# ECEN 4872/5827 Lecture Notes

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1 ECEN 4872/5827 Lecture Ntes Lecture #5 Objectives fr lecture #5: 1. Analysis f precisin current reference 2. Appraches fr evaluating tlerances 3. Temperature Cefficients evaluatin technique 4. Fundamentals f Cmmn Mde ejectin ati (CM) 5. Fundamentals f Pwer Supply ejectin ati (PS) Analysis f Tlerances and Temperature Cefficient fr a Precisin Current eference Circuit Circuit: The gal is t have 1mA. f the p-amp is ideal, and the BJT has very large current gain, v() v( ), and therefre Taking int accunt the p-amp ffset vltage, the finite BJT current gain β, and the p-amp input bias current B, we have (frm previus lecture): β s B (1) β 1 Neglecting B, we get: β s (2) β 1 Nte that if B were significant, the effect f B culd be canceled by adding a resistr (f the same resistance value, ) in series with the psitive p-amp input.

2 Given the tlerances and temperature cefficients fr, OS, β, and, ur bjective is t find the tlerance and the temperature cefficient f. T slve this prblem, we linearize as a functin f, OS, β, and arund the nminal value : β s (3) β s β s Evaluating the abve partial derivatives using (2),, yields: β s 1 β (4) 2 2 (1 β ) Simplificatin and rearrangement f the abve equatin yields: 1 β β β s (5) Numerical example: β ±50% β ±0.1% s ±0.25% ±1% Appraches t Evaluating Tlerances There are tw general appraches fr evaluating individual tlerances Wrst Case Apprach Standard Deviatin Apprach The Wrst Case apprach, in general, is a very cnservative apprach where the maximum abslute values f the individual parameter tlerances are simply added: Tlerance ± ndividualtlerances ±1.85% (in this numerical example) (6) f the result f interest depends n a large number f individual parameters in a circuit, the wrst-case apprach leads t a very cnservative result. n such cases, a statistical

3 methd can be applied. Assuming that the individual parameter values can be cnsidered as independent randm variables with nrmal distributins, the standard deviatin f the result can be fund as 2 σ σ 1.2% (in this numerical example) (7) Ntes n the precisin current reference: The precisin current reference circuit is designed starting frm a precisin vltage reference, and using an p-amp and a transistr in a negativefeedback cnfiguratin t set the reference current. The input ffset vltage OS f the p-amp can have a significant effect n the tlerance f. n the precisin current reference design, the largest cntributr t the tlerance f is the resistr tlerance. n the numerical example, we assumed the resistr tlerance f ±1%, which can easily be accmplished using a discrete resistr. Fr resistrs that can be realized n an integrated circuit, the abslute tlerances are usually much wrse (e.g. ±20%), which can be a significant prblem in C design. Temperature Cefficient General Cmment: Unlike tlerances which are due t randm variatins f cmpnent parameters, temperature cefficients usually have knwn signs, making it pssible t cancel temperature cefficients. This cancellatin apprach is used t make precisin band-gap vltage references, which will be addressed later in class. Fractinal Temperature Cefficient f the reference current can be fund as: TCF ( ) (8) T and is expressed in % per degree C, r parts per millin (ppm) per degree C. Nte that TC F includes the term, which can be cmputed as in (3)-(5), taking int accunt the temperature cefficients f the circuit parameters. The result is: where TC F 1 1 s ( ) TCF ( β ) TCF ( ) TCF ( ) β T s is the temperature drift f the ffset vltage, ±10µ/ C in the T numerical example (9)

4 β TC F ( β ) 1%/ C ppm/ C β T TC F ( ) 100 ppm/ C T TC F ( ) 1000 ppm/ C T Using (9) and the numerical values fr the precisin current reference circuit in Fig. 1, we get: 1 ppm ppm 10µ ppm TCF ( ) (10,000) 100 ± (10) 100 C C C 1.26 C Nte that the temperature cefficient f the current reference is affected mainly by the temperature cefficient f the resistr : 1000 ppm TCF ( ) TCF ( ) (11) C Cmmn-mde ejectin ati (CM) The cmmn mde input is the vltage applied simultaneusly t bth the psitive and the negative inputs. t is accunted fr as fllws: Fr small-signal inputs: v( ) v( ) Cmmn Mde input: v cm 2 Differential Mde input: v id v( ) v( ) v( ) v( ) A vid Acmvcm A ( ( ) ( )) Acm 2 The cmmn mde rejectin rati is defined as: The CM is usually expressed in db: CM A CM (12) Acm A [ db] 20lg (13) ACM The value f the CM can generally be fund in data sheets fr the particular pamp. t is f interest t relate the cncept f CM in terms f the input ffset vltage. Cnsider the circuit shwn belw:

5 Figure 2: Open-lp p-amp with CM input; ref v cm. deally, the CM is zer, and the utput is zer, but with a real p-amp that has a finite CM this is nt the case. n the circuit f Fig. 2, the utput vltage can be set t zer, 0 by applying a differential vltage v id. n circuit frm, this lks as fllws: Figure 3: Open-lp p-amp with CM input and a differential input added t set the utput t zer. ref v cm n the circuit f Fig.3, we have Ntes: A v A v 0 (14) id cm cm 1 1 vcm id s A CM (15) A v cm id Equatin (15) shws that CM can be cmputed by finding hw the input ffset vltage depends n the input cmmn-mde vltage, which in sme cases can be easier t find than by definitin (12). Based n (15), we can als cnclude that the effects f finite CM are similar t the effects f the input ffset vltage, except that s shuld be cnsidered a functin f the cmmn-mde input vltage. A quick example: Suppse that CM 80dB (CM 10 4 ), and that the input cmmn-mde vltage has an amplitude f CM 10. Frm (15), it fllws that the amplitude f the equivalent input ffset vltage prduced by the cmmn-mde input as a 3 result f the finite CM is CM 10 1 m Pwer Supply ejectin atin (PS) The PS indicates hw much the supply vltage disturbances effect the uput f the pamp. Neglecting finite CM, CM CM A ( ( ) ( )) A ( DD ) A ( SS ) (16)

6 Here, A and A are the small-signal gains frm the psitive supply rail DD t the utput, and frm the negative supply rail SS t the utput, respectively. Similar t CM, PS can be defined in terms f the gains r in terms f the dependence f the input ffset vltage n the supply vltages, PS 1 A OS A (17) PS 1 A OS A (18) n p-amp data sheets, the tw PS values are usually expressed in db.

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