Frequency response: Resonance, Bandwidth, Q factor

1 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()1 VRRCHVsLCRC = + ( ) At the Frequency for which the term 21LC 0 = the magnitude becomes ()1H = ( ) The dependence of ()H on Frequency is shown on Figure 2 for which L=47mH and C=47 F and for various values of R. Spring 2006, Chaniotakis and Cory 1 Figure 2. The Frequency 01LC = is called the resonance Frequency of the RLC network.

2 The impedance seen by the source Vs is 11 ZRjLjCRjLC =++ =+ ( ) Which at 01LC == 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()RVt and the current through it ()cos()RIt are in phase, the power is 2( )cos()cos()cos()RRRRptVtItVIt == ( ) Spring 2006, Chaniotakis and Cory 2 And the average power becomes 21()212 RRRPVIIR == ( ) Notice that this power is a function of Frequency since the amplitudes and RVRI are Frequency dependent quantities.

3 The maximum power is dissipated at the resonance Frequency 02max()12 SVPPR === ( ) Spring 2006, Chaniotakis and Cory 3 Bandwidth. At a certain Frequency the power dissipated by the resistor is half of the maximum power which as mentioned occurs at 01LC =. 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/ 214 VPR= ( ) Figure 3 shows in graphical form the various frequencies of interest. 1/2 Figure 3 Therefore, the power occurs at the frequencies for which ()()222121 RCLCRC = + ( ) Equation ( ) has two roots 2120122 RRLL = ++ ( ) 2220122 RRLL =++ ( ) Spring 2006, Chaniotakis and Cory 4 The bandwidth is the difference between the half power frequencies 2 BandwidthB1 == ( ) By multiplying Equation ( ) with Equation ( ) we can show that 0 is the geometric mean of 1 and 2.

4 012 = ( ) 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 resonanceSDEQE == ( ) which represents the ratio of the energy stored to the energy dissipated in a circuit.

5 The energy stored in the circuit is 21122S2 ELICVc=+ ( ) For sin()VcAt = the current flowing in the circuit is cos()dVcICCAdtt ==. The total energy stored in the reactive elements is 22222211cos()sin ()22 SELCAtCAt =+ ( ) At the resonance Frequency where 01LC == the energy stored in the circuit becomes 212 SECA= ( ) Spring 2006, Chaniotakis and Cory 5 The energy dissipated per period is equal to the average resistive power dissipated times the oscillation period. 22222000221222 DCARC0 ERIRAL === ( ) And so the ratio Q becomes 001 LQRRC == ( ) The quality factor increases with decreasing R The bandwidth decreases with decreasing R By combining Equations ( ), ( ), ( ) and ( ) we obtain the relationship between the bandwidth and the Q factor .

6 0 LBRQ == ( ) Therefore: A band pass filter becomes more selective (small B) as Q increases. Spring 2006, Chaniotakis and Cory 6 Similarly 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)jLZjLLCR = + ( ) 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. 21LC =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()RRSRCLSZIIZZZjLIRLCRj L =++= + ( ) And the transfer function becomes ()()222()RSILHIRLCRL == + ( ) Again for L=47mH and C=47 F and for various values of R the transfer function is plotted on Figure 5.

7 For the parallel circuit the half power frequencies are found by letting 1()2H = Spring 2006, Chaniotakis and Cory 7 ()()22212 LRLCRL = + ( ) Solving Equation ( ) for we obtain the two power frequencies. 21201122 RCRC 1 = ++ ( ) 22201122 RCRC 1 =++ ( ) Figure 5 And the bandwidth for the parallel RLC circuit is 211 PBRC = = ( ) The Q factor is 000 PRQRCBL === ( ) Spring 2006, Chaniotakis and Cory 8 Summary of the properties of RLC resonant circuits.

8 Series Parallel Circuit RRCVR+-VsI LCRIs(t)IR(t) Transfer function ()()222()1 VRRCHVsLCRC = + ()()222()RSILHIRLCRL == + Resonant Frequency 01LC = 01LC = power frequencies 2120122 RRLL = ++ 2220122 RRLL =++ 21201122 RCRC 1 = ++ 22201122 RCRC 1 =++ Bandwidth 21 SRBL = = 211 PBRC = = Q factor 0001 SLQBRRC === 000 PRQRCBL === Spring 2006, Chaniotakis and Cory 9 Example: A very useful circuit for rejecting noise at a certain Frequency such as the interference due to 60 Hz line power is the band reject filter sown below. VR+-CLVs Figure 6 The impedance seen by the source is 21jLZRLC =+ ( ) When 01LC == the impedance becomes infinite.

9 The LC combination resembles an open circuit. If we take the output across the resistor the magnitude of the transfer function is ()()2222(1)()VRRLCHVsRRLCL == + ( ) Consideration of the Frequency limits gives 0,0,()1()0,()HHH 1==== ( ) which is a band-stop notch filter. If we are interested in suppressing a 60 Hz noise signal then 1260LC = ( ) Spring 2006, Chaniotakis and Cory 10 For L=47mH, the corresponding value of the capacitor is C=150 F. The plot of the transfer function with the above values for L and C is shown on Figure 7 for various values of R.

10 Figure 7 Since the capacitor and the inductor are in parallel the bandwidth for this circuit is 1 BRC= ( ) If we require a bandwidth of 5 Hz, the resistor R=212 . In this case the pot of the transfer function is shown on Figure 8. Figure 8 Spring 2006, Chaniotakis and Cory 11