4.3- Modeling the Diode Forward Characteristic

Transcription

1 2/8/2012 3_3 Modelng the ode Forward Characterstcs 1/ Modelng the ode Forward Characterstc Readng Assgnment: pp How do we analyze crcuts wth juncton dodes? 2 ways: Exact Solutons ffcult! Approxmate Solutons Easy (relatvely). A. Exact Solutons The juncton dode equaton often results n an unsolvable transcendental equaton! HO: TRANSCENENTAL SOLUTIONS OF JUNCTION IOE CIRCUITS B. Approxmate Solutons Jm Stles The Unv. of Kansas ept. of EECS

2 2/8/2012 3_3 Modelng the ode Forward Characterstcs 2/3 To obtan a quck (but less accurate) soluton, we replace all juncton dodes wth approxmate crcut models. 3 knds of models: 1. Ideal ode 2. Constant Voltage rop (CV) 3. Pecewse-Lnear (PWL) HO:THE IEAL IOE MOEL To mprove on the deal dode model, we smply add a voltage source! HO: THE CONSTANT VOLTAGE ROP MOEL Let s try a crcut analyss example wth the CV model. EXAMPLE: JUNCTION IOECIRCUIT ANALYSIS WITH THE CV MOEL A more accurate but much more complex model s the Pecewse Lnear Model (PWL). Jm Stles The Unv. of Kansas ept. of EECS

3 2/8/2012 3_3 Modelng the ode Forward Characterstcs 3/3 HO: THE PIECEWISE LINEAR MOEL There are at least two good approaches for constructng an accruate juncton dode PWL model. HO: CONSTRUCTING THE PWL MOEL Let s try an example for constructng a PWL model. EXAMPLE: CONSTRUCTING A PWL MOEL It s unfathomably mportant that you learn how to correctly mplement these models to analyze juncton dode crcuts! EXAMPLE: JUNCTION IOE MOELS Jm Stles The Unv. of Kansas ept. of EECS

4 2/6/2012 Transcendental Solutons present 1/8 Transcendental Solutons of Juncton ode Crcuts In a prevous example, we were able to use the juncton dode equaton to algebracally analyze a crcut and fnd numerc solutons for all crcut currents and voltages. However, we wll fnd that ths type of crcut analyss s, n general, often mpossble to acheve usng the juncton dode equaton! Q: Impossble!?! If we have an explct mathematcal descrpton of each devce n a crcut (whch we do for a juncton dode), can t we can use KVL and KCL to analyze any crcut. A: Although we can always determne a numercal soluton, t s often mpossble to fnd ths soluton algebracally. Jm Stles The Unv. of Kansas ept. of EECS

5 2/6/2012 Transcendental Solutons present 2/8 One equaton and one unknown so what s the bg deal Consder ths smple juncton dode crcut: From KVL: V v v 0 S R V v R 0 S V v S R V S + - v R vr Lkewse, from the juncton dode equaton: v I s nv e T 1 Equatng these two, we have a sngle equaton wth a sngle unknown (v ): V S v R I s e v nv T 1 Jm Stles The Unv. of Kansas ept. of EECS

6 2/6/2012 Transcendental Solutons present 3/8 Q: Rght! You have 1 equaton wth 1 unknown. Try solvng ths! Just solve ths equaton for v, and then you can determne all other unknown voltages and currents (.e., and v R ). A: But that s the problem! What s the algebrac soluton of v for the equaton: Q:???? V S v R I s e v nv T 1 A: The above equaton s mystcally known as a transcendental equaton. It s an algebrac expresson for whch there s no algebrac soluton! Jm Stles The Unv. of Kansas ept. of EECS

7 2/6/2012 Transcendental Solutons present 4/8 There s a soluton, however Examples of transcendental equatons nclude: 2 x cos x, y ln y, or 4 x 2 x Q: But, we could buld ths smple juncton dode crcut n the lab. V S + - v R vr Therefore v, and v R must have some numerc value, rght!?! A: Absolutely! For every value of source voltage V S, resstance R, and juncton dode parameters n and I s, there s a specfc numercal soluton for v, and v R. However, we cannot fnd ths numercal soluton wth algebrac methods! Jm Stles The Unv. of Kansas ept. of EECS

8 2/6/2012 Transcendental Solutons present 5/8 The soluton requres a computer Q: Well then how the heck do we fnd soluton?? A: We use what s know as numercal methods, often mplementng some teratve approach, typcally wth the help of a computer. Ths generally nvolves more work than we wsh to do when analyzng juncton dode crcuts (despte the help of Fortran and the teletype)! Jm Stles The Unv. of Kansas ept. of EECS

9 2/6/2012 Transcendental Solutons present 6/8 But, we dd t before wthout a computer! Q: So just how do we analyze juncton dode crcuts?? A: We replace the juncton dodes wth crcut models that approxmate juncton dode behavor! Q: Wat! I recall an earler example when analyzed a juncton dode crcut, but we dd not use approxmate models nor numercal methods to fnd the answer! A: Ths s absolutely correct; we dd not use approxmate models or numercal methods to solve that problem. However, f you look back at that example, you wll fnd that the problem was a bt contrved. Jm Stles The Unv. of Kansas ept. of EECS

