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Chapter 12: Parallel LC & Harmonics

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Parallel Resonance: Comparison ParameterSeriesParallel Z at ResonanceLowestHighest I at ResonanceHighestLowest Effect below resonanceCapacitiveInductive Effect above resonanceInductiveCapacitive Notice the trend …? Lets investigate the trend for parallel resonant circuits 2

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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 3

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Parallel Resonance: Vector Analysis See figure 12-2 Recall: – Z = Impedance 4

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Parallel Resonance: Formula Same equation as for series resonance! – f r = Resonant frequency – L = Inductance – C = Capacitance 5

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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 6

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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 7

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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 8

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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 9

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Harmonics: Waveform addition 10

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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 11

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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 12

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Frequency Spectra: From Sound to Light Two types of transmission – Electromagnetic waves – Sound pressure (compression & rarefaction) Major frequency spectra – Sound – Radio – Light 13

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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) 14

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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 15

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Frequency Spectra: Sound Pressure SourceSPL (dB) Threshold of hearing0(at 1kHz) *perfect silence is -17dB Rustling Leaves10-20 Very Calm Room20-30 Conversation at 1m40-60 50City of Ottawa Bylaw for Noise Complaints as measured in your residence Average factory70 EPA-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc. Hearing Damage85Damage to hearing (need not be continuous) Traffic on a busy road90 Vuvuzela at 1m120Also, jet engine at 100m Threshold of pain130 Explosive shock-wave>194The theoretical limit for SPL 16

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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 17

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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 10 0 to 100 9 or put another way, from 1 to 100 billion To convert between frequency and wavelength: λ = c f C = speed of light, 3x10 8 (ms -1 ) λ = wavelength (meters) f = frequency (Hz, or s -1 ) 18

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E.M. Frequency Spectra: Bands FrequencyWavelength (m)Abrv.Common Radio “Bands” & Examples 3Hz – 30Hz10 5 – 10 4 kmELFMilitary Submarine Radio 30Hz – 300Hz10 4 – 10 3 kmSLFSubmarine Radio 300Hz – 3000Hz10 3 – 10 2 kmULFUsed in mines 3kHz – 30kHz100-10kmVLFNear-surface submarine 30kHz – 300kHz10-1kmLFStandard Time Broadcast & Subs 300kHz – 3MHz1km-100mMF180m & AM radio 3MHz – 30MHz100m-10mHF80,40,30,20,17,15,12,(CB),10m 30MHz – 300MHz10m-1mVHF6,2,1.25m & FM radio & Air-band 300MHz – 3GHz1m-10cmUHF70cm, (FRS, GMRS) 3GHz – 30GHz10cm-1cmSHFWiFi, modern Radar 30GHz – 300GHz1cm-1mmEHFRadio astronomy 19

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E.M. Frequency Spectra: Light & Beyond 20

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Questions? 21

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