29. Network Functions for Circuits Containing Op Amps

Transcription

1 9. Network Functons for Crcuts Contanng Op Amps Introducton Each of the crcuts n ths problem set contans at least one op amp. Also each crcut s represented by a gven network functon. These problems can be solved by analyzng the crcut to determne ts network functon and then comparng the network functon of the crcut to the gven network functon. Network functons are descrbed n Secton 3.3 of Introducton to Electrc Crcuts by.c. Dorf and J.A Svoboda. Op amps are dscussed n Chapter 6. Also, Table 0.7- summarzes the correspondence between the tme-doman and the frequency doman. Worked Examples Example : Consder the crcut shown n Fgure. The nput to the crcut s the voltage of the voltage source, v (t). The output s the voltage across the 0 kω resstor, v o (t). The network functon that represents ths crcut s gven to be H ( + o ) () Determne the values of the capactance, C, and of the resstances, and. Fgure The crcut consdered n Example.

2 Soluton: The crcut has been represented twce, by a crcut dagram and also by the gven network functon. The unknown capactance and resstances, C, and, appear n the crcut dagram, but not n the gven network functon. We can analyze the crcut to determne ts network functon. Ths second network functon wll depend on the unknown capactance and resstances. We wll determne the value of the capactance and resstances by equatng the two network functons. A network functon s the rato of the output phasor to the nput phasor. Phasors exst n the frequency doman. Consequently, our frst step s to represent the crcut n the frequency doman, usng phasors and mpedances. Fgure shows the frequency doman representaton of the crcut from Fgure. Fgure The crcut from Fgure, represented n the frequency doman, usng mpedances and phasors. To analyze the crcut n Fgure, we wrte a node equaton at the node labeled as node a. (The node voltage at node a s zero volts because the voltages at the nput nodes of an deal op amp are equal. The current enterng the nvertng nput of the op amp s zero, so there are four currents n ths node equaton, the currents n the mpedances correspondng to,, C and C.) ( ) ( ) o( ) o( ) C C Dong a lttle algebra gves Then, + C ( ) + C ( ) o + C + C o

3 Fnally, the network functon of the crcut s H ( ) + C o + C C + + C () The network functons gven n Equatons and must be equal. That s Equatng coeffcents gves: C 0 H + + C 500 0, 5000, and C 500 C 9 Now usng C nf 0 F gves and kω, ( ) 9 ( )( ) kω 0 C nf Example : Consder the crcut shown n Fgure 3. The nput to the crcut s the voltage of the voltage source, v (t). The output s the node voltage at the output termnal of the op amp, v o (t). Ths crcut s an example of a frst order low-pass flter. The network functon that represents a frst order low-pass flter has the form o k H (3) + p Ths network functon depends on two parameters, k and p. The parameter k s called the dc gan of the frst order low-pass flter and p s the pole of the frst order low-pass flter. Determne the values of k and of p for the frst order low-pass flter n Fgure 3. 3

4 Fgure 3 The crcut consdered n Example. Soluton: We wll analyze the crcut to determne ts network functon and then put the network functon nto the form gven n Equaton 3. A network functon s the rato of the output phasor to the nput phasor. Phasors exst n the frequency doman. Consequently, our frst step s to represent the crcut n the frequency doman, usng phasors and mpedances. Fgure 4 shows the frequency doman representaton of the crcut from Fgure 3. Fgure 4 The crcut from Fgure 3, represented n the frequency doman, usng mpedances and phasors. To analyze the crcut n Fgure 4, we wrte a node equaton at the node labeled as node a. (The current enterng the non-nvertng nput of the op amp s zero, so there are two currents n ths node equaton, the currents n the mpedances correspondng to 0 kω resstor and 0. µf capactor.) 4

5 ( ) ( ) a a Dong a lttle algebra gves a Then 3 ( ) + 6 ( 0 0 ) a( ) + a( ) a( ) The op amp together wth the 0 kω and 40 kω resstors comprse a nonnvertng amplfer so 40 5 o a Fnally, the network functon s o 5 H ( ) ( ) (4) Comparng the network functons gven by Equatons 3 and 4 gves k 5 / and p 50 rad/s 5