Chapter 12:Parallel LC & Harmonics

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Parallel Resonance: Comparison

Parameter Series Parallel

Z at Resonance Lowest Highest

I at Resonance Highest Lowest

Effect below resonance Capacitive Inductive

Effect above resonance Inductive Capacitive

• Notice the trend …?• Lets investigate the trend for parallel resonant

circuits

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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

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Parallel Resonance: Vector Analysis

• See figure 12-2

• Recall:– Z = Impedance

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Parallel Resonance: Formula

• Same equation as for series resonance!– fr = Resonant frequency– L = Inductance– C = Capacitance

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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

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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

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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

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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 fundamentalSecond (even) harmonic of the fundamentalFundamental harmonic frequency

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Harmonics: Waveform addition

-1.5

-1

-0.5

0

0.5

1

1.5

Sin(x/4)Sin(x/2)/2A+B

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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

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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

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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

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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)

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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

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Frequency Spectra: Sound PressureSource SPL (dB)

Threshold of hearing 0 (at 1kHz) *perfect silence is -17dB

Rustling Leaves 10-20

Very Calm Room 20-30

Conversation at 1m 40-60

50 City of Ottawa Bylaw for Noise Complaints as measured in your residence

Average factory 70 EPA-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc.

Hearing Damage 85 Damage to hearing (need not be continuous)

Traffic on a busy road 90

Vuvuzela at 1m 120 Also, jet engine at 100m

Threshold of pain 130

Explosive shock-wave >194 The theoretical limit for SPL

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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

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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 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)

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E.M. Frequency Spectra: Bands

Frequency Wavelength (m) Abrv. Common Radio “Bands” & Examples

3Hz – 30Hz 105 – 104km ELF Military Submarine Radio

30Hz – 300Hz 104 – 103km SLF Submarine Radio

300Hz – 3000Hz 103 – 102km ULF Used in mines

3kHz – 30kHz 100-10km VLF Near-surface submarine

30kHz – 300kHz 10-1km LF Standard Time Broadcast & Subs

300kHz – 3MHz 1km-100m MF 180m & AM radio

3MHz – 30MHz 100m-10m HF 80,40,30,20,17,15,12,(CB),10m

30MHz – 300MHz 10m-1m VHF 6,2,1.25m & FM radio & Air-band

300MHz – 3GHz 1m-10cm UHF 70cm, (FRS, GMRS)

3GHz – 30GHz 10cm-1cm SHF WiFi, modern Radar

30GHz – 300GHz 1cm-1mm EHF Radio astronomy

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E.M. Frequency Spectra: Light & Beyond

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Questions?