Texas Instruments Incorporated Amplifiers: Op Amps Calculating figure in op amps

By James Karki (Email: [email protected]) Member, Group Technical Staff, High-Performance Linear

Introduction Figure 1. Non-inverting noise analysis diagram Noise figure is commonly used in commu- nications systems because it provides a simple method to determine the impact of system noise on sensitivity. Today, the performance of wide-band op amps is making them viable alternatives S Op Amp to more traditional open-loop amplifiers N N I ini I A SO like monolithic microwave integrated NI circuits (MMICs) and discrete transistors NO e in communications design. ni NO Recognizing the need to specify wide- RS RT band op amps in RF engineering termi- i nology, some manufacturers do provide ii eS eT noise figure, but they seem to be the exception rather than the rule. Op amp manufacturers typically specify e e noise performance by giving the input- G F RG RF referred voltage and current noise. The noise figure depends on these parameters, the circuit topology, and the value of external components. If you have all this information, noise figure can be calculated. Review of noise figure Noise figure (NF) is the equivalent of noise factor Assuming that the input is terminated in the same

(F): NF (dB) = 10log(F). impedance as the source, NI = kT = –174 dBm/Hz, where Noise factor of a device is the power ratio of the signal- k is Boltzman’s constant and T = 300 ). Once we to-noise ratio (SNR) at the input (SNRI) divided by the find the input of the device, it is a SNR at the output (SNRO): simple matter to plug it into Equation 2 and calculate F. SNR F = I . (1) NF in op amps SNRO Op amps specify input-referred voltage and current noise. Using these two parameters, adding the noise of the exter- The output signal (SO) is equal to the input signal (SI) nal resistors, and calculating the total input-referred noise × times the gain: SO = SI G. The output noise is equal to based on the circuit topology, we can calculate the input the noise delivered to the input (NI) from the source plus spectral density and use it in Equation 2. the input noise of the device (NA) times the gain: In this discussion, the terms “op amp” and “amplifier” × NO = (NI + NA) G. Substituting into Equation 1 and mean different things. “Op amp” refers to only the active simplifying, we get device itself, whereas “amplifier” includes the op amp and  S  associated passive resistors that make it work as a usable  I  SNR N N amplifier stage. In other words, the amplifier is everything F ==I  I  =+ A .1 (2) shown in Figures 1–3 except R , and the op amp is only SNR  GS×  N S O  I  I the components within the dashed triangles. In this way,  GN()+ N  IA the plane marked NA and NI is the input to the amplifier. This is the point to which the noise sources must be referred so that Equation 2 can be used.

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Analog Applications Journal 4Q 2003 www.ti.com/sc/analogapps Analog and Mixed-Signal Products Amplifiers: Op Amps Texas Instruments Incorporated

The noise from the source and the input Figure 2. Inverting noise analysis diagram noise of the amplifier are referred to the same point. Because the impedance is the

same, expressing the ratio between NA and NI as a voltage ratio squared is equivalent to the power ratio. An op amp is a voltage- driven device, so using voltage-squared Op Amp terms makes the calculations easier. In the ini following discussion, voltage-squared terms SO are used for N and N . NO A I e Op amps use negative feedback to control ni NO the gain of the amplifier. One result is that RT the voltage across the input terminals is driven to zero. This is often referred to as iii eT a “virtual short.” It is used in the following analysis* and referred to as “amplifier SI action,” since it is a by-product of the op N NA N amp doing its job as an amplifier. I I eF eG Superposition is used throughout the analysis, wherein all sources except the one RG RF under consideration are defeated—voltage RS R sources are shorted and current sources M are opened.

Non-inverting amplifier eS eM Of the three basic op amp circuits, it is easiest to find the input-referred noise for the non-inverting op amp amplifier, so it will be discussed first. Figure 1 shows a noise analysis diagram for a non-inverting op amp Figure 3. Fully differential noise analysis diagram amplifier with the noise sources identified. The source resistance RS generates a √ N NA RG R noise voltage equal to 4kTRS. The noise I F voltage delivered to the amplifier input from

the source is divided by the resistors RS and eG eF RT. Therefore, SI 2  R  NI NkTR= 4  T  . IS RR+  Op Amp ST ini SO R R RT is typically used to terminate the input S M NO so that R = R , in which case N = kTR . e T S I S ni NO The amplifier’s voltage noise is a combina-

tion of eni, ini, and iii with associated imped- eS eM ances eT, eG, and eF. These are all referred iii

eF eG

RG RF

*The virtual-short concept simplifies the analysis. Much more work is required to obtain the same results by other means such as nodal analysis. 32