10 2/6/2012 Transcendental Solutons present 7/8 Professors: they re so trcky! 1 Recall that effectvely, we were gven the voltage across one dode as part of the problem statement. We were then asked to fnd the source voltage V S. + V - s v 1 v R R 65. ma 01K. 2 v 2 Ths was a bt of an academc problem, as n the real world t s unlkely that we would somehow know the voltage across the dode wthout knowng the value of the voltage source that produced t! Thus, problems lke ths prevous example are sometmes used by professors to create juncton dode crcut problems that are solvable, wthout encounterng a dreaded transcendental equaton! Jm Stles The Unv. of Kansas ept. of EECS

11 2/6/2012 Transcendental Solutons present 8/8 Transcendental s the norm; crcut models are the soluton In the real world, we typcally know nether the dode voltage nor the dode current drectly transcendental equatons are most often the sad result! Instead of applyng numercal technques, we wll fnd t much faster (albet slghtly less accurate) to apply approxmate crcut models. Jm Stles The Unv. of Kansas ept. of EECS

12 2/6/2012 The Ideal ode Model present 1/5 The Ideal ode Model One way to analyze juncton dode crcuts s smply to assume the juncton dodes are deal. In other words: Replace: v wth: v v We know how to analyze deal dode crcuts (recall sect. 4.1)! Jm Stles The Unv. of Kansas ept. of EECS

13 2/6/2012 The Ideal ode Model present 2/5 Ths s why we studed secton 4.1 IMPORTANT NOTE!!! PLEASE REA THIS CAREFULLY: Make sure you analyze the resultng crcut precsely as we dd n secton 4.1: 1. You assume the same deal dode modes, 2. you enforce the same deal dode values, 3. you analyze n the exact same manner, 4. and you check the same deal dode results, precsely as before. Once we replace the juncton dodes wth deal dodes, we have an deal dode crcut no juncton dodes are nvolved! Jm Stles The Unv. of Kansas ept. of EECS

14 2/6/2012 The Ideal ode Model present 3/5 That s the thng about approxmatons they gve you answers that are only approxmately correct Q: But, deal dodes are not juncton dodes; won t we get the wrong answer??? A: YES!!! arn rght we won t! However, the answers, albet ncorrect, wll be close to the actual values. In other words, our answers wll be approxmately correct. We approxmate a juncton dode as an deal dode. Our answers are therefore approxmatons!! Jm Stles The Unv. of Kansas ept. of EECS

15 2/6/2012 The Ideal ode Model present 4/5 For example For example, say we replace the juncton dode n a crcut wth the deal dode model. We now have an deal dode crcut! Say we then assume, enforce, analyze and check, and fnd that the deal dode current and voltage are: 247mA and v 0 v 0 247mA 247mA Thus, we conclude that the juncton dode current s approxmately 247 ma, and the juncton dode current s approxmately zero. v 0 Jm Stles The Unv. of Kansas ept. of EECS

16 2/6/2012 The Ideal ode Model present 5/5 Q: What? It s that approxmaton thng agan they re always n error I thought that f the juncton dode current s postve, then the juncton dode voltage s approxmately 0.7 V not zero volts! A: Yes, the deal dode model provdes a course approxmaton wth perhaps sgnfcant error partcularly f the juncton dode s operatng n the forward bas regon. deal dode model juncton dode v Jm Stles The Unv. of Kansas ept. of EECS

17 2/6/2012 The Constant Voltage rop Model present 1/16 The Constant Voltage rop (CV) Model Q: We know f sgnfcant postve current flows through a juncton dode, the dode voltage wll be some value near 0.7 V. Yet, the deal dode model provdes an approxmate answer of v =0 V. Isn t there a more accurate model? A: Yes! Consder the Constant Voltage rop (CV) model. For the CV model, we approxmate a juncton dode as: 0 f v 0.7V CV model juncton dode v 0.7V f 0 v Note ths s a farly accurate statement of juncton dode behavor! 0.7 V Jm Stles The Unv. of Kansas ept. of EECS

18 2/6/2012 The Constant Voltage rop Model present 2/16 Models have more than one components Q: Yes, but what s the crcut devce for the CV model I don t know of any component that behaves lkes ths? A: Our crcut models do not have to be a sngle devce. Instead, these models typcally consst of two or more devces! From KVL we fnd: For example, consder an deal dode n seres wth a 0.7 V voltage source: V v 0.7 v V 0.7 V I v 07. V And from KCL: I Jm Stles The Unv. of Kansas ept. of EECS

19 2/6/2012 The Constant Voltage rop Model present 3/16 V s the voltage across the entre model! Thus, f the deal dode n ths crcut s forward based ( 0 and v 0), the model voltage and current s: v V V 0.7 and I 0 I 0 V V Or, f the deal dode n ths crcut s reverse based ( v 0 and 0 ), the model voltage and current s: I 0 and v V V 0.7 I 0 V 07. v V Jm Stles The Unv. of Kansas ept. of EECS

20 2/6/2012 The Constant Voltage rop Model present 4/16 Smells lke the CV model! In summary, we fnd that for ths crcut model: I I 0 f V 0.7V V 0.7V f I 0 V v 07. V Q: Hey! Isn t ths precsely the expresson for the CV model, only wth V and I?? v 0 f v 0.7V CV model juncton dode v 0.7V f 0 v 0.7 V Jm Stles The Unv. of Kansas ept. of EECS

