30. Bode Plots. Introduction


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1 0. Bode Plots Introduction Each of the circuits in this problem set is represented by a magnitude Bode plot. The network function provides a connection between the Bode plot and the circuit. To solve these problems, we first observe the features of the Bode plot to determine the corresponding network function. Next, we calculate the network function of the circuit by representing and analyzing the circuit in the frequencydomain. Finally, we compare the two network functions. Network functions are described in Section. of Introduction to Electric ircuits by R.. Dorf and J.A Svoboda. Bode plots are described in Section.4. Also, Table 0.7 summarizes the correspondence between the timedomain and the frequency domain. Worked Examples Example : onsider the circuit shown in Figure. The input to the circuit is the voltage of the voltage source, v i (t). The output is the node voltage at the output terminal of the op amp, v o (t). The network function that represents this circuit is = o i () The corresponding magnitude Bode plot is also shown in Figure. Determine the values of the capacitances, and 2. Figure The circuit and Bode plot considered in Example.
2 Solution: The network function provides a connection between the circuit and the Bode plot. We can determine the network function from the Bode plot and we can also analyze the circuit to determine its network function. The values of the capacitances are determined by equating the coefficients of these two network functions. First, we make some observations regarding the Bode plot shown in Figure :. There are two corner frequencies, at 80 and 500 rad/s. The corner frequency at 80 rad/s is a pole because the slope of the Bode plot decreases at 80 rad/s. The corner frequency at 500 rad/s is a zero because the slope increases at 500 rad/s The coroner frequencies are separated by log0 = decades. The slope of the Bode plot is = 40 db/decade between the corner frequencies At low frequencies, i.e. at frequencies smaller than the smallest corner frequency, the slope is  20 db/decade so the network function includes a factor (. onsequently, the network function corresponding to the Bode plot is: j j + k j = k 500 = () j + + j where k is a constant that is yet to be determined. Next, we analyze the circuit shown in Figure to determine its network function. A network function is the ratio of the output phasor to the input phasor. Phasors exist in the frequency domain. onsequently, our first step is to represent the circuit in the frequency domain, using phasors and impedances. Figure 2 shows the frequency domain representation of the circuit from Figure. ) Figure 2 The circuit from Figure, represented in the frequency domain, using impedances and phasors. 2
3 To analyze the circuit in Figure 2, we first write a node equation at the node labeled as node a. (The current entering the noninverting input of the op amp is zero, so there are two currents in this node equation, the currents in the impedances corresponding to 25 kω resistor and capacitor.) i a( ) a( ) = 25 0 where is the node voltage at node a. Doing a little algebra gives a Then i = a a a i ( ) = + ( 25 0 ) = + i 25 0 Next, we write a node equation at the node labeled as node b. (The current entering the inverting input of the op amp is zero, so there are two currents in this node equation, the currents in the impedances corresponding to 0 kω resistor and capacitor 2.) ( ) ( ) ( ) a a o = 0 2 Doing some algebra gives o = 0 a 2 ( a ) ( + ( 0 0 )) ( ) 2 a = o Finally, ( ) i o = ( 0 0 ) + o = = i ( 0 0 ) (2) The network functions given in Equations and 2 must be equal. That is
4 k + j j 0 0 = = j ( 0 0 ) Equating coefficients gives: so 80 = and 500 = ( 25 0 ), 2 ( 0 0 ) ( ) k = = ( 0 0 ) = = 0. µ F and = = 0.2 µ F Example 2: onsider the circuit shown in Figure. The input to the circuit is the voltage of the voltage source, v i (t). The output is the node voltage at the output terminal of the op amp, v o (t). The network function that represents this circuit is = o i () The corresponding magnitude Bode plot is also shown in Figure. Determine the values of the capacitances, and 2. Figure The circuit considered in Example 2. 4
5 Solution: The network function provides a connection between the circuit and the Bode plot. We can determine the network function from the Bode plot and we can also analyze the circuit to determine its network function. The values of the capacitances are determined by equating the coefficients of these two network functions. First, we make some observations regarding the Bode plot shown in Figure :. There are two corner frequencies, at 40 and 60 rad/s. Both corner frequencies are poles because the slope of the Bode plot decreases at both the corner frequencies. 2. Between the corner frequencies the gain is = 26 db = 0 = 20 /.. At low frequencies, i.e. at frequencies smaller than the smallest corner frequency, the slope is 20 db/decade so the network function includes a factor ( ). onsequently, the network function corresponding to the Bode plot is: ( ) k = + j + j () Next, we will analyze the circuit shown in Figure to determine its network function. A network function is the ratio of the output phasor to the input phasor. Phasors exist in the frequency domain. onsequently, our first step is to represent the circuit in the frequency domain, using phasors and impedances. Figure 4 shows the frequency domain representation of the circuit from Figure. Figure 4 The circuit from Figure, represented in the frequency domain, using impedances and phasors. 5
6 To analyze the circuit in Figure 4, we write a node equation at the node labeled as node a. In doing so, we will treat the series impedances, 20 kω and, as an single equivalent impedance equal to (The node voltage at node a is zero volts because the voltages at the input nodes of an ideal op amp are equal. The current entering the inverting input of the op amp is zero, so there are three currents in this node equation.) Doing some algebra gives ( ) ( ) ( ) i o o + + = Finally, ( ) i( ) ( )( ) ( ) + ( 20 0 ) o( ) 0 = ( 2( )) o i = + ( ) o = = ( + j 20 0 )( + j 2( )) i (4) The network functions given in Equations and 4 must be equal. That is ( ) ( ) k = = j j Equating coefficients gives: ( j 20 0 )( j 2( )) so also = ( 20 0 ), = 2 ( ) and k = ( ) = =.25 µ F and ( ) = = nf ( ) 2 6 k = = = 0.5 6
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