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This chapter explores parallel resonance in electrical circuits, highlighting the characteristics when inductive and capacitive reactance cancel each other out, resulting in purely resistive behavior. It presents the formulas for resonant frequency and imparts an understanding of how circulating current behaves in such conditions. Additionally, it delves into harmonics, defining them as integer multiples of fundamental frequency, and discusses their role in sound, music, and various waveforms. The relationship between frequency spectra in sound and electromagnetic waves is also analyzed.
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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)