Analog and Mixed-Signal Products www.ti.com/sc/analogapps 4Q 2003 Analog Applications Journal Texas Instruments Incorporated Amplifiers: Op Amps to the input by their respective scaling factors and Figure 2 shows a noise analysis diagram for an inverting summed to find NA; i.e., op amp amplifier with the noise sources identified. To find the input-referred noise, it is easiest in some Ncecicicecece=+++++2 2 2 2 2 2 , A1 niniiiT2 3 4 5 G6 F (3) cases to find the output noise and then divide by the signal gain of the amplifier. where c through c are the scaling factors. 1 6 The noise voltage delivered to the input from the source The op amp’s input voltage noise is e . It appears ni is divided by the resistors R and R in parallel with R . directly at the amplifier’s input and its scaling factor is 1 S M G Therefore, or unity, so that c e2 = e2 . 1 ni ni 2 The op amp’s non-inverting input current noise is i .  RR  ni NkTR= 4  MG  . It develops a voltage through the parallel combination of IS ++ RRSM()() R G RR MG RS and RT, which appears directly at the amplifier’s input, so that R is typically selected so that R || R = R , in which 2 M M G S  RR  case N = kTR . ci22= i ST . I S 2 ni ni  +  The amplifier’s input-referred voltage noise is a combi-  RRST nation of eni, ini, and iii with associated impedances eT, eG, The op amp’s inverting input current noise is i . It devel- eF, and eM. These are all referred to the input by their ii respective scaling factors and summed to find N ; i.e., ops a voltage through the parallel combination of RF and A RG at the op amp’s inverting input. By amplifier action, Ncecicicececece=++++++2 2 2 2 2 2 2 , (4) this voltage appears at the amplifier’s input, so that A1 niniiiT2 3 4 5 G6 F7 M 2 where c through c are the scaling factors.  RR  1 7 ci22= i FG . The op amp’s input voltage noise, e , at the op amp’s 3 ii ii  +  ni  RRFG non-inverting input appears at the amplifier output as a function of the amplifier noise gain, The noise voltage term eT associated with RT is equal to √ RF 4kTRT. It is divided by the resistors RS and RT, so that 1+ , RR 2 R + SM  R  G RR+ ce2 = 4 kTR S . SM 4 TT +   RRST and is then referred back to the amplifier input as a func- 2 If RT = RS, then c4eT = kTRT. tion of the signal gain, RF/RG. Thus, The noise voltage term eG associated with RG is equal to √   2 4kTRG. This noise is divided by the resistors RF and RG and applied to the op amp’s inverting input. Again by  R R  22=+ G G  amplifier action, noise from R appears at the amplifier’s ce1 ni e ni .** G  R RRSM input, so that  F R +   G RR+  2 SM  R  ce2 = 4 kTR  F  . 5 GG+ The op amp’s non-inverting input current noise is ini. It  RRFG develops a voltage through RT that appears directly at the amplifier’s input, so that The noise voltage term eF associated with RF is equal to √ 2 4kTRF and appears at the amplifier’s output. Dividing by   the signal gain gives us  RR RR  22=+ TG TG  2 ci2 ni i ni .  R   R RRSM ce2 = 4 kTR G .  F R +  6 FF +   G RR+   RRFG SM With all the terms in Equation 3 quantified, we can take It is hard to see how to calculate the op amp’s inverting input current noise, i . Basically, due to amplifier action, the sum to find NA and use NA along with NI in Equation 2 ii to find F. the inverting node is at ground so that no current is drawn through the input resistor R . The noise current flows Inverting amplifier G through RF, producing a voltage at the output equal to iiiRF. Finding the input-referred noise of an inverting op amp Referring to the amplifier’s input results in c i2 = i2(R )2. amplifier is more cumbersome than finding that of a non- 3 ii ii G The noise voltage term eT associated with RT is equal to inverting op amp amplifier. The main problem is that the √4kTR . Just like e , it appears at the output as a function signal gain of the amplifier and the noise gain are different. T ni

