Resonance In AC Circuits

3.1 Introduction M M M h An example of resonance in the form of mechanical : oscillation Potential energy change to kinetic energy than kinetic energy will change back to potential energy. If there is no lost of energy cause by friction potential energy is equal to kinetic energy. mgh = ½ mv ² It will oscillate for a long time. E p =mgh E k= ½ mv ² v

Resonance in electrical circuit C L i i E p= ½ CV ² E m= ½ LI ² Potential energy stored in capacitor change to magnetic energy that stored in inductor. Then magnetic energy change back to potential energy stored in capacitor. If there is no lost of energy by resistor potential energy equal to magnetic energy ½ CV ² = ½ LI ² It will oscillate for a long time.

Characteristic of resonance circuit The frequency response of a circuit is maximum The voltage V s and current I are in phase The impedances is purely resistive. Power factor equal to one Circuit reactance equal zero because capacitive and inductive are equal in magnitude

At frequency resonance, (1) V I jX L - jX C CV² = LI² We know V = I * X L or V = I * X C = I * ωL = I * 1/ωC = I * 2חfL = I / 2חfC Substitute into 1 C (2ח fLI )² = LI² f² = L C (2חfL)² f = 1 2ח√LC ½ CV ² = ½ LI ²

Ideal case ( no resistance) Practical ( energy loss due to resistance) i i t t

Main objective we analysis resonance circuits to find five resonance parameters : a)Resonance frequency, ω o Angular frequency when value of current or voltage is maximum b)Half power frequency, ω 1 and ω 2 Frequency where current (or voltage) equal I max /√2 (or V max /√2 ). c)Quality factor, Q Ratio of its resonant frequency to its bandwidth d)Bandwidth, BW Difference between half power frequency

3.3 Series Resonance Circuits R VRVR V L R j X L - j X C V By KVL : V = V R + V L + V C = V R + jV L – jV C At resonance X L = X C Hence V = V R + 0 = V R = I R * R V c Figure 1

Series Resonance Circuits R VRVR V L j X L - j X C V V c Figure 1 Z = R + j X L - jX C = R + j (X L – X C ) X L = 2πf L X C = 1 2πf C where XLXL R XCXC f0f0 f |Z| (X L -X C )

f f0f0 |I| |Z| |I| = |V| |Z|

Resonance parameter for series circuit a)Resonant frequency,ω o The resonance condition is ω o L = 1 / ω o C or ω o = 1 / √ LC rad/s since ω o = 2Пf o f o = 1/ 2ח√LC Hz b)Half power frequencies At certain frequencies ω = ω 1, ω 2, the half power frequencies are obtain by setting Z = √2R √ R² + ( ω L – 1/ ω C)² = √ 2R

Solving for ω, we obtain ω 1 = - R/2L + √ (R/2L)² + 1/LC rad/s ω 2 = R/2L + √ (R/2L)² + 1/LC rad/s Or in term of resonant parameter, ω 1 = ωo [ - 1/ 2Q + √ (1/ 2Q)² + 1 ] rad/s ω 2 = ωo [ 1/ 2Q + √ (1/ 2Q)² + 1 ] rad/s c)Quality factor, Q Ratio of its resonant frequency to its bandwidth.

Q = V L V S = [ I ] x X L [ I ] x R = ω L ; Q = X L R R = 2 חf r L R Q = V C V = [ I ] x X C [ I ] x R = 1 ; Q = X C ω C R R = 1 2ח f r CR or

d)Bandwidth, BW BW = ω 2 – ω 1 = ω o [ √ 1+ (1/ 2Q)² + 1/ 2Q ] - ωo [ √ 1+ (1/ 2Q)² - 1/ 2Q ] = ω o [ 1/ 2Q + 1/2Q ] = ω o [2/ 2Q] = ω o / Q Q = ω o L /R = 1/ ω o CR thus, BW = R / L = ω o / Q or, BW = ω o ²CR

