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Chapter 12: Parallel LC & Harmonics. Parallel Resonance: Comparison. Notice the trend …? Lets investigate the trend for parallel resonant circuits. Parallel Resonance: Characteristics. At resonance: Inductive and capacitive reactance are equal and effectively cancel each other
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Parallel Resonance: Comparison • Notice the trend …? • Lets investigate the trend for parallel resonant circuits
Parallel Resonance: Characteristics • At resonance: • Inductive and capacitive reactance are equal and effectively cancel each other • Result is purely resistive character • Impedance equals resistance • Current is at its lowest
Parallel Resonance: Vector Analysis • See figure 12-2 • Recall: • Z = Impedance
Parallel Resonance: Formula • Same equation as for series resonance! • fr = Resonant frequency • L = Inductance • C = Capacitance
Formula: Worked Example • What is the resonant frequency of the circuit above? • (2π*sqrt(10e-6 x 100e-3))-1 • At a frequency of 160Hz in the above circuit, what relative current would you expect? • Minimum current since the circuit is at resonance ≈ 160Hz
Parallel Resonance: Circulating Current • As current flows initially, electrical potential is stored in the capacitor and magnetic potential is stored in the inductor • As the current drops, the inductor acts to resist the change in current, allowing the magnetic field to collapse, causing charge to develop on capacitor • Without any losses (ie. Ideal components) the circulating current would continue resonating indefinitely
Preview: Oscillators • Imagine water sloshing around between two tanks which are connected by a large pipe • Voltage stored in a capacitor and magnetic potential stored in an inductor behave in an analogous manner • With minimal input, a rhythmic flip/flop of energy can occur with the resulting flow of energy producing a sinusoidal wave
Harmonics: Introduction • Fundamental frequency is (generally) the lowest frequency in a related grouping • One may define a frequency instead • Harmonics are integer multiples of the fundamental frequency • Eg: Frequencies as follows: • 30kHz • 20kHz • 10kHz Third (odd) harmonic of the fundamental Second (even) harmonic of the fundamental Fundamental harmonic frequency
Harmonics: Square Waves • Square waves may be synthesized by adding a large number of odd harmonics to achieve a relatively “flat” crest • In practice, this is achieved by analog function generators cascading mixing and multiplication stages
Harmonics: Speech & Music • The human voice differs between individuals primarily as a result of differences in harmonic content • Musical instruments all exploit harmonics • Simplest examples are string instruments such as the piano which have a fundamental frequency of 256Hz for “middle C” • Richness of music is the interaction of multiple harmonics which are mathematically related
Frequency Spectra: From Sound to Light • Two types of transmission • Electromagnetic waves • Sound pressure (compression & rarefaction) • Major frequency spectra • Sound • Radio • Light
Frequency Spectra: Sound • Average human hearing extends from 20Hz to 20kHz • The majority of human voice exists between 300Hz and 3kHz • The bandwidth is therefore 3000-300 = 2.7kHz • Telephone and SSB radio take advantage of this fact • Sound intensity (volume) measured as decibels (dB)
Frequency Spectra: Sound • Decibels expresses a ratio between the threshold of hearing at 1kHz and the frequency of interest • We can only hear a difference in sound volume of 3dB • Double the intensity since 3dB = 2x
Frequency Spectra: Radio • Now lets take a look at the other form of frequency production: electromagnetic • Unlike sound, we can only perceive a very small range of EM frequencies • Light • Infrared
E.M. Frequencies: Primer • All electromagnetic frequencies between 3Hz and 300GHz are considered to be in the radio spectrum • That is a substantial range, from 100 to 1009 or put another way, from 1 to 100 billion • To convert between frequency and wavelength: λ = c f C = speed of light, 3x108(ms-1) λ = wavelength (meters) f = frequency (Hz, or s-1)
