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Sinusoids and Phasors
The mathematics of everything that goes round and round or up and down. The sinusoid as the shadow of a turning arm, the adding of two sinusoids by phasors, interference, beats, the standing wave as a product, and the phasor diagram of an AC circuit.
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Why this chapter
Why this chapter exists
You may go on with the physics now. When a lesson needs this tool, it will tell you and bring you here.
A swinging pendulum, a sound wave at your ear and the mains voltage in your home all vary as sines. So does the light at one point of a screen. Physics keeps asking one question about them: what happens when two such sines are added? This chapter answers it with one picture, an arrow that turns at a steady rate. The arrow turns the addition of sines into the addition of arrows, and it makes interference, beats, standing waves and AC circuits one idea.
You will learn
By the end of this chapter you will
- Write any sinusoid as a turning arm, and read its amplitude, angular frequency and phase from the arm.
- Add two sinusoids of one frequency with a phasor triangle, and find the amplitude and the phase of the sum.
- Find the resultant amplitude and intensity of two interfering waves, and the beat frequency of two nearby notes.
- Write two opposite waves as a standing wave, and draw the phasor diagram of a series LCR circuit.
Before you begin
Before you begin
You should know the sine and the cosine of an angle, angles in radians, and how to add two vectors tip to tail. Nothing about waves or circuits is assumed; each lesson brings its own physics example.
Lessons
1
The Sinusoid from a Rotating Arm
An arm that turns at a steady rate has a tip whose height rises and falls as a sine. The amplitude, the angular frequency and the phase are the length, the rate of turning and the angle of the arm.
The sinusoid from a rotating armA cosine is a sine a quarter turn aheadTime is the angle turned divided by omega
2
Adding Sinusoids of the Same Frequency
Two sinusoids of one frequency add to one sinusoid of that frequency. Its phasor is the tip-to-tail sum of their phasors, and one triangle gives its amplitude and its phase.
Adding sinusoids of the same frequencyA sine plus a cosine is one sinusoid
3
Interference by Phasors
Two waves that meet at a point add like two phasors. In phase they reinforce, in opposite phase they cancel, and in between the phasor triangle gives the amplitude. The intensity goes as the square of the amplitude.
Interference by phasorsIntensity goes as the square of the amplitudePath difference sets the phase difference
4
Adding Sinusoids of Nearby Frequencies
Two sinusoids of slightly different frequencies drift in and out of step. The sum-to-product identity writes their sum as a fast oscillation inside a slow envelope, and the loudness rises and falls \(|f_1 - f_2|\) times a second.
Adding sinusoids of nearby frequenciesThe envelope and the beat frequencyFinding an unknown frequency from beats
5
The Standing Wave as a Product
Two equal waves travelling opposite ways add to a product of a function of position and a function of time. The product does not travel. Its first factor fixes the nodes and antinodes, and at each point the two waves add as two phasors.
The standing wave as a productNodes and antinodes from the first factorEach point adds two phasors
6
Phasor Diagrams for AC
In a series AC circuit every voltage is a sinusoid of the same frequency, so the voltages add as phasors. One diagram gives the impedance, the phase angle, resonance and the power factor.
Phasor diagrams for acResonance in the phasor diagramThe power factor from the diagram
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- amplitude
- The largest value of a sinusoid, \(A\). It is the length of the phasor.
- angular frequency
- The rate at which the phasor turns, \(\omega\), in radians per second. It is related to the period by \(\omega = 2\pi/T\) and to the frequency by \(\omega = 2\pi f\).
- antinode
- A point of a standing wave with the largest amplitude, \(2A\), midway between two nodes.
- beat frequency
- The number of beats in one second, \(|f_1 - f_2|\).
- beats
- The periodic rise and fall of the amplitude, or the loudness, when two sinusoids of slightly different frequencies are added.
- constructive interference
- Two waves meeting in phase, so that their amplitudes add, \(A = a_1 + a_2\).
- destructive interference
- Two waves meeting in opposite phase, so that their amplitudes subtract, \(A = |a_1 - a_2|\).
- envelope
- The slowly changing outline \(\pm 2A\cos(\pi\Delta f\,t)\) that encloses the fast oscillation of two nearby frequencies.
- impedance
- The ratio \(Z = V/I\) of the supply voltage to the current in an AC circuit. For a series LCR circuit, \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).
- initial phase
- The angle \(\phi\) of the phasor at \(t = 0\).
- intensity
- The energy a wave carries across unit area in unit time. It is proportional to the square of the amplitude.
- lag
- A sinusoid lags another when its phasor is behind in the anticlockwise sense. It reaches each peak later.
- lead
- A sinusoid leads another when its phasor is ahead in the anticlockwise sense. It reaches each peak earlier.
- node
- A point of a standing wave that never moves, where the amplitude \(2A\sin kx\) is zero.
- period
- The time \(T\) for one full turn of the phasor, after which the sinusoid repeats.
- phase
- The angle of the phasor at time \(t\), \(\omega t + \phi\). It says where in its cycle the sinusoid is.
- phase difference
- The fixed angle between the phasors of two sinusoids of the same angular frequency.
- phasor
- An arrow of length \(A\) that turns anticlockwise at the steady rate \(\omega\). Its projection on the vertical axis is the sinusoid \(A\sin(\omega t + \phi)\).
- power factor
- The factor \(\cos\phi\) in the average power \(P = VI\cos\phi\) of an AC circuit, where \(\phi\) is the angle between the voltage and the current.
- reactance
- The opposition of an inductor, \(X_L = \omega L\), or of a capacitor, \(X_C = 1/\omega C\), to an alternating current, in ohms. Its voltage is a quarter turn from the current.
- resonance
- The state of a series LCR circuit in which \(X_L = X_C\). The current is largest and in phase with the supply voltage.
- resultant
- The arrow from the start of the first phasor to the tip of the last, when phasors are drawn tip to tail. It is the phasor of the sum.
- sinusoid
- A quantity that varies with time as \(y = A\sin(\omega t + \phi)\), with constant \(A\), \(\omega\) and \(\phi\). A cosine is a sinusoid too.
- standing wave
- The sum of two identical waves travelling in opposite directions, \(y = [2A\sin kx]\cos\omega t\). Its shape does not travel.
- superposition
- The principle that the displacement caused by two waves at a point is the sum of the displacements that each wave alone would cause.