2 WaveA wave is a disturbance. Unlike water waves, electrical waves cannot be seen directly but they have similar characteristics. All periodic waves can be constructed from sine waves, which is why sine waves are fundamental.
3 Sine WavesThe sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform.Electrical sine waves are named from the mathematical function with the same shape.
5 Sine WavesSine waves are characterized by the amplitude and period. The amplitude is the maximum value of a voltage or current; the period is the time interval for one complete cycle.ExampleAThe amplitude (A) of this sine wave is20 VTThe period is50.0 ms
6 Sine WavesThe period of a sine wave can be measured between any two corresponding points on the waveform.TTTTATTBy contrast, the amplitude of a sine wave is only measured from the center to the maximum point.
7 FrequencyFrequency ( f ) is the number of cycles that a sine wave completes in one second.Frequency is measured in hertz (Hz).ExampleIf 3 cycles of a wave occur in one second, the frequency is3.0 Hz1.0 s
9 Period and FrequencyThe period and frequency are reciprocals of each other.andThus, if you know one, you can easily find the other.(The 1/x key on your calculator is handy for converting between f and T.)ExampleIf the period is 50 ms, the frequency is0.02 MHz = 20 kHz.
10 Generation of a Sine Wave Sinusoidal voltages are produced by ac generators and electronic oscillators.When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated.DCBAWhen the loop is moving perpendicular to the lines of flux, the maximum voltage is induced.When the conductor is moving parallel with the lines of flux, no voltage is induced.
11 AC Generator (Alternator) Generators convert rotational energy to electrical energy. A stationary field alternator with a rotating armature is shown. The armature has an induced voltage, which is connected through slip rings and brushes to a load. The armature loops are wound on a magnetic core (not shown for simplicity).Small alternators may use a permanent magnet as shown here; other use field coils to produce the magnetic flux.
12 AC Generator (Alternator) By increasing the number of poles, the number of cycles per revolution is increased. A four-pole generator will produce two complete cycles in each revolution.
13 Function Generator Readout Typical controls: Function selection FrequencyRangeAdjustOutputsOutput level (amplitude)Duty cycleDC offsetCMOS output
14 Sine Wave Voltage and Current There are several ways to specify the voltage of a sinusoidal voltage waveform. The amplitude of a sine wave is also called the peak value, abbreviated as VP for a voltage waveform.ExampleVPThe peak voltage of this waveform is20 V.
15 Sine Wave Voltage and Current The voltage of a sine wave can also be specified as either the peak-to-peak or the rms value. The peak-to-peak is twice the peak value. The rms value is times the peak value.ExampleThe peak-to-peak voltage isVrms40 V.VPPThe rms voltage is14.1 V.
16 Sine Wave Voltage and Current For some purposes, the average value (actually the half-wave average) is used to specify the voltage or current. By definition, the average value is as times the peak value.ExampleThe average value for the sinusoidal voltage isVavg12.7 V.
23 Angular MeasurementAngular measurements can be made in degrees (o) or radians. The radian (rad) is the angle that is formed when the arc is equal to the radius of a circle. There are 360o or 2p radians in one complete revolution.
26 Angular MeasurementBecause there are 2p radians in one complete revolution and 360o in a revolution, the conversion between radians and degrees is easy to write. To find the number of radians, given the number of degrees:To find the number of degrees, given the radians:
28 Sine Wave EquationInstantaneous values of a wave are shown as v or i. The equation for the instantaneous voltage (v) of a sine wave iswhereVp =Peak voltageq =Angle in rad or degreesExampleIf the peak voltage is 25 V, the instantaneous voltage at 50 degrees is19.2 V
35 PhasorThe sine wave can be represented as the projection of a vector rotating at a constant rate. This rotating vector is called a phasor. Phasors are useful for showing the phase relationships in ac circuits.
36 Phase ShiftThe phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference. To show that a sine wave is shifted to the left or right of this reference, a term is added to the equation given previously.wheref =Phase shift
37 Phase Shift …and the equation has a negative phase shift Example of a wave that lags the reference…and the equation has a negative phase shiftv = 30 V sin (q - 45o)Notice that a lagging sine wave is below the axis at 0o
38 Phase Shift Notice that a leading sine wave is above the axis at 0o Example of a wave that leads the referenceNotice that a leading sine wave is above the axis at 0ov = 30 V sin (q + 45o)…and the equation has a positive phase shift
39 Power in Resistive AC Circuits The power relationships developed for dc circuits apply to ac circuits except you must use rms values when calculating power. The general power formulas are:
40 Power in Resistive AC Circuits Assume a sine wave with a peak value of 40 V is applied to a 100 W resistive load. What power is dissipated?ExampleSolutionVrms = x Vp = x 40 V = V8 W
53 Triangular and Sawtooth Wave Triangular and sawtooth waveforms are formed by voltage or current ramps (linear increase/decrease)Triangular waveforms have positive-going and negative-going ramps of equal duration.The sawtooth waveform consists of two ramps, one of much longer duration than the other.
54 HarmonicsAll repetitive non-sinusoidal waveforms are composed of a fundamental frequency (repetition rate of the waveform) and harmonic frequencies.Odd harmonics are frequencies that are odd multiples of the fundamental frequency.Even harmonics are frequencies that are even multiples of the fundamental frequency.