21 2/6/2012 The Constant Voltage rop Model present 5/16 Just lke we dd for the deal dode model A: It s! Ths crcut s the CV crcut model, and we use t to analyze juncton dode crcuts n precsely the same manner as wth the deal dode model: Replace: v wth: v 07. v v 07. V In other words, replace the juncton dode wth two devces an deal dode n seres wth a 0.7 V voltage source. Jm Stles The Unv. of Kansas ept. of EECS

22 2/6/2012 The Constant Voltage rop Model present 6/16 Gve me three steps.. To fnd approxmate current and voltage values of a juncton dode crcut, follow these 3 steps: Step 1 - Replace each juncton dode wth the two devces of the CV model. Note you now a have an IEAL dode crcut! There are no juncton dodes n the crcut, and therefore no juncton dode knowledge need be (or should be) used to analyze t. Step 2 - Now analyze the IEAL dode crcut on your paper. etermne and v for each deal dode. IMPORTANT NOTE!!! PLEASE REA THIS CAREFULLY: Make sure you analyze the resultng crcut precsely as we dd n secton 4.1. You assume the same IEAL dode modes, you enforce the same IEAL dode values, and you check the same IEAL dode results, precsely as before. Once we replace the juncton dodes wth the CV model, we have an IEAL dode crcut no juncton dodes are nvolved! Jm Stles The Unv. of Kansas ept. of EECS

23 2/6/2012 The Constant Voltage rop Model present 7/16 Step 3 gves the approxmate answer Step 3 etermne the approxmate values and v of the juncton dode from the deal dode values and v : v v v 07. V v 07. V Jm Stles The Unv. of Kansas ept. of EECS

24 2/6/2012 The Constant Voltage rop Model present 8/16 The juncton voltage s always 0.7 hgher Note therefore, f the IEAL dode (note here I sad IEAL dode) s forward based ( 0 ), then the approxmaton of the juncton dode current wll lkewse be postve ( 0 ), and the approxmaton of the juncton dode voltage (unlke the deal dode voltage of v 0) wll be: v v V However, f the IEAL dode s reversed based ( 0 ), then the approxmaton of the juncton dode current wll lkewse be zero ( = 0), and the approxmaton of the juncton dode voltage (unlke the deal dode voltage of v 0 ) wll be: v v V Note that the approxmate juncton dode voltage s always 0.7V more than the deal dode voltage don t forget to add 0.7 to your calculated value of v! Jm Stles The Unv. of Kansas ept. of EECS

25 2/6/2012 The Constant Voltage rop Model present 9/16 If the voltage s postve, why sn t there current? Q: Wat a second! Say the deal dode n the CV model turns out to be reverse based, wth: v 0.1V and 0. The approxmate juncton dode current and voltage would thus be: v v V 0 The juncton dode voltage s 0.6 V ths nstead sounds lke forward bas! How does ths make sense? Jm Stles The Unv. of Kansas ept. of EECS

26 2/6/2012 The Constant Voltage rop Model present 10/16 That ambguous transston! A: Remember, an deal dode must be unambguously forward or reversed based. But, a juncton dode operates n ambguously defned regons, whose boundares are a bt murky. Thus, a result lke the example above, where v 0.6V (the forward bas regon?) and 0 (the reverse bas regon?), s ndcatve of juncton dode operatng n the ambguous transton regon between forward and reverse bas. CV model juncton dode Transton Regon v 0.7 V Jm Stles The Unv. of Kansas ept. of EECS

27 2/6/2012 The Constant Voltage rop Model present 11/16 Works qute well for reverse bas regon In other words, the juncton dode current s not suffcently large to be unambguously n the forward bas regon, but nether s t suffcently negatve to be unambguously n the reverse bas regon! Q:But stll, a juncton dode wth a voltage of v 0.6V would not have zero current and vce versa. Ths numercal result would seem to exhbt a bunch of error! A: True enough! If we plot both the juncton dode curve and the CV model curve, we see that they overlap very well n the reverse bas regon, meanng the CV model s very accurate f the juncton dode s operatng n that regon: Reverse Bas Regon CV model juncton dode v 0.7 V Jm Stles The Unv. of Kansas ept. of EECS

28 2/6/2012 The Constant Voltage rop Model present 12/16 Works reasonably well n the forward bas regon And, the two curves overlap reasonably well n the forward bas regon, meanng the CV model s reasonably accurate f the juncton dode s operatng n that regon: Reverse Bas Regon CV model juncton dode v 0.7 V Jm Stles The Unv. of Kansas ept. of EECS

29 2/6/2012 The Constant Voltage rop Model present 13/16 Not so great n the transton regon But, the two curves dverge sgnfcantly n the transton regon between reverse and forward bas t s here where the approxmatons provded by the CV model wll typcally exhbt the most (although often stll acceptable) error. CV model juncton dode Transton Regon v 0.7 V Jm Stles The Unv. of Kansas ept. of EECS

30 2/6/2012 The Constant Voltage rop Model present 14/16 No step 4 we re done! Q: OK, we re done wth step 3, what about step 4? A: There s no step 4. Once we use the results of our deal dode crcut analyss to estmate juncton dode voltage: v v 0.7 and juncton dode current: 0 we re done! Jm Stles The Unv. of Kansas ept. of EECS

