EEG 222 Resonance EEG 222 Dr. A. A. Amusan Department of Electrical & Electronics Engr. University of Lagos Learning Outcome 1. ο ο Understand the following concepts: Frequency response Resonance 2. Analyse a parallel RLC resonant circuit with respect to resonance frequency, impedance, output voltage/ current, power, bandwidth and Q-factor. 2. Analyse a series RLC resonant circuit with respect to resonance frequency, frequency, impedance, output voltage/ current, power, bandwidth and Q-factor. 4. Design filters based on RLC resonant circuits EEG 222 Frequency response • Behavior of our circuit can change dramatically depending on the frequency (or frequencies) of operation • Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. EEG 222 Loudspeakers frequency response and frequency response digitally fixed. Source wikipedia Resonance • A notable feature of the frequency response of a circuit is the sharp resonant peak (or dip) exhibited on the amplitude (magnitude ) characteristics. • Resonance occurs in any circuit with complex conjugate pair of poles i.e any circuit with at least one inductor and capacitor. • In a two-terminal electrical network containing at least one inductor and one capacitor, we define resonance as the condition which exists when the input impedance of the network is purely resistive. • In an electrical circuit, resonance condition exists when the inductive reactance and the capacitive reactance are of equal magnitude, causing electrical energy to oscillate between the magnetic field of the inductor and the electric field of the capacitor. • Resonance is used for tuning and filtering, because resonance occurs at a particular frequency for given values of inductance and capacitance. Resonance can be detrimental to the operation of communications circuits by causing unwanted sustained and transient oscillations that may cause noise, signal distortion, and damage to circuit elements. EEG 222 Analysis of series R-L-C resonant circuit ππ = 1 π πΆ ππΏ = π πΏ Z=π» π = Z=π» π = π πΌ π πΌ = π + ππΏ + ππ 1 = π + π πΏ + π πΆ Let s = ππ Z = π» ππ = π πΌ 1 = π + πππΏ + πππΆ 1 π» ππ = π + πππΏ + πππΆ EEG 222 Analysis of series R-L-C resonant circuit Recall, resonance condition is when the inductive reactance and the capacitive reactance are of equal magnitude. This implies that the imaginary part of the frequency response is zero at resonance. πΌπ π» ππ = 0 1 πππΏ + =0 πππΆ π πππΏ − =0 ππΆ 1 ππΆ = ππΏ 1 π0 = πΏπΆ 1 f0 = 2π πΏπΆ π 0 is the resonance angular frequency in (rad / seconds) f0 is the resonance frequency in Hertz EEG 222 Important points in series R-L-C resonant circuit At resonance for series rlc ο± the series LC circuit acts like a short circuit, hence all the circuit current flows through the resistor, R only. ο± Total impedance (H(jw)) of a series resonance circuit at resonance is R, which is the minimum circuit impedance. At resonance, Z = π» ππ = π is minimum for series resonance. ο± Circuit current and voltage is in phase at resonance, hence power factor is unity ο± Circuit current is maximum at resonance, while the circuit voltage is minimum EEG 222 Impedance of series R-L-C resonant circuit π = π + πππΏ − π = EEG 222 π½ ππΆ 1 π 2 + ππΏ − ππΆ 2 Current in series R-L-C resonant circuit πΌ= π π = π π + πππΏ − π½ ππΆ πΌ = π 1 π 2 + ππΏ − ππΆ 2 Note: Circuit current is maximum at resonance Circuit voltage behaves like impedance, which is minimum at resonance EEG 222 Power: series R-L-C circuit ο§ Power is dissipated in the resistor 2 π πππ 2 πππ£π = πΌπππ π = π π π ο§ Highest power is dissipated at resonance frequency where πΌ π0 = π ππππ₯ = π π0 = ππππ 2 π π ππ 2 ×π = 2 π ×π = 2 1 ππ 2 π ο§ We define π1 and π2 as the half power frequencies. This is the frequencies at which the power dissipated becomes half the maximum value. π(π0 ) 1 ππ2 π π1 = π π2 = = 2 4 π ο§ Half power current value πΌ π1 = πΌ π2 = EEG 222 πΌ π0 2 = ππ π 2 Half power frequencies – series RLC cct ππ πΌ = π 2 1 + ππΏ − ππΆ 1 1 + ππΏ − ππΆ 1 2 π + ππΏ − ππΆ π 2 2 2 = = ππ π 2 1 π 2 2 = 2π 2 2 π2 πΏπΆ − ππ πΆ − 1 = 0 π= π πΆ± π 2 πΆ 2 +4πΏπΆ 2πΏπΆ π π 2 π2 = 2πΏ + EEG 222 2πΏ 1 ππΏ − = π 2 ππΆ 1 ππΏ − = ±π ππΆ π2 πΏπΆ − 1 = ±ππ πΆ or π2 πΏπΆ + ππ πΆ − 1 = 0 or + 1 πΏπΆ π= and −π πΆ± π 2 πΆ 2 +4πΏπΆ 2πΏπΆ π1 = −π 2πΏ + π 2 2πΏ 1 + πΏπΆ Half power frequencies – series RLC cct contd. π2 = π 2πΏ + and π1 = π 2 2πΏ −π 2πΏ + + 1 πΏπΆ π 2 2πΏ + 1 πΏπΆ Also, π0 = π1 π2 π π2 − π1 = πΏ The resonant frequency is the geometrical mean of half power frequencies EEG 222 Bandwidth in series R-L-C resonant π1 and π2 are the half power frequencies π 2 = 2ππ2 π 1 = 2ππ1 At half power, amplitude (current) is decreased 1 1 by 3 dB or 2, while power is decreased by 2 Bandwidth = π2 − π1 EEG 222 Q-factor – RLC series The quality factor (Q-factor) is a quantitative measure of the sharpness of the resonance peak. 2π × ππππ ππππππ¦ π π‘ππππ ππ π‘βπ πππππ’ππ‘ π= πΈπππππ¦ πππ π ππππ‘ππ ππ π‘βπ πππππ’ππ‘ ππ πππ ππππππ ππ πππ ππππππ For series RLC cct. Energy is stored in L or C 1 1 2 2 πΈπ π‘ππππ = πΏπΌπΏ = πΏπΌ 2 2 Note inductor current is same as circuit current 1 2πf0 × πΏπΌ2 π0 πΏ 2 π= = 1 2 π πΌ π 2 EEG 222 Q-factor – RLC series (Contd.) The quality factor (Q-factor) is also the ratio of voltage across inductor or capacitor to voltage across resistor at resonance. VπΏ XπΏ π0 πΏ π= = = Vπ π π π= VπΆ XπΆ 1 = = Vπ π π0 π πΆ π0 πΏ 1 1 πΏ π= = = π π0 π πΆ π πΆ BW (rad / s) = π2 − π1 = π= EEG 222 π πΏ π0 πΏ 1 1 πΏ π0 = = = π π0 π πΆ π πΆ BW (rad / s) Q-factor – RLC series (Contd.) The larger the Q, the sharper the resonance peak, hence the more selective the circuit , that is, it better allows certain frequencies while it rejects all others . BUT Larger Q will give smaller bandwidth and vice versa For circuits with large Q, typically Greater than 10, half power frequency is approximately symmetrical around the resonance, thus π2 ≈ π0 + π΅π 2 π΅π π1 ≈ π0 − 2 High Q desired in filter ccts. π0 π1 π2 BW Q EEG 222 Another Application of Resonance Wireless power transfer EEG 222 Exercises 1. A series RLC circuit has R=2 ohms, L=1mH, and C=0.4 pF. If the input voltage is 20π ππππ‘ V, Determine (a) resonant and half power frequencies (b) Q-factor and bandwidth (c) amplitude of the current at resonant and half power frequen cies. ππππ ππππ ππππ ππππ Ans π0 = 50 π π1 = −49 π π2 = 51 π BW = 2 π Q=25, πΌ(π0 ) = 10π΄ πΌ(π1 , π2 ) = 7.071π΄ 2. Given a series RLC circuit with R=4 ohms, L=25mH with input V= 100π ππππ‘ (a) Determine the value of C to give a Q-factor of 50 (b) π0 π1 π2 BW (c) average power dissipated at π0 π1 π2 ππππ ππππ ππππ πππ Ans C=0.625 µF π0 = 8 π π1 = 7.920 π π2 = 8.080 π BW = 160 π π(π0 ) = 1250π π(π1 , π2 ) = 625π EEG 222 Analysis of parallel R-L-C resonant circuit ππ = 1 π πΆ ππΏ = π πΏ π=π» π = π=π» π = πΌ 1 1 1 = + + π π ππΏ ππ πΌ 1 1 = + + πππΆ π π π πΏ Let s = ππ π = π» ππ = πΌ 1 1 = + + πππΆ π π πππΏ π» ππ = EEG 222 1 π + πππΆ − π ππΏ Analysis of parallel R-L-C resonant circuit Recall, resonance condition exists when the inductive reactance and the capacitive reactance are of equal magnitude i.e πΌπ π» ππ = 0 1 1 ππΆ = π = ππΏ 0 πΏπΆ πππΆ − π =0 ππΏ f0 = 1 2π πΏπΆ π 0 is the resonance angular frequency in (rad / seconds) f0 is the resonance frequency in Hertz At resonance for parallel rlc: ο§ the parallel LC tank circuit acts like an open circuit with the circuit current being determined by the resistor, R only. ο§ So the total impedance of a parallel resonance circuit at resonance becomes just the value of the conductance created by the resistor ο§ the circuit admittance is minimum (or impedance is maximum) since the total 1 susceptance of the LC parallel tank is zero. πΌπ π» ππ = 0 Y = π» ππ = π ο§ power factor is unity EEG 222 Impedance of parallel R-L-C resonant circuit At resonance, circuit Impedance is maximum and = R (or admittance is minimum) π(π) = EEG 222 1 1 1 + πππΆ + π πππΏ Current in parallel R-L-C resonant circuit πΌ = ππ = π πΌ =π 1 1 + πππΆ + π πππΏ 1 1 + ππΆ − π 2 ππΏ 2 Note: Circuit current is minimum at resonance Circuit voltage behaves like impedance, which maximum is at resonance EEG 222 Parameters in parallel R-L-C resonant circuit • Resonance frequency is same as with series rlc π0 = 1 πΏπΆ π1 π2 is not……. EEG 222 Half power frequencies – parallel RLC cct 1 1 + πππΆ + π πππΏ 1 1 π= + π ππΆ − π ππΏ 1 π π= 1 + π ππ πΆ − π ππΏ πΌ = ππ = π 1 At half power frequencies, π = π 1 ± π π ππ πΆ − = ±1 ππΏ π2 π πΏπΆ − π = ±ππΏ π2 π πΏπΆ − ππΏ − π = 0 or π2 π πΏπΆ + ππΏ − π = 0 π= πΏ± πΏ2 +4π 2 πΏπΆ 2π πΏπΆ π2 = EEG 222 1 2π πΆ + 1 2 2π πΆ or + 1 πΏπΆ π= −πΏ± πΏ2 +4π 2 πΏπΆ 2π πΏπΆ and π1 = −1 2π πΆ + 1 2 2π πΆ 1 + πΏπΆ Half power frequencies – parallel RLC cct π2 = 1 2π πΆ 1 2 2π πΆ + and π1 = −1 2π πΆ + 1 + πΏπΆ 1 2 2π πΆ + 1 πΏπΆ Also, π0 = π1 π2 1 π2 − π1 = π πΆ The resonant frequency is the geometrical mean of half power frequencies EEG 222 Q-factor – parallel RLC cct The quality factor (Q-factor) is the ratio of current through the inductor or capacitor to current through resistor at resonance. 1 π0 πΏ IπΏ ππ΅πΏ π= = = 1 Iπ ππΊ π π= π = π0 πΏ π= IπΆ VBπΆ = = π0 π πΆ Iπ ππΊ π πΆ = π0 π πΆ = π π0 πΏ πΏ 1 BW (rad / s) = π2 − π1 = π πΆ π πΆ π0 π= = π0 π πΆ = π = π0 πΏ πΏ BW (rad / s) For Q > 10, EEG 222 π2 ≈ π0 + π΅π 2 π1 ≈ π0 − π΅π 2 Exercises 10sinωt 8kΩ 0.2mH In the circuit, determine: (a) π0 π1 π2 BW and Q 8µF (b)Average power dissipated at π0 π1 π2 ππππ ππππ Ans: π0 = 25 π π1 = 24.992 π ππππ πππ π2 = 25.008 π BW = 15.625 π Q=1600 π(π0 ) = 6.25ππ π(π1 , π2 ) = 3.125π 100kΩ EEG 222 20mH 5nF In the circuit, determine: (a) π0 π1 π2 BW and Q ππππ ππππ Ans: π0 = 100 π π1 = 99 π π2 = 101 ππππ π BW = 2000 πππ π Q=50 Determine the resonance frequencies in the circuits below Vmcosωt 10Ω 2H 0.1F 2Ω 100mH Vmcosωt 0.5mF 20Ω EEG 222 Overview of filters based on RLC circuits LPF HPF EEG 222 Series BPF Shunt BPF Shunt BSF Series BSF
