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**CHAPTER 4 RESONANCE CIRCUITS**

Tunku Muhammad Nizar Bin Tunku Mansur Pegawai Latihan Vokasional Pusat Pengajian Kejuruteraan Sistem Elektrik

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**Content Series Resonance Parallel Resonance Important Parameters**

Resonance Frequency, o Half-power frequencies, 1 and 2 Bandwidth, Quality Factor, Q Application

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Introduction Resonance is a condition in an RLC circuit in which the capacitive and reactive reactance are equal in magnitude, thereby resulting in a purely resistive impedance. Resonance circuits are useful for constructing filters and used in many application.

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**Series Resonance Circuit**

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At Resonance At resonance, the impedance consists only resistive component R. The value of current will be maximum since the total impedance is minimum. The voltage and current are in phase. Maximum power occurs at resonance since the power factor is unity.

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**Series Resonance Total impedance of series RLC Circuit is At resonance**

The impedance now reduce to The current at resonance

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Resonance Frequency Resonance frequency is the frequency where the condition of resonance occur. Also known as center frequency. Resonance frequency

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Half-power Frequency Half-power frequencies is the frequency when the magnitude of the output voltage or current is decrease by the factor of 1 / 2 from its maximum value. Also known as cutoff frequencies.

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Bandwidth, Bandwidth, is define as the difference between the two half power frequencies. The width of the response curve is determine by the bandwidth.

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**Current Response Curve**

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**Voltage Response Curve**

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**Quality Factor (Q-Factor)**

The ratio of resonance frequency to the bandwidth The “sharpness” of response curve could be measured by the quality factor, Q.

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Q-Factor Vs Bandwidth Higher value of Q, smaller the bandwidth. (Higher the selectivity) Lower value of Q larger the bandwidth. (Lower the selectivity)

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High-Q It is to be a high-Q circuit when its quality factor is equal or greater than 10. For a high-Q circuit (Q 10), the half-power frequencies are, for all practical purposes, symmetrical around the resonant frequency and can be approximated as

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**Maximum Power Dissipated**

The average power dissipated by the RLC circuit is The maximum power dissipated at resonance where Thus maximum power dissipated is

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**Power Dissipated at 1 and 2**

At certain frequencies, where ω = ω1 and ω2, the dissipated power is half of maximum power Hence, ω1 and ω2 are called half-power frequencies.

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**Example 14.7 If R=2Ω, L=1mH and C=0.4 F, calculate**

Resonant frequency, ωo Half power frequencies, ω1 and ω2 Bandwidth, Amplitude of current at ωo, ω1 and ω2.

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Solution Resonant frequency Bandwidth Quality Factor

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Solution Since Q 10 , we can regard this as high-Q circuit. Hence

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Solution Current, I at = o Current, I at = 1 , 2

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Practice Problem 14.7 A series connected circuit has R=4Ω and L=25mH. Calculate Value of C that will produce a quality factor of 50. Find 1 , 2 and . Determine average power dissipated at = o , 1 and 2. Take Vm = 100V

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Solution Value of C that will produce Q = 50 Bandwidth

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Solution Since Q 10 , we can regard this as high-Q circuit. Hence

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Solution Power dissipated at = o Power dissipated at = 1 , 2

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Parallel Resonance

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Parallel Resonance The total admittance Resonance occur when

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**At Resonance At resonance, the impedance consists only conductance G.**

The value of current will be minimum since the total admittance is minimum. The voltage and current are in phase.

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**Parameters in Parallel Circuit**

Parallel resonant circuit has same parameters as the series resonant circuit. Resonance frequency Half-power frequencies

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**Parameters in Parallel Circuit**

Bandwidth Quality Factor

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**Example 14.8 If R=8kΩ, L=0.2mH and C=8F, calculate**

ωo Q and ω1 and ω2 Power dissipated at ωo, ω1 and ω2.

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Solution Resonant frequency Bandwidth Quality Factor

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Solution Since Q 10 , we can regard this as high-Q circuit. Hence

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Solution Power dissipated at = o Power dissipated at = 1 , 2

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Practice Problem 14.8 A parallel resonant circuit has R=100kΩ, L=25mH and C=5nF. Calculate o 1 and 2 Q

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Solution Resonant frequency Bandwidth Quality Factor

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Solution Since Q 10 , we can regard this as high-Q circuit. Hence

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APPLICATION

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PASSIVE FILTERS A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuates others A filter is a Passive Filters if it consists only passive elements which is R, L and C. Filters that used resonant circuit Bandpass Filter Bandstop Filter

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BANDPASS FILTER A bandpass filter is designed to pass all frequencies within ω1 ωo ω2

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BANDPASS FILTER SERIES RLC CIRCUIT

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BANDPASS FILTER PARALLEL RLC CIRCUIT

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BANDSTOP FILTER A bandstop or bandreject filter is designed to stop or reject all frequencies within ω1 ωo ω2

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BANDSTOP FILTER SERIES RLC CIRCUIT

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BANDSTOP FILTER PARALLEL RLC CIRCUIT

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EXERCISE

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Problem A series bandpass filter has f1 and f2 equal to 10kHz and 11 kHz. Determine fo and Q R and L if C = 1nF Im if Vin = 24V Value of maximum power dissipated

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