31 2/6/2012 The Constant Voltage rop Model present 15/16 What about CHECK? Q: Hold on! The math of the CV model was condtonal; don t you remember: 0 f v 0.7V v 0.7V f 0 Shouldn t we now CHECK these nequaltes? A: Recall that we never assumed anythng about the juncton dode thus there s nothng to CHECK. Of course n step 2, we had to ASSUME and CHECK somethng about the deal dode n the CV crcut model. But, once we determne for certan (n step 2) the deal dode voltage and current, there are no more assumptons to check! Jm Stles The Unv. of Kansas ept. of EECS

32 2/6/2012 The Constant Voltage rop Model present 16/16 The smart thng to do! However, t s always smart do a santy check on your fnal answer: * If your juncton dode voltage estmate s less than 0.7 volts, then the juncton dode current estmate must be zero. * If your juncton dode current estmate s postve, then the juncton dode voltage estmate must be 0.7 V. * Your juncton dode voltage estmate can never be greater than 0.7 volts, nor can your juncton dode current estmate ever be negatve. If any of these statements are not true for your estmates, then you have smply made a mstake go back and correct your error! Jm Stles The Unv. of Kansas ept. of EECS

33 2/8/2012 Example Another Juncton ode Model Example 1/9 Example: Juncton ode Crcut Analyss wth the CV model Consder now ths crcut: R 1 =1K R 3 =1K 0.5 ma R 2 =1K v Usng the CV model, let s estmate the voltage across, and current through, the juncton dode. Step 1 s to replace the juncton dode wth the CV model: R 1 =1K R 3 =1K 0.5 ma R 2 =1K v 07. V Jm Stles The Unv. of Kansas ept. of EECS

34 2/8/2012 Example Another Juncton ode Model Example 2/9 Now we have an IEAL dode crcut, and therefore Step 2 s to analyze t precsely as we dd n secton 4.1!! ASSUME the IEAL dode s forward based (why not?). ENFORCE the equalty condton that v 00. V (a short crcut). R 1 =1K R 3 =1K 1 3 v R 1 2 v R ma R 2 =1K v R V Now we ANALYZE ths IEAL dode crcut: 1 R 1 =1K R 3 =1K 3 v R 1 2 v R ma R 2 =1K v R V Frst, from KCL: 05. ma 1 And a second applcaton of KCL: Jm Stles The Unv. of Kansas ept. of EECS

35 2/8/2012 Example Another Juncton ode Model Example 3/ And fnally a thrd dose of KCL: Now, we play the Ohm s Law card: and v R 1 05 R v R 1 R3 3 3 And now KVL: 1 R 1 =1K R 3 =1K ma R 2 =1K v R 1 2 v R 3 v R V 0v v R2 R v v R2 R3 Combnng wth the results from Ohm s Law: Jm Stles The Unv. of Kansas ept. of EECS

36 2/8/2012 Example Another Juncton ode Model Example 4/ v v R2 R One equaton and one unknown (.e., )! Now solvng for IEAL dode current : ma Therefore: 05. R 1 =1K 01. R 3 =1K ma 06. R 2 =1K 07. V Our ANALYSIS passes the santy test! But now, we must CHECK the nequalty assocated wth the IEAL dode assumpton: =- 0.1 ma > 0? Jm Stles The Unv. of Kansas ept. of EECS

37 2/8/2012 Example Another Juncton ode Model Example 5/9 Ykes! We made the wrong assumpton! Let s MOIFY our assumpton and try agan. Now ASSUME the IEAL dode s reverse based. ENFORCE the equalty that 00. ma (an open crcut). R 1 =1K R 3 =1K ma R 2 =1K v R 1 2 v R 3 v R 2 0 v 07. V Now we ANALYZE the IEAL dode crcut. 1 R 1 =1K R 3 =1K 3 v R 1 2 v R ma R 2 =1K v R 2 0 v 07. V From KVL: 0v v v R2 R3 v v v 07. R2 R3 Jm Stles The Unv. of Kansas ept. of EECS

38 2/8/2012 Example Another Juncton ode Model Example 6/9 From Ohm s Law: and v R 1 R v R 1 R Thus, combnng wth the KVL result: v Now for KCL! 1 R 1 =1K R 3 =1K ma R 2 =1K v R 1 2 v R 3 v R 2 0 v 07. V Frst, from KCL: 05. ma 1 And a second applcaton of KCL: 0 3 And fnally a thrd dose of KCL: ma Jm Stles The Unv. of Kansas ept. of EECS

39 2/8/2012 Example Another Juncton ode Model Example 7/9 Therefore, the IEAL dode voltage s: v V 2 3 Now for the santy test: 05. R 1 =1K 0 R 3 =1K It passed! v R ma 05. R 2 =1K V Now, we CHECK to see f our assumpton s correct (t better be!): v Q: Great! So we re all done? =- 0.2 V < 0? A: Holy smokes no! We need to fnd the approxmate values of the juncton dode voltage and juncton dode current. Q: What do you mean? I thought we just dd that! The dode current s zero and the dode voltage s -0.2 volts. Rght? Jm Stles The Unv. of Kansas ept. of EECS

