal detector operates on the principle of resonance in AC circuits.
Transformers
Transformers are used to change (transform) alternating voltage from one value to another, either higher or lower.
Transformer Basics
A transformer operates on the principle of mutual induction and consists of two insulated coils wound on a soft-iron core.
Induced EMF in Transformers
When an alternating voltage (vp) is applied to the primary coil, it generates an alternating magnetic flux that induces an electromotive force (emf) in the secondary coil.
Transformer Efficiency and Power Relations
Assuming a 100% efficient transformer (no energy losses), the power input equals the power output.
Step-Up and Step-Down Transformers
In a step-up transformer, the secondary coil has more turns than the primary (Ns > Np), resulting in higher voltage (Vs > Vp) and lower current (Is < Ip). Conversely, in a step-down transformer, the secondary coil has fewer turns (Ns < Np).
Real-World Transformer Limitations
In actual transformers, energy losses occur due to factors such as flux leakage.
Transmission of Electrical Energy
The large-scale transmission of electrical energy is done using transformers.
Capacitive Reactance and Power in Capacitors
An ac voltage v = vm sin(ωt) applied to a capacitor drives a current i = im sin(ωt + π/2).
Series RLC Circuit Analysis
In a series RLC circuit driven by voltage v = vm sin(ωt), the current is given by i = im sin(ωt + φ).
Wattless Current in Purely Inductive or Capacitive Circuits
In purely inductive or capacitive circuits, cos(φ) = 0, meaning no power is dissipated despite current flow.
Phasors in AC Circuit Analysis
The phase relationship between current and voltage in an AC circuit can be represented using phasors.
Reactance and Impedance
Reactance is the opposition to AC current. Inductive reactance (X_L) is given by X_L = ωL, and capacitive reactance (X_C) is given by X_C = 1/(ωC).
Quality Factor and Power Factor
The quality factor (Q) is dimensionless and is defined as Q = ω_0L/R for a series RLC circuit. The power factor is also dimensionless and is defined as cos(φ).
Key Points on AC Circuits
1. RMS values are typically used for AC voltage and current. 2. The power rating in AC circuits refers to average power. 3. Power consumed in an AC circuit is never negative. 4. AC amperes are defined based on Joule heating effects, not direct current attraction.
Power Factor in RLC Circuits
The power factor in an RLC circuit indicates how effectively the circuit converts electric power into mechanical power.
Generators and Motors
In generators and motors, the roles of input and output are reversed: - In a motor: Electric energy is the input; Mechanical energy is the output. - In a generator: Mechanical energy is the input; Electric energy is the output.