**The gain is actually –RF/RG; but since it is squared, the minus sign is ignored in this analysis. 33

Analog Applications Journal 4Q 2003 www.ti.com/sc/analogapps Analog and Mixed-Signal Products Amplifiers: Op Amps Texas Instruments Incorporated

of the amplifier noise gain and is then referred back to the where c1 through c6 are the scaling factors. amplifier input as a function of the signal gain, so that In this analysis it is assumed that the two input resistors 2 R are equal and that the two feedback resistors R are   G F equal.  R R  ce2 =+ kTR  G G  . The op amp’s input voltage noise, eni, at the op amp’s 4 TT R RR  input appears at the amplifier output as a function of the F R + SM  G +  amplifier noise gain,  RRSM RF The noise voltage term eG associated with RG is equal to 1+ , √ RRSM 4kTRG. It is divided by the resistors RG and RS in parallel R + G 2()RR+ with RM en route to the amplifier’s input, so that SM 2   and is then referred back to the amplifier input as a func-  R  tion of the signal gain, R /R . Thus, 2 =  G  F G ce5 GG4 kTR .  RRSM 2  R +     G RR+    SM R R ce22=+ e  G G  . The noise voltage term e associated with R is equal to 1 ni ni R RR  F F  F R + SM  √ G + 4kTRF and appears directly at the amplifier’s output.  2()RRSM Dividing by the signal gain gives us 2 Since the input resistors are equal and the feedback  R  2 = G ce6 FF4 kTR   . resistors are equal, the op amp’s non-inverting input cur-  RF  rent noise, ini, and inverting input current noise, iii, have the same scaling factors. Due to amplifier action, the input The noise source eM associated with the input termina- √ nodes of the op amp are ac grounds so that no current is tion matching resistor RM is equal to 4kTRM. It is divided drawn through the input resistors RG. All the noise cur- by the resistors RM and RS in parallel with RG, so that rent flows through RF, producing a voltage at the output 2 equal to i  RR  niRF or iiiRF. Referring to the amplifier’s input ce2 = 4 kTR  SG  . results in c i2 = i2 (R )2 and c i2 = i2(R )2. 7 MM ++ 2 ni ni G 3 ii ii G RRMS() R G RR SG The noise voltage term eG associated with each RG is √ equal to 4kTRG. It is divided by the resistors RG and one- With all the terms in Equation 4 quantified, we can take half R in parallel with R , so that the sum to find N and use N along with N in Equation 2 S M A A I 2 to find F.     R Fully differential amplifier ce2 =×24 kTR  G  . 4 GG RR  Fully differential op amp amplifiers are very similar to R + SM  G + inverting op amp amplifiers, and the analysis follows very  2()RRSM closely. Figure 3 shows the noise analysis diagram. The source resistance generates thermal noise equal to The noise voltage term eF associated with each RF is √ equal to √4kTR and appears directly at the amplifier’s 4kTRS. The noise voltage delivered to the input from the F output. Dividing by the signal gain gives us source is divided by the resistors RS and RM in parallel with 2R . Therefore, 2 G  R  ce2 = 4 kTR  G  .   2 5 FF  2RRMG RF  +  RRMG2 NkTR= 4   . The noise source eM associated with the input termina- IS 2RR  tion matching resistor R is equal to √4kTR . It is divided R + MG M M  S +   RRMG2  by the resistors RM and RS in parallel with 2RG, so that  2RR  2 RM is typically selected so that RM || 2RG = RS, in which SG case N = kTR .  RR+ 2  I S ce2 = 4 kTR  SG . The amplifier’s input-referred voltage noise is a combi- 6 MM 2RR  R + SG nation of eni, ini, and iii with associated impedances eG, eF,  M +   RRSG2  and eM. These are all referred to the input by their respec- tive scaling factors and summed to find NA; i.e., As before, with all the terms in Equation 5 quantified, Ncecicicecece=+++++2 2 2 2 2 2 , (5) NA can be calculated and used with NI in Equation 2 to A1 niniiiG2 3 4 5 F6 M find the noise factor.

