Frequency response: Resonance, Bandwidth, Q factor Resonance. Let’s continue the exploration of the frequency response of RLC circuits by investigating the series RLC circuit shown on Figure 1. RRCVR+-VsI Figure 1 The magnitude of the transfer function when the output is taken across the resistor is ()()222()1VR RCHVsLC RCωωωω≡=−+ (1.1) At the frequency for which the term 21LCω0−= the magnitude becomes () 1Hω= (1.2) The dependence of ()Hω on frequency is shown on Figure 2 for which L=47mH and C=47µF and for various values of R. 6.071/22.071 Spring 2006, Chaniotakis and Cory 1Figure 2. The frequency 01LCω= is called the resonance frequency of the RLC network. The impedance seen by the source Vs is 11ZRjLjCRj LCωωωω=+ +⎛⎞=+ −⎜⎟⎝⎠ (1.3) Which at 01LCωω== becomes equal toR. • Therefore at the resonant frequency the impedance seen by the source is purely resistive. • This implies that at resonance the inductor/capacitor combination acts as a short circuit. • The current flowing in the system is in phase with the source voltage. The power dissipated in the RLC circuit is equal to the power dissipated by the resistor. Since the voltage across a resistor()cos( )RVtω and the current through it ()cos( )RItω are in phase, the power is 2( ) cos( ) cos( )cos ( )RRRRpt V tI tVI tωωω== (1.4) 6.071/22.071 Spring 2006, Chaniotakis and Cory 2And the average power becomes 21()212RRRPVIIRω== (1.5) Notice that this power is a function of frequency since the amplitudes and RVRI are frequency dependent quantities. The maximum power is dissipated at the resonance frequency 02max ( )12SVPPRωω=== (1.6) 6.071/22.071 Spring 2006, Chaniotakis and Cory 3Bandwidth. At a certain frequency the power dissipated by the resistor is half of the maximum power which as mentioned occurs at 01LCω= . The half power occurs at the frequencies for which the amplitude of the voltage across the resistor becomes equal to 12 of the maximum. 2max1/214VPR= (1.7) Figure 3 shows in graphical form the various frequencies of interest. 1/ 2 Figure 3 Therefore, the ½ power occurs at the frequencies for which ()()222121RCLCRCωωω=−+ (1.8) Equation (1.8) has two roots 2120122RRLLωω⎛⎞=− + +⎜⎟⎝⎠ (1.9) 2220122RRLLωω⎛⎞=+ +⎜⎟⎝⎠ (1.10) 6.071/22.071 Spring 2006, Chaniotakis and Cory 4The bandwidth is the difference between the half power frequencies 2Bandwidth B1ωω==− (1.11) By multiplying Equation (1.9) with Equation (1.10) we can show that 0ω is the geometric mean of 1ω and 2ω. 012ωωω= (1.12) As we see from the plot on Figure 2 the bandwidth increases with increasing R. Equivalently the sharpness of the resonance increases with decreasing R. For a fixed L and C, a decrease in R corresponds to a narrower resonance and thus a higher selectivity regarding the frequency range that can be passed by the circuit. As we increase R, the frequency range over which the dissipative characteristics dominate the behavior of the circuit increases. In order to quantify this behavior we define a parameter called the Quality Factor Q which is related to the sharpness of the peak and it is given by maximum energy stored22total energy lost per cycle at resonanceSDEQEππ== (1.13) which represents the ratio of the energy stored to the energy dissipated in a circuit. The energy stored in the circuit is 21122S2ELI CVc=+ (1.14) For sin( )Vc A tω= the current flowing in the circuit is cos( )dVcICCAdttωω==. The total energy stored in the reactive elements is 222 2 2 211cos ( ) sin ( )22SELCA t CA tωω=+ω (1.15) At the resonance frequency where 01LCωω== the energy stored in the circuit becomes 212SECA= (1.16) 6.071/22.071 Spring 2006, Chaniotakis and Cory 5The energy dissipated per period is equal to the average resistive power dissipated times the oscillation period. 22222000221222DCARC0ERI R ALωπππωω⎛⎞⎛⎞== =⎜⎜⎟⎝⎠⎝⎠ω⎟ (1.17) And so the ratio Q becomes 001LQRRCωω== (1.18) • The quality factor increases with decreasing R • The bandwidth decreases with decreasing R By combining Equations (1.9), (1.10), (1.11) and (1.18) we obtain the relationship between the bandwidth and the Q factor. 0LBRQω== (1.19) Therefore: A band pass filter becomes more selective (small B) as Q increases. 6.071/22.071 Spring 2006, Chaniotakis and Cory 6Similarly we may calculate the resonance characteristics of the parallel RLC circuit. LCRIs(t)IR(t) Figure 4 Here the impedance seen by the current source is //2(1 )jLZjLLCRωωω=−+ (1.20) At the resonance frequency and the impedance seen by the source is purely resistive. The parallel combination of the capacitor and the inductor act as an open circuit. Therefore at the resonance the total current flows through the resistor. 21LCω−=0 If we look at the current flowing through the resistor as a function of frequency we obtain according to the current divider rule 21111()RRSRCLSZIIZZZjLIRLCR j Lωωω=++=−+ (1.21) And the transfer function becomes ()()222()RSILHIRLCR Lωωωω==−+ (1.22) Again for L=47mH and C=47µF and for various values of R the transfer function is plotted on Figure 5. For the parallel circuit the half power frequencies are found by letting 1()2Hω= 6.071/22.071 Spring 2006, Chaniotakis and Cory 7()()22212LRLCR Lωωω=−+ (1.23) Solving Equation (1.23) for ω we obtain the two ½ power frequencies. 21201122RC RCω1ω⎛⎞=− + +⎜⎟⎝⎠ (1.24) 22201122RC RCω1ω⎛⎞=+ +⎜⎟⎝⎠ (1.25) Figure 5 And the bandwidth for the parallel RLC circuit is 211PBRCωω=−= (1.26) The Q factor is 000PRQRCBLωωω== = (1.27) 6.071/22.071 Spring 2006, Chaniotakis and Cory 8Summary of the properties of RLC resonant circuits. Series Parallel Circuit RRCVR+-VsI LCRIs(t)IR(t) Transfer function ()()222()1VR RCHVsLC RCωωωω≡=−+ ()()222()RSILHIRLCR Lωωωω==−+ Resonant frequency 01LCω= 01LCω= ½ power frequencies 2120122RRLLωω⎛⎞=− + +⎜⎟⎝⎠ 2220122RRLLωω⎛⎞=+ +⎜⎟⎝⎠ 21201122RC RCω1ω⎛⎞=− + +⎜⎟⎝⎠ 22201122RC RCω1ω⎛⎞=+ +⎜⎟⎝⎠ Bandwidth 21SRBLωω=−= 211PBRCωω=−= Q factor 0001SLQBRRCωωω== = 000PRQRCBLωωω== = 6.071/22.071 Spring 2006, Chaniotakis and Cory 9Example: A very useful circuit for rejecting noise at a certain frequency such as the