NOTE: The items d) and e) of Question 4 gave you bonus marks.

Transcription

1 MAE 40 Linear ircuit Summer 2007 Final Solution NOTE: The item d) and e) of Quetion 4 gave you bonu mark. Quetion [Equivalent irciut] [4 mark] Find the equivalent impedance between terminal A and B in the following circuit. L A B Let compute one term at a time. Firt followed by Finally Z() //[L//(/)] //(/) (/) ( ) (/) ( ), ( ) L//(/) L ( ) 2 L(L)( ). ( ) Z() //[L//(/)] 2 2 L(L)() () 2 L(L)() () 2 2 L (L)( ) (L)(L2)().

2 L A B Z() /////(L/) o all there i to do i to compute the parallel aociation ( 2 L ) Z() ()( 2 L ). 2 L 2

3 Quetion 2 [Nodal Analyi in the -Domain] [6 mark] Tranform the following circuit to the -domain and formulate node-voltage equation. Aume initial condition and reference ground a indicated in the figure. The current ource i contant. L i L (0) I v (0) Tranform to the -domain uing ource of current for initial condition, a we will need to write node-voltage equation. i L (0) A L B I v (0) Now write node-voltage equation by inpection: [ L 2 L L L ]( ) VA () V B (S) ( ) i L(0) Ii L (O). v (0) 3

4 Quetion 3 [Tranient Analyi in the -Domain] [6 mark] The witch in the next circuit ha been left in poition A for a long time and i moved to poition B at t 0. Find v c (t) for t 0. The voltage ource i contant. A B V L v The firt tep i to determine v (0). Thi i obtained by noticing that if the witch i on A for a long time then the current in, and conequently in, hould be zero, therefore, v (0) V. Now tranform the circuit after the witch i moved to B to the -domain a in the following diagram. B L I() v (0) Meh analyi applied to the diagram provide (L/)I() v (0) v (0) I() (L/) V 2 L Noticing that v () v (0) ( I() Therefore, applying the Laplace invere we have ) V 2 L L 2 L V { } v c (t) VL 2 V co(ωt)u(t), ω : /L 2 /L V L. 4

5 Quetion 4 [ircuit Variable and OpAmp ircuit Deign] When two different metal wire are placed in contact (creating a junction) a voltage appear that i proportional to the junction temperature and the material propertie. A pair of wire made with different material connected at one end a in the next figure i known a a thermocouple, and i a very popular temperature enor. No voltage appear on junction made of ame material becaue of temperature. The point B and B are at the ame temperature T B. B metal x A B metal y A good model for the thermocouple junction i a a voltage ource with voltage v K KT, where K and K are contant that depend only on the material ued in the junction and T i the junction temperature. A imple circuit model for the above thermocouple i given in the next diagram, where x and y repreent the reitance of the wire, which are eentially function of the cro ection area and length of the thermocouple. It i fair to aume that x y. B x A B y v K KT A A thermocouple made with metal x being copper and metal y being contantan can meaure temperature in the range -200 o to 350 o with K 43µ V/ o. The voltage v i meaured from the copper terminal () to the contantan terminal (). A you will ee oon, the value of K i not important. a) [3 mark] A friend of your uggeted that you can meaure the temperature of point A (T A ) by imply connecting a voltmeter with copper lead to the point B and B and meaure the reulting voltage in and (internal to the voltmeter), a in the next figure. The point B and B are at the ame temperature T B. The point and are at the ame temperature T. Draw the circuit diagram correponding to thi etup and how that he/he i not correct: thi etup can only meaure V V K(T A T B ). (Hint: remember that a voltage appear on all junction made with different material!) V copper copper B B copper contantan A 5