40 2/8/2012 Example Another Juncton ode Model Example 8/9 A: NO! We have only determned the current and voltage of the IEAL dode voltage n our CV model. These are not the estmated values of the juncton dode n our crcut! Instead, we estmate the juncton dode voltage by calculatng the voltage across the entre CV model (.e., deal dode and 0.7 V source): v = = 0.5V What an nterestng result! Although the IEAL dode n the CV model s reversed based, our juncton dode voltage estmate s postve v =0.5 V!!! We lkewse estmate the current through the juncton dode by determnng the current through the PWL model (OK, the current through the model s also the current through the deal = Hopefully, ths example has convnced you as to the necessty of carefully, patently and precsely applyng the juncton dode models models that nclude IEAL dodes only. 0 v 05. V V Jm Stles The Unv. of Kansas ept. of EECS

41 2/8/2012 Example Another Juncton ode Model Example 9/9 Then, you must use the model results to carefully, patently and precsely determne approxmate values for the juncton dode. Each and every step of ths process s requred to acheve the correct answer I ll fnd out later n the semester f you have been payng attenton! Jm Stles The Unv. of Kansas ept. of EECS

42 2/8/2012 The Pecewse Lnear Model present 1/6 The Pece-Wse Lnear Model Q: The CV model approxmates the forward based juncton dode voltage as v 07. V regardless of the juncton dode current. Ths of course s a good approxmaton, but n realty, the juncton dode voltage ncreases (logarthmcally) wth ncreasng dode current. Isn t there a more accurate model? A: Yes! Consder the Pece-Wse Lnear (PWL) model. PWL model 1 R juncton dode v V 0 Jm Stles The Unv. of Kansas ept. of EECS

43 2/8/2012 The Pecewse Lnear Model present 2/6 The PWL crcut model Replace: v wth: v v R V 0 PWL model 1 R juncton dode v V 0 In other words, replace the juncton dode wth three devces an deal dode, n seres wth some voltage source (not 0.7 V!) and a resstor. Jm Stles The Unv. of Kansas ept. of EECS

44 2/8/2012 The Pecewse Lnear Model present 3/6 Gve me three steps.. To fnd approxmate current and voltage values of a juncton dode crcut, follow these 3 steps: Step 1 - Replace each juncton dode wth the three devces of the PWL model. Note you now a have an IEAL dode crcut! There are no juncton dodes n the crcut, and therefore no juncton dode knowledge need be (or should be) used to analyze t. Step 2 - Now analyze the IEAL dode crcut on your paper. etermne and v for each deal dode. IMPORTANT NOTE!!! PLEASE REA THIS CAREFULLY: Make sure you analyze the resultng crcut precsely as we dd n secton 4.1. You assume the same IEAL dode modes, you enforce the same IEAL dode values, and you check the same IEAL dode results, precsely as before. Once we replace the juncton dodes wth the PWL model, we have an IEAL dode crcut no juncton dodes are nvolved! Jm Stles The Unv. of Kansas ept. of EECS

45 2/8/2012 The Pecewse Lnear Model present 4/6 Step 3 gves the approxmate answer Step 3 etermne the approxmate values the IEAL dode values and v : and v of the juncton dode from v V0 v + + v R V 0 Jm Stles The Unv. of Kansas ept. of EECS

46 2/8/2012 The Pecewse Lnear Model present 5/6 The PWL model when the deal dode s forward based Note therefore, f the IEAL dode (note here I sad IEAL dode) s forward based ( 0 ), then the approxmaton of the juncton dode current wll lkewse be postve ( 0 ), and the approxmaton of the juncton dode voltage (unlke the deal dode voltage of v 0) wll be: v v V r 0 d 00. V0 rd V0 R Thus, t s apparent that f the IEAL dode s forward based ( 0 ), then the juncton dode voltage estmate must be greater than voltage source V : 0 0 R V 0 0 = 0 + > 0 v V R V Jm Stles The Unv. of Kansas ept. of EECS

47 2/8/2012 The Pecewse Lnear Model present 6/6 The PWL model when the deal dode s reverse based However, f the IEAL dode s reversed based ( 0 ), then the approxmaton of the juncton dode current wll lkewse be zero 0), and the approxmaton of the juncton dode voltage (unlke the deal dode voltage of v 0 ) wll be: v v V0 R v V0 0 v v V0 0 0 V 0 Thus, t s apparent that f the IEAL dode s reverse based ( v < 0 ), then the juncton dode voltage estmate must be greater R than voltage source V : 0 = + 0 < 0 v v V V NOTE: o not check the resultng juncton dode approxmatons. You do not assume anythng about the juncton dode, so there s nothng to check regardng the juncton dode answers. Jm Stles The Unv. of Kansas ept. of EECS

48 2/8/2012 Constructng the PWL Juncton ode Model present 1/19 Constructng the PWL Juncton ode Model Q: Wat a mnute! How the heck are we supposed to use the PWL model to analyze juncton dode crcuts? v v You have yet to tell us the numerc values of voltage source V O and resstor R! R V 0 A: That s rght! The reason s that the proper values of voltage source V 0 and resstor R are up to you to determne! Jm Stles The Unv. of Kansas ept. of EECS

