Alternating Current notes

MDCAT Physics

This chapter explains alternating current, phase difference, RMS values, and the behaviour of AC in resistors, capacitors, inductors and LCR circuits. It also covers power factor, resonance, rectification, choke coils, and the properties, uses and spectrum of electromagnetic waves.

Alternating Current and Its Phase

Alternating current is a current whose magnitude and direction change periodically with time. In a simple AC generator, the current varies sinusoidally and may be written as I = I0 sin ωt, where I0 is the peak current and ω is angular frequency.

The phase of an alternating quantity describes its position in a cycle at a particular instant. Two alternating quantities are in phase when they reach maximum and zero values together. They are out of phase when one reaches these values earlier or later than the other.

The general equation of alternating current is I = I0 sin(ωt + φ).
I0 is the peak or maximum current, ω is angular frequency, t is time, and φ is the phase constant.
Angular frequency is related to frequency by ω = 2πf.
The time period is T = 1/f.
In a purely resistive circuit, current and voltage are in phase, so their phase difference is 0°.
A phase difference of 90° is equal to π/2 radians.
Current leads voltage when current reaches its corresponding value earlier than voltage.
Current lags voltage when current reaches its corresponding value later than voltage.

RMS Value, Mean Square Value and Heating Effect

The value of AC that produces the same heating effect in a resistor as a particular value of direct current is called the root mean square value, or RMS value. AC voltmeters and ammeters generally indicate RMS values.

For a sinusoidal current, the average current over a complete cycle is zero because the positive and negative halves cancel. However, the average of i2 is not zero, so heating depends on i2 and is independent of the direction of current.

For sinusoidal current, Irms = I0/√2.
For sinusoidal voltage, Vrms = V0/√2.
Mean square current is Irms2 = I02/2.
If the peak current is 7√2 A, its RMS current is 7 A.
The heating power in a resistor is P = Irms2R = Vrms2/R.
Heating effect does not depend on the direction of current because it depends on I2R.
For constant RMS voltage across a pure resistor, mean power is Vrms2/R and does not depend on frequency.
For I = 5 sin(100πt) A and R = 10 Ω, Irms = 5/√2 A and mean power is 125 W.

AC Through a Pure Resistor

In a pure resistor, the opposition to current is resistance R. The voltage and current rise, fall, become zero and reverse direction at the same times. Therefore, there is no phase difference between them.

The instantaneous power p = vi changes during a cycle, but its average value is positive. Electrical energy is converted into heat in the resistor.

For a pure resistor, V = IR applies to instantaneous, RMS and peak values when corresponding quantities are used.
Impedance of a pure resistor is Z = R.
Phase angle for a pure resistive circuit is θ = 0°.
Power factor is cos θ = 1.
Mean power is P = VrmsIrms = Irms2R = Vrms2/R.
A resistor stores no energy permanently. It changes electrical energy into heat.
The current direction changes in AC, but the heating effect remains positive in both half cycles.

AC Through a Pure Inductor

An inductor opposes a change in current because a changing current produces a changing magnetic field and an induced emf. This induced emf opposes the change in current according to Lenz's law.

In a pure inductive circuit, current lags behind voltage by 90°, or π/2 radians. The inductor temporarily stores energy in its magnetic field and returns it to the circuit during another part of the cycle.

Inductive reactance is XL = ωL = 2πfL.
The unit of inductance is henry, H.
The internal resistance of an ideal or pure inductor is zero.
The impedance of a pure inductor is Z = XL.
In a pure inductive circuit, current lags voltage by 90°.
The average power consumed by a pure inductor is zero.
Energy stored in an inductor is U = 1/2 LI2.
Increasing frequency increases the inductive reactance.
A choke coil is an inductor used to offer large opposition to AC with very small power loss.

AC Through a Pure Capacitor

A capacitor consists of conductors separated by an insulating material. It stores energy in its electric field. When connected to AC, it repeatedly charges and discharges as the voltage changes.

In a pure capacitive circuit, current leads voltage by 90°. A capacitor allows AC to pass more easily as frequency increases because its capacitive reactance decreases.

Capacitive reactance is XC = 1/(ωC) = 1/(2πfC).
The unit of capacitance is farad, F.
The impedance of a pure capacitor is Z = XC.
In a pure capacitive circuit, current leads voltage by 90°.
Increasing frequency decreases capacitive reactance.
The average power consumed by a pure capacitor is zero.
Energy stored in a capacitor is U = 1/2 CV2.
A capacitor blocks steady DC after it has become fully charged.
Both an ideal inductor and an ideal capacitor have zero average power consumption, but the inductor stores energy magnetically while the capacitor stores energy electrically.

Series LCR Circuit, Impedance and Power Factor

A series LCR circuit contains resistance R, inductance L and capacitance C connected in series with an AC source. The same current passes through all three components, but their voltages have different phase relationships.

The net reactance is the difference between inductive and capacitive reactance. The impedance is obtained by combining resistance and net reactance using a right-angled phasor triangle.

