Understanding Sinusoidal Waveforms

Sinusoidal Waveform

• Alternating current (AC) is electric current that changes direction; flow duration in one direction equals a half-period, and all half-periods are the same.

Sine Wave Generation

• AC waveforms are sinusoidal or non-sinusoidal.
• AC waveforms can be created via mechanical alternators, electronic inverters, or oscillators.
• Alternators produce sinusoidal output.
• Electronic circuits generate both types with square waves being the simplest and cheapest to produce.
• Sine waves, symmetrical waveforms, symbolize voltage or current, swinging equally around a level and encountered very often in AC circuits.
• A sinusoidal waveform relates directly to an alternator's rotary motion; a sine wave trace on a linear time base denotes its sinusoidal nature.
• Instantaneous voltage or current amplitude is proportional to the sine value at the rotation angle; for instance, at 30° rotation, it's 0.500 × max amplitude, at 45° rotation 0.707 × max, amplitude at 60° 0.866 × max, and amplitude at 90° 1.000 × max.
• In AC, electrons alternate direction, causing continuous current and voltage variations, depicted graphically as a sine wave.
• Sine waves are charted using vertical axes showing magnitude/direction, and horizontal axes showing time/angle of rotation.

360 Degree Cycle of a Sine Wave

• Waveforms above the time axis indicate positive current flow, below indicates negative flow.
• A complete cycle spans 360°, split evenly between positive and negative.
• Sinusoidal sine waves illustrate induced EMF values versus time during a 360° loop rotation.
• AC current, unlike DC, alternates direction with amperage varying over time versus DC's constant amperage with sine waves representing AC waveforms most often.
• Faster flux cutting by a conductor results in greater electromotive force and potential difference, peaking in sine waves when wires move perpendicularly across the magnetic field.
• Each sine wave cycle includes identical voltage variations.
• Positive alternation represents current movement in one direction, dictated by generated terminal voltage polarity.
• Negative alternation represents current movement in the opposite direction because generated voltage terminal polarities reverse.
• Amplitude measures the distance from zero to each alternation's maximum value, which is equal for both positive and negative alternations.

Simple AC Generator

• The armature conductors run parallel at 0°, and no voltage is induced because they don't cut through magnetic lines of flux.
• From 0° to 90° rotation, increasing lines of flux are cut and induced voltage builds positively.
• From 90° to 180°, the armature cuts fewer lines; induced voltage decreases from positive maximum to zero.
• From 180° to 270°, conductors cut more lines in opposite direction; voltage induces negatively, peaking at 270°.

Sine Wave Terminology

• One cycle is a complete sequence of voltage or current change, moving from zero through positive and negative peaks, and returning to zero.
• One alternation is half an AC cycle where voltage or current rises or falls from zero to a peak and back.

AC Frequency

• AC frequency measures cycles completed per second, in hertz (Hz).
• 1 Hz equals one cycle per second, so one loop rotation per second produces 1 Hz and two rotations produce 2 Hz.
• AC or voltage complete cycles each second define frequency, measured in hertz.

Sine Wave Period

• A waveform's period is the time it takes for one cycle, representing duration, and if two cycles occur per second, each cycle lasts 0.5 s.
• Calculating period from frequency: at 50 Hz, the period is 1 s ÷ 50 = 0.02 s, and at 400 Hz, it is 1 s + 400 = 0.0025 s.
• Faster coil spinning increases frequency, produces sine waves faster, and reduces period time, with period expressed as a time fraction in seconds.
• More poles increase the number of cycles per revolution, and a two-pole generator has one cycle per revolution, while a four-pole generator has two.

Peak Voltage or Current

• A sine wave's peak value occurs twice per cycle; once at the positive and once at the negative maximum value.
• Peak-to-Peak voltage or current measures magnitude between peak positive and negative values.
• Instantaneous values represent voltage or current at any point on a sine wave, varying from zero to peak, as illustrated at 90°, 150°, and 240°, where peak voltage equals 100 volts.

Instantaneous Value Formula

• The instantaneous value of a sign wave is calculated using: 𝑉INST = 𝑉PK × sinØ
• Ø represents the angle of rotation in the cycle.

Root, Mean, Square Voltage or Current

• Root Mean Square (RMS) indicates AC power needed for equal heating effect, calculated as the square root of the mean (average).
• VRMS or IRMS symbols denote RMS value, with 120 V voltage cited as an RMS example.
• Sine wave RMS value is 0.707 times the peak value, voltmeters measure averages displaying AC as RMS.

Average AC Voltage or Current

• The average value of AC voltage or current measures the mean of instantaneous values in one alternation and is between zero and peak since voltage increases and decreases from zero to peak in one alternation.
• Computation shows one sine wave alternation averages to 0.637 times peak, and one alternation measures values between 0° and 180°.
• Don't confuse single alternation averages with complete cycle averages of 0, since voltage alternates between positive and negative.

Multimeter Measurement Values

• Multimeters measure averages but show RMS, which is the accepted sine wave value, with True Value RMS meters available at higher costs.

Phase

• Phase relates to recurring AC changes of voltage or current.
• Electrical phase is measured in degrees, with 360° as a full cycle.
• Sinusoidal voltage is proportional to the phase's cosine or sine; 'in phase' occurs when voltage and current sine waves intersect the zero line concurrently.
• Being 'in phase' also applies to two-or-more perfectly paralleled AC power sources.

Sine Waves in Phase

• A voltage sine wave applied to a resistance results in a sine wave current, following Ohm's law where current is proportional to the applied voltage.
• Sine waves of voltage and corresponding current are superimposed, and voltage/current increase/reverse in direction together.
• In-phase sine waves are precisely in sync, reaching maximum/minimum points at the same time and direction.
• Some circuits feature many in-phase sine waves, generating multiple voltage drops that align in phase with each other and the circuit current.

Sine Waves Out of Phase

• Phase A starts at 0°, and as it reaches its positive peak at 90°, Phase B transitions through OV, with phase difference existing if waves don't hit maximum and minimum points simultaneously.

Three-Phase AC Power

• Three-phase generators have three conductors for three AC power phases (120° apart).

Various AC Signal Types

• AC voltages shift over time between positive/negative values.
• Waveforms with repeated shapes include sine, square, triangular, and sawtooth waves.
• Alternating waves have equal areas above and below the t-axis.

Square Wave

• Basic square wave generators are switches (on/off) that are suitable for low-speed operations.
• Modern processors apply PC clocks, i.e., very fast square wave generators whose current CPU clocks surpass 4.0 GHz or 4,000,000,000 clock pulses per second, but this cannot be achieved using simple switches.

Sawtooth wave

• A simple sawtooth wave can be generated using a DC circuit that measures the voltage at a capacitor that is is gradually charged and rapidly discharged once the peak voltage is reached.
• Sawtooth waves are recognized by time variations between rise and fall in value.

Triangular Wave

• Triangular and sawtooth waves differ based on wave shape timing where sawtooth waves exhibit unequal rise/fall but triangular waves have matching periods.
• Integrators process square waves into triangular waves.
• Integration affects functions where, for instance, it gives velocity with acceleration, is achieved via calculus, and relies on charging/discharging capacitors in electronics.