49 2/8/2012 Constructng the PWL Juncton ode Model present 2/19 The PWL crcut model To see why t s up to you to determne, consder the current voltage relatonshp of the PWL model: 0 1 v R V 0 R for for v V v 0 V 0 1 R v V 0 Jm Stles The Unv. of Kansas ept. of EECS

50 2/8/2012 Constructng the PWL Juncton ode Model present 3/19 em ex plus bee Note that when the deal dode n the PWL model s forward based, the currentvoltage relatonshp s smply the equaton of a lne! 1 V 0 v R R y m x b = + Compare the above to the forward based juncton dode approxmaton: s v Ie n VT An exponental equaton! Jm Stles The Unv. of Kansas ept. of EECS

51 2/8/2012 Constructng the PWL Juncton ode Model present 4/19 An exponental s not a lne! An exponental functon and the equaton of a lne are very dfferent the two functons can approxmately match only over a lmted regon: 1 Q: Lmted match!? Then why even bother wth ths PWL model? juncton dode V 0 R v PWL model regon of greatest model accuracy A: Remember, the PWL model s more accurate than our two alternatves the deal dode model and the CV model. At the very least, the PWL model (unlke the two alternatves) shows an ncreasng voltage v wth ncreasng. Jm Stles The Unv. of Kansas ept. of EECS

52 2/8/2012 Constructng the PWL Juncton ode Model present 5/19 Four ways to construct the PWL model Moreover, f we select the values of V 0 and R properly, the PWL can very accurately match the actual (exponental) juncton dode curve over a decade or more of current (e.g., accurate from = 1mA to 10mA, or from = 20mA to 200mA). Q: Yes well I asked you a long tme ago what R and V 0 should be, but you stll have not gven me an answer! A: OK. We now know that the values of R and V 0 specfy a lne. We also know there are 4 potental ways to specfy a lne: 1. Specfy two ponts on the lne. 2. Specfy one pont on the lne, as well as ts slope m. 3. Specfy one pont on the lne, as well as ts y-ntercept b. 4. Specfy both ts slope and ts y-ntercept b. We wll fnd that the frst two methods are the most useful. Let s address them one at a tme. Jm Stles The Unv. of Kansas ept. of EECS

53 2/8/2012 Constructng the PWL Juncton ode Model present 6/19 Method 1: Specfy two ponts on the lne The obvous queston here s: Whch two ponts? Q: Whch two ponts? A: Hopefully t s equally obvous that the two ponts should be ponts lyng on the juncton dode exponental curve (after all, t s ths curve that we are attemptng to approxmate!). Typcally, we pck two current values separated by about a decade (.e., 10 tmes). 1 For example, we mght select I 1 =10 ma and I 2 =100 ma. We wll fnd that the resultng PWL model wll be farly accurate over ths regon R V 0 V 2 v Jm Stles The Unv. of Kansas ept. of EECS

54 2/8/2012 Constructng the PWL Juncton ode Model present 7/19 You must use the juncton dode equaton! Q: I ve got a queston! How do we fnd the correspondng voltage values V 1 and V 2 for these two currents? A: Remember, we are selectng two ponts on the exponental juncton dode curve. Thus, we can use the juncton dode equaton to determne the correspondng voltages: I I 1 2 v nv ln and v nv ln 1 T 2 T I I s s Or, f the dode manufacturer provdes us wth a test pont nstead of scale current Is : I I 1 2 V V nv ln and V V nv ln 1 test T 2 test T I I test test Jm Stles The Unv. of Kansas ept. of EECS

55 2/8/2012 Constructng the PWL Juncton ode Model present 8/19 Ths should brng back fond memores Now, the rest s smply Mddle School mathematcs. If our PWL lne ntersects these two ponts, then: I I V 0 R V R 1 V 0 V 2 R R I 2 1 R I 1 V 0 V 1 V 2 v Thus, we can solve the above two equatons to determne the two unknown values of V 0 and R, such that our PWL lne wll ntersect the two specfed ponts on the juncton dode curve. Jm Stles The Unv. of Kansas ept. of EECS

56 2/8/2012 Constructng the PWL Juncton ode Model present 9/19 The slope of the lne s: Mddle school math m R I V 2 I V 1 V R I 2 V 1 I 2 1 And then we use our PWL lne equaton to fnd V : 0 V v Rr or V v Rr d d (note these two equatons are KVL!). Jm Stles The Unv. of Kansas ept. of EECS

57 2/8/2012 Constructng the PWL Juncton ode Model present 10/19 Method 2: specfy one pont and the slope he PWL crcut model Now let s examne another way of constructng our PWL model. We frst specfy just one pont that the PWL lne must ntersect. Let s denote ths pont as (I, V ) and call ths pont our bas pont. Of course, we want our bas pont to le on the exponental juncton dode curve,.e.: V nvt I I Ise or equvalently V nvt ln I s Now, nstead of specfyng a second ntersecton pont, we merely specfy drectly the PWL lne slope (.e., drectly specfy the value of R!): Q: But I have no dea what the value of ths slope should be!?! 1 m R Jm Stles The Unv. of Kansas ept. of EECS