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Analog and Mixed-Signal Products www.ti.com/sc/analogapps 4Q 2003 Analog Applications Journal Texas Instruments Incorporated Amplifiers: Op Amps

Table 1. Comparison of calculated vs. measured noise figure

eni ini iii RF RG RT RM CALCULATED NF MEASURED NF OP AMP CONFIGURATION (nV) (pA) (pA) (Ω)(Ω)(Ω)(Ω) (dB) (dB) THS3202 Non-inverting 1.65 13.5 20 255 49.9 49.9 — 13.6 13.0 THS3202 Inverting 1.65 13.5 20 255 49.9 — — 11.6 11.5 THS4501 Fully differential 7 1.7 1.7 392 392 — 56.2 30.1 30.6

Conclusion amplifiers configured as previously detailed were measured The input-referred voltage noise and current noise, along with an Agilent N8973A noise figure analyzer. Table 1 with the circuit configuration and component values, can shows that the results are good, with the input current be used to calculate noise figure. This is a tedious task at and voltage noise specifications given as typical values. best. Setting up a spreadsheet for each topology where Related Web sites component values and op amp specs can be entered is analog.ti.com recommended. In this way, various scenarios can be quickly www.ti.com/sc/device/THS3202 tested. Verification by testing the circuit with a noise figure www.ti.com/sc/device/THS4501 analyzer is always suggested. As an example of how well the theory outlined in this article matches test results, the noise figure of three op amp

Appendix—Summary of noise terms in op amp amplifiers

Signal input noise (NI) terms AMPLIFIER NOISE SOURCE NOISE CONTRIBUTION CONFIGURATION 2  R  Non-inverting Source thermal noise 4kTR T S  +   RRST

2  RR  Inverting Source thermal noise NkTR= 4  MG  IS ++ RRSM()() R G RR MG

  2 2RRMG  RR+ 2  Fully differential Source thermal noise 4kTR  MG S  2RR  R + MG  S +   RRMG2 

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Analog Applications Journal 4Q 2003 www.ti.com/sc/analogapps Analog and Mixed-Signal Products Amplifiers: Op Amps Texas Instruments Incorporated

Appendix—Summary of noise terms in op amp amplifiers (Continued)

Device input noise (NA) terms AMPLIFIER NOISE SOURCE NOISE CONTRIBUTION CONFIGURATION 2 Op amp input-referred voltage noise eni 2  RR  Op amp non-inverting input-referred current noise i2 ST ni  +   RRST 2  RR  Op amp inverting input-referred current noise i2 FG ii  +   RRFG 2 Non-inverting  R  Termination resistor thermal noise voltage 4kTR S T  +   RRST 2  R  Gain resistor thermal noise voltage 4kTR F G  +   RRFG 2  R  Feedback resistor thermal noise voltage 4kTR G F  +   RRFG

  2  R R  Op amp input-referred voltage noise e2  G + G  ni  R RR  F R + SM  G +   RRSM

  2  RR RR  Op amp non-inverting input-referred current noise i2  TG+ TG  ni  R RR  F R + SM  G +   RRSM

2 2 Op amp inverting input-referred current noise iii(RG)   2  R R  Inverting Non-inverting bias matching resistor thermal noise voltage 4kTR  G + G  T  R RR  F R + SM  G +   RRSM

  2  R  Gain resistor thermal noise voltage 4kTR  G  G  RR  R + SM  G +   RRSM

  2 RG Feedback resistor thermal noise voltage 4kTRF    RF 

2  RR  4kTR  SG  Inverting termination matching resistor thermal noise voltage M ++ RRMS() R G RR SG

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Analog and Mixed-Signal Products www.ti.com/sc/analogapps 4Q 2003 Analog Applications Journal Texas Instruments Incorporated Amplifiers: Op Amps

Appendix—Summary of noise terms in op amp amplifiers (Continued)

Device input noise (NA) terms (Continued) AMPLIFIER NOISE SOURCE NOISE CONTRIBUTION CONFIGURATION   2   R R  Op amp input-referred voltage noise e2 G + G ni R RR   F R + SM  G +  2()RRSM

2 2 Op amp non-inverting input-referred current noise ini(RG) 2 2 Op amp inverting input-referred current noise iii(RG)   2   R 24× kTR  G  Fully differential Gain resistor thermal noise voltage G  RR  R + SM  G +  2()RRSM

2  R  × G Feedback resistor thermal noise voltage 24kTRF    RF 

  2 2RRSG  RR+ 2  Termination matching resistor thermal noise voltage 4kTR  SG M  2RR  R + SG  M +   RRSG2 

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