6 Becaue the junction B ha two different metal the circuit diagram of the above etup i a follow: z B x A z B y v K KT A v K KT B Becaue no current flow into the voltmeter, then a tated. V V ( K KT A ) ( K KT B ) K(T A T B ) K(T A T B ) One way of overcomming the above problem i to let the temperature T B be known. A popular approach i to have the junction B be immered in a bath of water and ice, in which the temperature i exactly 0.0 o (known a the triple point of water) o that T A T B (V V )/K (V V )/K. b) [4 mark] Aume that T B i in a cold bath at the triple point of water (aume T B 0) and deign an OpAmp circuit to be connected at - that output a voltage v 0 αt A, where α 0 m V/ o. Note that thi circuit hould make the meaurement independent of the wire reitance. If the OpAmp i powered with 0V and 0V what i the temperature range that you can meaure accurately with your circuit? Becaue T A (V V )/K our circuit have to implement the function with gain v 0 αt A (α/k)(v V ) α K 0 03 V/ o V/ o One poible anwer i to ue a non-inverting amplifier etup 6

7 2 V V v 0 with gain Thi configuration ha a high input impedance o the effect of the ohmic wire reitance i minized. Poible (unrealitic) choice for and 2 that could be ued are KΩ, 2 23KΩ. We hould be able to read temperature while the OpAmp tay in the linear range. So we look for the aturation point: T low 0 T low 000 o, T high 0 T high 000 o. A the thermocouple i accurate in the range [200,350] o we hould be able to read accurately the entire cale of the thermocouple. Indeed, we could have ued a higher gain to enhance the circuit reolution, perhap uing two tage of amplification. Another way of overcomming the temperature reference problem i to directly meaure the temperature T B. The jutification for thi i that T B i the temperature of a controlled environment, ay your workbench, while T A may be an extreme temperature you re trying to meaure. Therefore, you could ue a temperature enor to meaure T B that i le expenive or perhap acurate only on ambient temperature. One uch device i called a termitor, which i a reitor whoe reitance varie with the temperature. Termitor are typically accurate and approximately linear from 0 o to a dozen degree above ambient temperature. c) [2 mark] You have a linear termitor with a reitance of 30KΩ at 0 o and a reitance of 0KΩ at 20 o. Show that the relationhip between the termitor reitance ( T ) and the termitor temperature (T B ) i T (30 T B ) 0 3 Ω. Up to what temperature do you think thi termitor i acurate (or at leat linear)? Why? 7

8 Becaue the termitor i linear it hould atify T at B b for ome a and b. Evaluating T at T B 0 o and T B 20 o yield a 0b b Ω, a 20b Ω from where b a (0 0 3 b)/20 ( )/ Ω/ o. Hence T (30 T B ) 0 3 Ω. You hould tart being upiciou about thi termitor model when it reache temperature for which T become near zero, that i, near (30 T B ) T B 30 o. You hould then worry about uing it in a hot day or improve your linear aumption :). d) [Bonu: 4 mark] Uing the above relationhip between the termitor reitance and temperature find value for the component, 3 and 4 o that the following circuit produce v 0 αt B, where α 0 mv/ o and T B i the temperature of the termitor and the junction B. T 0 V 3 4 v 0 Firt recognize that the above circuit i a differential amplifier where v 0 K v K 2 v 2, K T, K 2 T ( ) T

9 and v v 2 0V. Uing thi fact v 0 0(K K 2 ) ( T 4 0 ) T ( ) 4 ( T ) T ( 3 4 ) ( 4 3 T ) Now ubtitute for T 0 βt B, where Ω and β 0 3 For v 0 αt B we need to choe 0 v 0 ( 3 4 ) ( β 3 T B ) 0 ( 3 4 ) ( ) 0β 3 ( 3 4 ) T B , 0β 3 ( 3 4 ) 0β ) α ( 4 3 Thi fixe the choice of ince 0βα (000 30)KΩ 970KΩ Poible choice of 3 and 4 are 3 970KΩ, KΩ, e) [Bonu: 4 mark] Deign an OpAmp circuit that ha a output voltage v 0 α(t A T B ), where α 0 mv/ o and T B i meaured uing the termitor a in item d). (Hint: ue the circuit you deigned in item d)!) The implet olution i to reue the circuit developed in item b) and d) through a differential amplifier configuration with v 0 K v K 2 v 2, K 2, K with v αt B a in item d) and v 2 αt A a in item b). Thi require K 2, K

10 Poible choice of, 2, 3 and 4 are KΩ. The final circuit diagram i a follow. V V K 23K T 00K 00K 00K 00K v 0 α(t A T B ) 0 V 970K 970K 30K 0