3.4 Parallel Resonance Circuits Resonance can be divided into 2: a) Ideal parallel circuit b) Practical parallel circuit At least 3 important information that is needed to analyze to get the resonances parameter: In resonance frequency, ω o the imaginary parts of admittance,Y must be equal to zero. When in lower cut-off frequency, ω 1 and in higher cut-off frequency, ω 2 the magnitude of admittance,Y must be equal to √2/R. i +v-+v- RCL Ideal Parallel RLC circuit ωoωo

Resonance parameter for ideal RLC parallel circuit R-jX c jX L Ideal Parallel RLC circuit YTYT Y T = Whereas G(ω) is the real part called the conductance and B(ω) is the imaginary parts called the susceptance.

a) Resonant frequency,ω o Angular resonance frequency is when B(ω)=0. b) Lower cut-off angular frequency, ω 1 Produced when the imaginary parts = (-1/R)

c)Higher cut-off angular frequency, ω2. Produced when the imaginary parts = (1/R) d)Quality Factor, Q e)Bandwidth, BW

Duality Concept R VRVR V L j X L - j X C V V c Figure 1 Series circuit Parallel circuit i +v-+v- RCL Ideal Parallel RLC circuit Z = Z 1 + Z 2 + Z 3 Z = R + j X L - jX C Y = Y 1 + Y 2 + Y 3 Y = Y Z = R + j (ωL – )

Duality Concept R VRVR V L j X L - j X C V V c Figure 1 Series circuit Parallel circuit i +v-+v- RCL Ideal Parallel RLC circuit Y Z = R + j (ωL – ) R LC CL

Duality Concept R VRVR V L j X L - j X C V V c Figure 1 Series circuit Parallel circuit i +v-+v- RCL Ideal Parallel RLC circuit ω 1 = - R/2L + √( (R/2L)² + 1/LC) rad/s ω 2 = R/2L + √( (R/2L)² + 1/LC) rad/s ω 1 = - 1/2RC + √ ((1/2RC)² + 1/LC) rad/s ω 2 = 1/2RC + √( (1/2RC)² + 1/LC) rad/s

Resonance parameter for practical RLC parallel circuit Practical Parallel RLC circuit i +V-+V- R1R1 C L I1I1 ICIC I1I1 ICIC Z1Z1 = R 1 + jX L = |Z 1 |/θ θ |I 1 |cosθ |I 1 |sinθ I

Resonance parameter for practical RLC parallel circuit Practical Parallel RLC circuit i +V-+V- R1R1 C L I1I1 ICIC I1I1 ICIC Resonance occur when |I 1 |sinθ = I C θ |I 1 |cosθ |I 1 |sinθ I

Resonance occur when |I 1 |sinθ = I C |I 1 |sinθ = I C |V| |Z 1 | x XLXL |V| = XCXC XLXL |Z 1 | 2 = XCXC 1 2πfrL2πfrL R 2 + (2πf r L) 2 =2πfrC2πfrC L = C (2πf r L) 2 = L C - R 2

Q factor XLXL R 2πfrL2πfrL = = Q = current magnification I C = |I 1 |sinθ I = |I 1 |cosθ tanθ = R I1I1 ICIC θ |I 1 |cosθ |I 1 |sinθ

Resonance parameter for practical RLC parallel circuit Ideal Parallel RLC circuit i +V-+V- R1R1 C L Second approach to analyze this circuit is by changing the series RL to parallel RL circuit. The purpose of this transformation is to make it much more easier to get the resonance parameter.

RL in series RlRl L RlRl jX p RL in parallel

By matching equation Z T and Y T above, we can get: Or By defining the quality factor, and R p and X p can be write as:

Resonance parameter a)Angular resonance frequency, ω o b)Lower cut-off angular frequency, ω 1 Produced when the imaginary parts = (1/R)

c) Higher cut-off angular frequency, ω 2. d) Quality Factor, Q e) Bandwidth, BW Produced when the imaginary parts = (1/R)