Net reactance is X = XL - XC.
For a series LCR circuit, impedance is Z = √[R2 + (XL - XC)2].
The current is Irms = Vrms/Z.
For an RL series circuit, tan θ = XL/R.
For an RC series circuit, tan θ = -XC/R when the usual sign convention is used.
For a series LCR circuit, tan θ = (XL - XC)/R.
Power factor is cos θ = R/Z.
Average power is P = VrmsIrms cos θ.
If power factor is 1, the circuit dissipates maximum power for a given RMS voltage and current relation because θ = 0°.
For R = 4 Ω and XL = 3 Ω in a series RL circuit, Z = √(42 + 32) = 5 Ω.

Electrical Resonance and Rectification

In a series LCR circuit, resonance occurs when inductive reactance equals capacitive reactance. At resonance, the effects of the inductor and capacitor cancel each other, leaving only resistance in the circuit.

Rectification is the conversion of AC into unidirectional or pulsating DC. Diodes are used because they allow current mainly in one direction.

The resonance condition is XL = XC.
The resonant angular frequency is ω0 = 1/√(LC).
The resonant frequency is f0 = 1/(2π√(LC)).
At resonance, impedance of a series LCR circuit is minimum and equals R.
At resonance, current is maximum: I = V/R.
The phase angle at resonance is 0° and the power factor is 1.
A series circuit has a sharp resonance when R is small.
A bridge rectifier uses four diodes and conducts during both half cycles of AC.
A bridge rectifier does not require a centre-tapped transformer and makes better use of the transformer secondary than an ordinary two-diode centre-tapped full-wave rectifier.

Electromagnetic Waves and Their Properties

Electromagnetic waves are produced by oscillating electric charges and their changing electric and magnetic fields. A changing electric field produces a magnetic field, and a changing magnetic field produces an electric field.

Electromagnetic waves can travel through vacuum, so they do not need a material medium. In vacuum, all electromagnetic waves travel with the same speed, c = 3 × 108 m/s. Their frequency and wavelength are related by c = fλ.

Electromagnetic waves are transverse waves.
The electric field and magnetic field are mutually perpendicular.
The direction of propagation is perpendicular to both the electric field and the magnetic field.
Energy and power are transmitted in the direction of propagation, perpendicular to both fields.
All electromagnetic waves have the same speed in vacuum or free space.
The speed of an electromagnetic wave in vacuum is approximately 3 × 108 m/s.
The relation between speed, frequency and wavelength is c = fλ.
No electromagnetic wave requires a medium for propagation.
X-rays and gamma rays are similar because both are electromagnetic waves, but they differ in origin and generally in wavelength and frequency.
For wavelength 3 × 10-5 m, frequency is f = c/λ = 1 × 1013 Hz.

Electromagnetic Spectrum and Applications

The electromagnetic spectrum is the complete range of electromagnetic radiation arranged according to wavelength or frequency. As frequency increases, wavelength decreases. Radio waves have the longest wavelengths, while gamma rays have the shortest wavelengths in the usual spectrum arrangement.

Different regions of the spectrum have different applications. Their common nature is that they are all transverse electromagnetic waves and travel at the same speed in vacuum.

The order from longest wavelength to shortest wavelength is radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays.
The order from lowest frequency to highest frequency is radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays.
Radio waves have the greatest wavelength among the listed electromagnetic waves.
Microwaves are used in radar, satellite communication and some navigation systems.
Infrared radiation is associated with thermal radiation and is used in remote controls and thermal imaging.
Visible light is the part of the spectrum detected by the human eye.
Ultraviolet radiation has higher frequency than visible violet light.
X-rays have wavelengths of about 10-10 m and are used for medical imaging.
Gamma rays have very high frequency and very short wavelength and are produced in nuclear processes.
Navstar is a modern satellite navigation system.

Key terms

Alternating current: A current that changes its magnitude and direction periodically with time.
Phase: The position of an alternating quantity in its cycle at a particular instant.
RMS value: The effective value of AC that produces the same heating effect as an equal value of DC.
Peak value: The maximum instantaneous value reached by an alternating current or voltage.
Reactance: The opposition offered by an inductor or capacitor to alternating current.
Inductive reactance: The opposition offered by an inductor to AC, given by XL = 2πfL.
Capacitive reactance: The opposition offered by a capacitor to AC, given by XC = 1/(2πfC).
Impedance: The total opposition offered by an AC circuit, combining resistance and reactance.
Power factor: The cosine of the phase angle between voltage and current, cos θ.
Resonance: The condition in a series LCR circuit when XL equals XC.
Choke coil: A coil with low resistance used to control AC by inductive reactance with little power loss.
Rectification: The process of converting AC into unidirectional or pulsating DC.
Electromagnetic wave: A transverse wave made of mutually perpendicular changing electric and magnetic fields.
Wavelength: The distance between two successive points in the same phase of a wave.
Frequency: The number of complete cycles or oscillations produced per second.
Navstar: A satellite navigation system that determines position using navigation satellites.

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Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

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