58 2/8/2012 Constructng the PWL Juncton ode Model present 11/19 A1: Not ths PWL model! Ths slope s too low juncton dode curve I Not ths one! V v Jm Stles The Unv. of Kansas ept. of EECS

59 2/8/2012 Constructng the PWL Juncton ode Model present 12/19 A2: Not ths PWL model ether. Ths slope s too hgh Not ths one! juncton dode curve I V v Jm Stles The Unv. of Kansas ept. of EECS

60 2/8/2012 Constructng the PWL Juncton ode Model present 13/19 A3: Thnk about t. Ths slope s just rght! Of all possble PWL models that ntersect the bas pont, the one that s most accurate s the one that has a slope equal to the slope of the exponental juncton dode curve (that s, at the bas pont)! Q: What! Not ths one! Ths one! Just how s t possble to determne the slope of the juncton dode curve at the bas pont?!? I Not ths one! V v Jm Stles The Unv. of Kansas ept. of EECS

61 2/8/2012 Constructng the PWL Juncton ode Model present 14/19 Nutn funner than calculus! A: Easy! We smply take the frst dervatve of the juncton dode equaton: d dv d v nv T Ie s dv v nv T Ie s nv T Q: Of course! Isn t ths equaton s the slope of the juncton dode curve at the bas pont? A: Actually no. The above equaton s not the slope of the juncton dode curve at the bas pont. Ths equaton provdes the slope of the curve as a functon dode voltage v. The slope of the juncton dode curve s n fact dfferent at every pont on the juncton dode curve. Jm Stles The Unv. of Kansas ept. of EECS

62 2/8/2012 Constructng the PWL Juncton ode Model present 15/19 Number we need a number In fact, as the equaton above clearly states, the slope of the juncton dode curve exponental ncreases wth ncreasng v! Q: Ykes! So what s the dervate equaton good for? A: Remember, we are nterested n the value of the slope of the curve at one partcular pont the bas pont. Thus, we smply evaluate the dervatve functon at that pont. The result s a numerc value of the slope at our bas pont! v d nv T m I e dv s v V v nv T Ie s nv T v V Ie s nv V nv T T Jm Stles The Unv. of Kansas ept. of EECS

63 2/8/2012 Constructng the PWL Juncton ode Model present 16/19 Note the numerator of ths result! Pretty darn smple We recognze ths numerator as smply the value of the bas current I : I I e s V nvt Therefore, we fnd that the slope at the bas pont s: m V Ie nvt s nvt I nv T Now, we want the slope of our PWL model lne to be equal to the slope of the juncton dode curve at our bas pont. Therefore, we desre: 1 I m R nv T Jm Stles The Unv. of Kansas ept. of EECS

64 2/8/2012 Constructng the PWL Juncton ode Model present 17/19 The small-sgnal PWL model Thus, rearrangng ths equaton, we fnd that the PWL model resstor value should be: nvt R I We lkewse can rearrange the PWL lne equaton to determne the value of the model voltage source V 0 : V V I R (KVL!) 0 Now, combnng the prevous two equatons, we fnd: V V I R 0 nv T V I I V nv T Jm Stles The Unv. of Kansas ept. of EECS

65 2/8/2012 Constructng the PWL Juncton ode Model present 18/19 In summary So, let s recap what we have learned about constructng a PWL model usng ths partcular approach. 1. We frst select a sngle bas pont (I, V ), a pont that les on the juncton dode curve,.e.: s V I Ie n VT 2. Usng the current and voltage values of ths bas pont, we can then determne drectly the PWL model resstor value: nvt R I Jm Stles The Unv. of Kansas ept. of EECS

66 2/8/2012 Constructng the PWL Juncton ode Model present 19/19 We ll use ths later 3. We can also drectly determne the value of the model voltage source: V V nv 0 T Ths method for constructng a PWL model produces a very precse match over a relatvely small regon of the juncton dode curve. We wll fnd that ths s very useful for many practcal dode crcut problems and analyss! Ths PWL model produced by ths last method (as descrbed by the equatons of the prevous page) s called the juncton dode smallsgnal model. We wll use the small-sgnal model agan make sure that you know what t s and how we construct t! Jm Stles The Unv. of Kansas ept. of EECS

67 2/8/2012 Example Constructng a PWL Model 1/5 Example: Constructng a PWL Model We measured a certan juncton dode n our lab, and determned that the current through ths dode s: and 10 ma when v 0 7. V 1 ma when v 0 6. V Say we wsh to construct a PWL model that wll approxmate ths partcular juncton dode. We want ths PWL mode to be partcularly accurate for dode currents from, say, approxmately 1 ma to about 10 ma. Recall that the resultng model wll relate juncton dode voltage v to juncton dode current as a lne of the form: 1 V 0 v r r d d We therefore need to determne the values of V 0 and R such that ths PWL model lne wll ntersect the two ponts I 1 = 1.0 ma, V 1 = 0.6 V and I 2 = 10.0 ma, V 2 = 0.7 V. Jm Stles The Unv. of Kansas ept. of EECS

68 2/8/2012 Example Constructng a PWL Model 2/5 (ma) 1 R 10 1 V v (V) The slope of ths lne must therefore be: m I I V V K mhos Thus our PWL model resstor value R must be: R = m = 9 = K Or n other words, R = 11.1 Ω. Q: Wow! That s a very small resstance value. Are you sure we calculated R correctly? Jm Stles The Unv. of Kansas ept. of EECS