55 HarmonicsA square wave is composed only of the fundamental frequency and odd harmonics (of the proper amplitude).
56 OscilloscopeThe oscilloscope is divided into four main sections.
59 Selected Key TermsA type of waveform that follows a cyclic sinusoidal pattern defined by the formula y = A sin q.Sine waveAlternating currentPeriod (T)Frequency (f)HertzCurrent that reverses direction in response to a change in source voltage polarity.The time interval for one complete cycle of a periodic waveform.A measure of the rate of change of a periodic function; the number of cycles completed in 1 s.The unit of frequency. One hertz equals one cycle per second.
60 Selected Key Terms Instantaneous value Peak valuePeak-to-peak valuerms valueThe voltage or current value of a waveform at a given instant in time.The voltage or current value of a waveform at its maximum positive or negative points.The voltage or current value of a waveform measured from its minimum to its maximum points.The value of a sinusoidal voltage that indicates its heating effect, also known as effective value. It is equal to times the peak value. rms stands for root mean square.
61 Selected Key TermsRadianPhaseAmplitudePulseHarmonicsA unit of angular measurement. There are 2p radians in one complete 360o revolution.The relative angular displacement of a time-varying waveform in terms of its occurrence with respect to a reference.The maximum value of a voltage or current.A type of waveform that consists of two equal and opposite steps in voltage or current separated by a time interval.The frequencies contained in a composite waveform, which are integer multiples of the pulse repetition frequency.
62 Quiz1. In North America, the frequency of ac utility voltage is 60 Hz. The period isa. 8.3 msb msc. 60 msd. 60 s
63 Quiz 2. The amplitude of a sine wave is measured a. at the maximum pointb. between the minimum and maximum pointsc. at the midpointd. anywhere on the wave
64 Quiz3. An example of an equation for a waveform that lags the reference isa. v = -40 V sin (q)b. v = 100 V sin (q + 35o)c. v = 5.0 V sin (q - 27o)d. v = 27 V
65 Quiz 4. In the equation v = Vp sin q , the letter v stands for the a. peak valueb. average valuec. rms valued. instantaneous value
66 Quiz5. The time base of an oscilloscope is determined by the setting of thea. vertical controlsb. horizontal controlsc. trigger controlsd. none of the above
67 Quiz 6. A sawtooth waveform has a. equal positive and negative going rampsb. two ramps - one much longer than the otherc. two equal pulsesd. two unequal pulses
68 Quiz7. The number of radians in 90o area. p/2b. pc. 2p/3d. 2p
69 Quiz8. For the waveform shown, the same power would be delivered to a load with a dc voltage ofa Vb Vc Vd V
70 Quiz 9. A square wave consists of a. the fundamental and odd harmonics b. the fundamental and even harmonicsc. the fundamental and all harmonicsd. only the fundamental
71 Quiz10. A control on the oscilloscope that is used to set the desired number of cycles of a wave on the display isa. volts per division controlb. time per division controlc. trigger level controld. horizontal position control
89 Discrete Fourier Transform Forward DFT:Inverse DFT:The complex numbers f0 … fN are transformed into complex numbers F0 … FnThe complex numbers F0 … Fn are transformed into complex numbersf0 … fN
90 DFT ExampleInterpreting a DFT can be slightly difficult, because the DFT of real data includes complex numbers.Basically:The magnitude of the complex number for a DFT component is the power at that frequency.The phase θ of the waveform can be determined from the relative values of the real and imaginary coefficents.Also both positive and “negative” frequencies show up.
94 Fast Fourier Transform Discrete Fourier Transform would normally require O(n2) time to process for n samples:Don’t usually calculate it this way in practice.Fast Fourier Transform takes O(n log(n)) time.Most common algorithm is the Cooley-Tukey Algorithm.
96 快速傅利葉轉換原理A complex nth root of unity is a complex number z such that zn = 1.n = e 2 i / n = principal n th root of unity.e i t = cos t + i sin t.i2 = -1.There are exactly n roots of unity: nk, k = 0, 1, , n-1.n2= n/2nn+k= nk2 = i314 = -10 = 175forms a group under multiplication (similar to additive group Z_n modulo n)6 = -i
97 Fourier Cosine Transform Any function can be split into even and odd parts:Then the Fourier Transform can be re-expressed as:
98 Discrete Cosine Transform (DCT) When the input data contains only real numbers from an even function, the sin component of the DFT is 0, and theDFT becomes a Discrete Cosine Transform (DCT) There are 8 variants however, of which 4 are common.
99 DCT Types DCT Type II Used in JPEG, repeated for a 2-D transform. Most common DCT.
100 DCT Types DCT Type IV Used in MP3. In MP3, the data is overlapped so that half the data from one sample set is reused in the next.Known as Modified DCT or MDCTThis reduces boundary effects.
101 Why do we use DCT for Multimedia? For audio:Human ear has different dynamic range for different frequencies.Transform to from time domain to frequency domain, and quantize different frequencies differently.For images and video:Human eye is less sensitive to fine detail.Transform from spacial domain to frequency domain, and quantize high frequencies more coarsely (or not at all)Has the effect of slightly blurring the image - may not be perceptable if done right.
102 Why use DCT/DFT?Some tasks are much easier to handle in the frequency domain that in the time domain.Eg: graphic equalizer. We want to boost the bass:Transform to frequency domain.Increase the magnitude of low frequency components.Transform back to time domain.
103 Fourier TransformFourier Series can be generalized to complex numbers, and further generalized to derive the Fourier Transform.Forward Fourier Transform:Inverse Fourier Transform:Note:
104 Fourier TransformFourier Transform maps a time series (eg audio samples) into the series of frequencies (their amplitudes and phases) that composed the time series.Inverse Fourier Transform maps the series of frequencies (their amplitudes and phases) back into the corresponding time series.The two functions are inverses of each other.