Frequency response: Resonance, Bandwidth, Q factor

Transcription

1 Frequency response: esonance, Bandwidth, Q factor esonance. Let s continue the exploration of the frequency response of circuits by investigating the series circuit shown on Figure. C + V - Figure The magnitude of the transfer function when the output is taken across the resistor is H ( ω) V ωc ( ω ) + ( ωc ) (.) At the frequency for which the term ω the magnitude becomes H ( ω ) (.) The dependence of H ( ω ) on frequency is shown on Figure for which L47mH and C47µF and for various values of. 6.7/.7 Spring 6, Chaniotakis and Cory

2 Figure. The frequency ω is called the resonance frequency of the network. The impedance seen by the source is Z + jωl+ jωc + j ωl ωc Which at ω ω becomes equal to. (.3) 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 circuit is equal to the power dissipated by the resistor. Since the voltage across a resistor( V cos( ω t) ) and the current through it ( cos( ω t) ) are in phase, the power is pt ( ) V cos( ωt ) cos( ωt) V cos ( ωt) (.4) 6.7/.7 Spring 6, Chaniotakis and Cory

3 And the average power becomes P( ω ) V (.5) Notice that this power is a function of frequency since the amplitudes frequency dependent quantities. V and are The maximum power is dissipated at the resonance frequency P max ( ω ω ) V S P (.6) 6.7/.7 Spring 6, Chaniotakis and Cory 3

4 Bandwidth. At a certain frequency the power dissipated by the resistor is half of the maximum power which as mentioned occurs at ω. The half power occurs at the frequencies for which the amplitude of the voltage across the resistor becomes equal to of the maximum. Vmax P/ (.7) 4 Figure 3 shows in graphical form the various frequencies of interest. / Figure 3 Therefore, the ½ power occurs at the frequencies for which ωc ( ω ) + ( ωc) (.8) Equation (.8) has two roots ω + + L L ω (.9) ω + + L L ω (.) 6.7/.7 Spring 6, Chaniotakis and Cory 4

5 The bandwidth is the difference between the half power frequencies Bandwidth B ω ω (.) By multiplying Equation (.9) with Equation (.) we can show that ω is the geometric mean of ω and ω. ω ωω (.) As we see from the plot on Figure the bandwidth increases with increasing. Equivalently the sharpness of the resonance increases with decreasing. For a fixed L and C, a decrease in corresponds to a narrower resonance and thus a higher selectivity regarding the frequency range that can be passed by the circuit. As we increase, the frequency range over which the dissipative characteristics dominate the behavior of the circuit increases. n 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 stored ES Q π π (.3) total energy lost per cycle at resonance E which represents the ratio of the energy stored to the energy dissipated in a circuit. The energy stored in the circuit is + (.4) ES L CVc dvc For Vc Asin( ωt) the current flowing in the circuit is C ωcacos( ωt). The dt total energy stored in the reactive elements is ω cos ( ω ) + sin ( ω ) (.5) ES L C A t CA t At the resonance frequency where ω ω the energy stored in the circuit becomes D ES CA (.6) 6.7/.7 Spring 6, Chaniotakis and Cory 5

6 The energy dissipated per period is equal to the average resistive power dissipated times the oscillation period. π ωca π C ED π A (.7) ω ω ωl And so the ratio Q becomes ωl Q (.8) ω C The quality factor increases with decreasing The bandwidth decreases with decreasing By combining Equations (.9), (.), (.) and (.8) we obtain the relationship between the bandwidth and the Q factor. Therefore: L ω B Q (.9) A band pass filter becomes more selective (small B) as Q increases. 6.7/.7 Spring 6, Chaniotakis and Cory 6

7 Similarly we may calculate the resonance characteristics of the parallel circuit. s (t) (t) L C Figure 4 Here the impedance seen by the current source is Z // jωl + ( ω ) jωl (.) At the resonance frequency ω 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. f we look at the current flowing through the resistor as a function of frequency we obtain according to the current divider rule S S Z + + Z Z Z C L jωl ω + jωl ( ) (.) And the transfer function becomes H ( ω) S ωl ( ω ) + ( ωl) Again for L47mH and C47µF and for various values of the transfer function is plotted on Figure 5. (.) For the parallel circuit the half power frequencies are found by letting H ( ω ) 6.7/.7 Spring 6, Chaniotakis and Cory 7

8 ωl ( ω ) + ( ωl) (.3) Solving Equation (.3) for ω we obtain the two ½ power frequencies. ω + + C C ω ω + + C C ω (.4) (.5) Figure 5 And the bandwidth for the parallel circuit is The Q factor is B P ω ω (.6) C Q ω ωc (.7) BP ωl 6.7/.7 Spring 6, Chaniotakis and Cory 8

9 Summary of the properties of resonant circuits. Series Parallel C Circuit + V - s(t) (t) L C Transfer function V H ( ω) ωc H ( ω) ωl ( ω ) + ( ωc ) S ( ω ) + ( ωl) esonant frequency ω ω ½ power frequencies ω + + L L ω ω + + L L ω ω + + C C ω ω + + C C ω L Bandwidth BS ω ω B P ω ω C Q factor ω ω L ω Q ωc B ω L Q B S ω C P 6.7/.7 Spring 6, Chaniotakis and Cory 9

10 Example: A very useful circuit for rejecting noise at a certain frequency such as the interference due to 6 Hz line power is the band reject filter sown below. L C + V - Figure 6 The impedance seen by the source is Z jωl + (.8) ω When ω ω the impedance becomes infinite. The combination resembles an open circuit. f we take the output across the resistor the magnitude of the transfer function is V H ( ω) ( ω ) ( ω ) + ( ωl) (.9) Consideration of the frequency limits gives ω, H ( ω) ω ω H ( ω) ω, H ( ω) which is a band-stop notch filter., (.3) f we are interested in suppressing a 6 Hz noise signal then π 6 (.3) 6.7/.7 Spring 6, Chaniotakis and Cory

11 For L47mH, the corresponding value of the capacitor is C5µF. The plot of the transfer function with the above values for L and C is shown on Figure 7 for various values of. Figure 7 Since the capacitor and the inductor are in parallel the bandwidth for this circuit is B (.3) C f we require a bandwidth of 5 Hz, the resistor Ω. n this case the pot of the transfer function is shown on Figure 8. Figure 8 6.7/.7 Spring 6, Chaniotakis and Cory