69 2/8/2012 Example Constructng a PWL Model 3/5 A: Typcally, we fnd that the resstor value n the PWL model s small. In fact, t s frequently less than 1 when we attempt to match the juncton dode curve n a hgh current regon (e.g., from =50 ma to =500 ma). Now that we have determned R, we can nsert ether pont nto the model lne equaton and solve for V 0. For example, the equatons: 1 V 0 1 V 0 I or 1 V I V R R R R become ether: or V V I R (00111) V V V I R (0 0111) V In other words, we can use ether pont to determne V 0. Jm Stles The Unv. of Kansas ept. of EECS

70 2/8/2012 Example Constructng a PWL Model 4/5 Our PWL model s therefore: v 0 for v 0.589V ma for v 0.589V v 0589V K Now, compare ths PWL model to the CV model: v 0589V K v 07V. 0.0K PWL CV Note that the CV model can be vewed as a PWL model wth V 0 = 0.7 V and zero resstance R = 0. Compare those values wth our model (V 0 = V and R = 11.1Ω ) not much dfference! Jm Stles The Unv. of Kansas ept. of EECS

71 2/8/2012 Example Constructng a PWL Model 5/5 Thus, the PWL model s not a radcal departure from the CV model (typcally V O s close to 0.7 V and r d s very small). Instead, the PWL can be vew as slght mprovement of the CV model. Jm Stles The Unv. of Kansas ept. of EECS

72 2/8/2012 Example Juncton ode Models 1/7 Example: Juncton ode Models Consder the juncton dode crcut, where the juncton dode has devce parameters I S = 10-9 ma, and n =1: +5 V v I numercally solved the resultng transcendental equaton, and determned the exact soluton: ma 0.05 K v V Now, let s determne approxmate values usng dode models! Frst, let s try the deal dode model. +5 V +5 V v v 0.05 K 0.05 K Jm Stles The Unv. of Kansas ept. of EECS

73 2/8/2012 Example Juncton ode Models 2/7 +5 V K Assume IEAL dode s on. Enforce v 0. Analyze the IEAL dode crcut. From KVL: ma 0.05 Check result: 100 ma 0? We therefore can approxmate the juncton dode current as the current through the deal dode model: 100 ma And approxmate the juncton dode voltage as the voltage across the deal dode model: v v 0 Compare these approxmatons to the exact solutons: ma and v V Close, but we can do better! Let s use the CV model. Jm Stles The Unv. of Kansas ept. of EECS

74 2/8/2012 Example Juncton ode Models 3/7 +5 V v V v 0.05 K 0.05 K V K Assume IEAL dode s on. Enforce v 0. Analyze the IEAL dode crcut. From KVL: 5.0 v ma 0.05 Check the result:? Jm Stles The Unv. of Kansas ept. of EECS

75 2/8/2012 Example Juncton ode Models 4/7 We therefore can approxmate the juncton dode current as the current through the CV model: ma And approxmate the juncton dode voltage as the voltage across the CV model: v 07. v V Compare these approxmatons to the exact solutons: 87.4 ma and v V Much better than before, but we can do even better! Let s use the PWL model. +5 V +5 V v v R V K 0.05 K Jm Stles The Unv. of Kansas ept. of EECS

76 2/8/2012 Example Juncton ode Models 5/7 Q: But, what values should we use for model parameters V 0 and R?? A: From the CV model, we know that s approxmately 86 ma. Therefore, let s create a PWL model that s accurate n the regon between, say: 50 ma 125 ma Frst, we determne v at I 1 = 50 ma and I 2 =125 ma. V nv ln I I 1 T 1 nv T 0.616V ln 50 S I S V nv ln I I 2 T 2 nv T 0.639V ln 125 S I S We now know two ponts lyng on the juncton dode curve! Let s construct a PWL model whose lne ntersects these two ponts. Recall that when the deal dode s forward based, applyng KVL to the PWL model results n: or equvalently: v V R 0 V 0 1 v R R Insertng the juncton dode values (V 1,I 1 ) and (V 2,I 2 ) nto ths PWL model equaton provdes: Jm Stles The Unv. of Kansas ept. of EECS

77 2/8/2012 Example Juncton ode Models 6/ V (0.05) R V (0.125) R 0 Two equatons and two unknowns!! Solvng, we get: V V and R K (small!!) 0 Therefore, the deal dode crcut s: +5 V Assume the IEAL dode s on K 06V K Enforce v = 0. Analyze the IEAL dode crcut. From KVL: 5.0 v 0.6 ( ) ma Check the result: 87.5 ma 0? Jm Stles The Unv. of Kansas ept. of EECS

78 2/8/2012 Example Juncton ode Models 7/7 We can therefore approxmate the juncton dode current as the current through the PWL model: 87.5 ma and approxmate the juncton dode voltage as the voltage across the PWL model: v v V R (0.087) V Now, compare these values to the exact values v = V and = 87.4 ma. The error of the PWL model estmates s just Volts and 0.1 ma! Each model provdes better estmates than the prevous one! (ma) v (V ) Ideal CV PWL Exact Jm Stles The Unv. of Kansas ept. of EECS