Frequency response: Passive Filters Let’s consider again the RC filter shown on Figure 1 RVsjCω1Vc+- Figure 1 When the output is taken across the capacitor the magnitude of the transfer function is 21()1( )HRCωω=+ (1.1) By letting 01RCω= the transfer function becomes 201()1Hωωω=⎛⎞+⎜⎟⎝⎠ (1.2) The overall characteristics of the transfer function may be determined by considering what happens at 0ω= and at ω→∞. 0, ( ) 1,()HHωωωω==→∞ →0 It is also interesting to look at the value of the transfer function at the frequency 0ω For 0ωω= the magnitude of the transfer function becomes 01()2Hωωω== (1.3) 6.071/22.071 Spring 2006, Chaniotakis and Cory 1The frequency 0ω is called the corner, cutoff, or the ½ power frequency. Also, by considering the definition of the dB we have ()() 20log ()dBHHωω= (1.4) Which at 0ωω= gives () 3dBHω=− dB (1.5) And so the frequency 0ω is also called the 3dB frequency. For our example RC circuit with R=10kΩ and C=47nF the Bode plot of the transfer function is shown on Figure 2. In this case the corner frequency equals 2,127 rad/sec and it is indicated on Figure 2. Figure 2 If the output is taken across the resistor, the magnitude of the transfer function becomes 020()1Hωωωωω=⎛⎞+⎜⎟⎝⎠ (1.6) In this case the limits are 0, ( ) 0,()HHωωωω==→∞ →1 6.071/22.071 Spring 2006, Chaniotakis and Cory 2The plot of this transfer function is shown on Figure 3 Figure 3 Filtering and Filters By investigating Figure 2 and Figure 3 we see that the magnitude of the output signal is a very strong function of frequency. The attenuation of the signal amplitude with frequency is also called filtering and the circuits that perform this operation are called filters. In general we say that filters are circuits which allow a specific range of frequencies to be passed (or rejected) as they are transmitted from an ac source to a load. Schematically the system is shown on Figure 4. AC source FilterLoad Figure 4 Filters in general fall into one of the following categories: • Low Pass: passes low frequencies (that is signals with low frequencies) and attenuates high frequencies • High Pass: passes high frequencies (that is signals with high frequencies) and attenuates low frequencies • Band Pass: passes frequencies in a certain range and attenuates frequencies outside this range • Band Stop: attenuates frequencies within a certain range and passes frequencies outside this range. 6.071/22.071 Spring 2006, Chaniotakis and Cory 3For the RC circuit, when the output is taken across the capacitor we obtain a Low Pass filter. By contrast when the output is taken across the resistor we have a High Pass filter. The corresponding plots are shown on Figure 5. Low Pass Filter High Pass Filter Figure 5 The transition frequency which indicates that range of frequencies that are allowed and those that are rejected is given by the cutoff frequency0ω. In practical situations the design of a High pass or Low pass filter is guided by the value of the cutoff or corner frequency 0ω. For our example RC circuit, with R=10kΩ and C=47nF, the cutoff frequency is 338 Hz. We may obtain a band pass filter by combining a low pas and a high pass filter. Consider the arrangement shown on Figure 6. RRCCVs Voab Figure 6 The transfer function may be calculated very easily if we first consider the equivalent circuit to the left of a-b as shown on Figure 7 RZThZCVTh Voab Figure 7 6.071/22.071 Spring 2006, Chaniotakis and Cory 4The voltage VTh is ZCVTh VsZCR=+ (1.7) And RZCZThRZC=+ (1.8) The transfer function now becomes 2()()Vo ZC RHVs R ZC R ZCω==++ (1.9) And upon simplification the magnitude becomes 222()1(3 )RHRRCCωωω=⎛⎞+−⎜⎟⎝⎠ (1.10) By looking at low and high values for ω we have 0, ( ) 0,()HHωωωω==→∞ →0 Also we notice that for 1RCω= the magnitude becomes 1()3Hω= The plot of Equation (1.10) is shown on Figure 8. This has the form of a band pass filter although the attenuation at the frequency 1/ is not desirable. We will next look at ways to improve this type of filter by considering the RLC circuit. RC Figure 86.071/22.071 Spring 2006, Chaniotakis and Cory 5Now let’s continue by exploring the frequency response of RLC circuits. RLCV+-c Vs The magnitude of the transfer function when the output is taken across the capacitor is ()()2221()1VcHVsLC RCωωω==−+ (1.11) Here again let’s look at the behavior of the transfer function, ()Hω, for low and high frequencies. 0, ( ) 1,()HHωωωω==→∞ →0 (1.12) There is another frequency that has a significant effect on the behavior of ()Hω. This is the frequency 0ω at which 20011 LCLCωω=⇒= (1.13) At this frequency the magnitude of the transfer function becomes 0()LCHRCω= (1.14) From the scaling given by Equation (1.12) we see that this circuit corresponds to a low pass filter. Indeed Figure 9 shows the plot for()Hω for R=2kΩ, L=47mH and C=47nF (the values we also used in the laboratory). 6.071/22.071 Spring 2006, Chaniotakis and Cory 6Figure 9 The cutoff frequency in this case is given by the frequency 0121, 276 rad/secLCω== . This is also indicated on the plot of Figure 9. The magnitude ()Hω at 0ωω= is inversely proportional to the resistor R . So let’s now investigate the behavior of the transfer function with R. Figure 10 shows the R dependence of the transfer function. We have plotted Figure 10 6.071/22.071 Spring 2006, Chaniotakis and Cory 7The peak observed at the frequency 0ω is called the resonance peak and the frequency 0ω is also referred to as the resonance frequency. A 0ω the transfer function becomes 2211()()VcHRCVsRCjLCLCeπω−== = (1.15) Which shows that there is 90 degree phase difference between Vc and Vs. The current flowing through the capacitor is IjCVcω= (1.16) And thus the phase difference between the current I and the source voltage Vs is zero. Resonance if defined as the condition at which the voltage and the current at the input of a circuit is in phase. 6.071/22.071 Spring 2006, Chaniotakis and Cory 8If we take the output across the inductor the magnitude of the transfer function is ()()2222()1VL LCHVsLC RCωωωω==−+ (1.17) In this case, consideration of the frequency limits gives 0, ( ) 0,()HHωωωω==→∞ →1 (1.18) And it corresponds to a high pass filter.