## 1 001111 2

Select R2 0.82 kQ and R1 equals 72.98 kQ. Since 72.98 kQ is not a standard 5 resistor value, R1 is selected as 75 kQ. The difference between the selected and calculated value of R1 has about a 3 effect on b, and this error shows up in the transfer function as an intercept rather than a slope error. The parallel resistance of R1 and R2 is approximately 0.82 kQ and this is much less than RG, which is 20 kQ, thus the earlier assumption that RG > > R1 R2 is justified. R2 could have been...

## 1021 rms versus PP Noise

Instantaneous noise voltage amplitudes are as likely to be positive as negative. When plotted, they form a random pattern centered on zero. Since noise sources have amplitudes that vary randomly with time, they can only be specified by a probability density function. The most common probability density function is Gaussian. In a Gaussian probability function, there is a mean value of amplitude, which is most likely to occur. The probability that a noise amplitude will be higher or lower than...

## 1024 Multiple Noise Sources

When there are multiple noise sources in a circuit, the total root-mean-square (rms) noise signal that results is the square root of the sum of the average mean-square values of the individual sources ETotalrms e1rms + e2rms + e2rms (10 2) Put another way, this is the only break that the designer gets when dealing with noise. If there are two noise sources of equal amplitude in the circuit, the total noise is not doubled (increased by 6 dB). It only increases by 3 dB. Consider a very simple...

## 103 Types of Noise

There are five types of noise in op amps and associated circuitry Some or all of these noises may be present in a design, presenting a noise spectrum unique to the system. It is not possible in most cases to separate the effects, but knowing general causes may help the designer optimize the design, minimizing noise in a particular bandwidth of interest. Proper design for low noise may involve a balancing act between these sources of noise and external noise sources.

## 1031 Shot Noise

The name shot noise is short for Schottky noise. Sometimes it is referred to as quantum noise. It is caused by random fluctuations in the motion of charge carriers in a conductor. Put another way, current flow is not a continuous effect. Current flow is electrons, charged particles that move in accordance with an applied potential. When the electrons encounter a barrier, potential energy builds until they have enough energy to cross that barrier. When they have enough potential energy, it is...

## 1032 Thermal Noise

Thermal noise is sometimes referred to as Johnson noise after its discoverer. It is generated by thermal agitation of electrons in a conductor. Simply put, as a conductor is heated, it will become noisy. Electrons are never at rest they are always in motion. Heat disrupts the electrons' response to an applied potential. It adds a random component to their motion (Figure 10-3). Thermal noise only stops at absolute zero. Like shot noise, thermal noise is spectrally flat or has a uniform power...

## 1034 Burst Noise

Burst noise, also called popcorn noise, is related to imperfections in semiconductor material and heavy ion implants. It is characterized by discrete high-frequency pulses. The pulse rates may vary, but the amplitudes remain constant at several times the thermal noise amplitude. Burst noise makes a popping sound at rates below 100 Hz when played through a speaker it sounds like popcorn popping, hence the name. Low burst noise is achieved by using clean device processing, and therefore is beyond...

## 1035 Avalanche Noise

Avalanche noise is created when a pn junction is operated in the reverse breakdown mode. Under the influence of a strong reverse electric field within the junction's depletion region, electrons have enough kinetic energy that, when they collide with the atoms of the crystal lattice, additional electron-hole pairs are formed (Figure 10-4). These collisions are purely random and produce random current pulses similar to shot noise, but much more intense. (InplBticri Rnijcri Da telKiii R Qttir...

## 104 Noise Colors

While the noise types are interesting, real op amp noise will appear as the summation of some or all of them. The various noise types themselves will be difficult to separate. Fortunately, there is an alternative way to describe noise, which is called color. The colors of noise come from rough analogies to light, and refer to the frequency content. Many colors are used to describe noise, some of them having a relationship to the real world, and some of them more attuned to the field of...

## 1043 Red Brown Noise

Red noise is not universally accepted as a noise type. Many sources omit it and go straight to brown, attributing red characteristics to brown. This has more to do with aesthetics than it does anything else (if brown noise is the low end of the spectrum, then pink noise should be named tan). So if pink noise is pink, then the low end of the spectrum should be red. Red noise is named for a connection with red light, which is on the low end of the visible light spectrum. But then this noise...

## 1051 The Noise Corner Frequency and Total Noise

Op amp noise is never specified as shot, thermal, or flicker, or even white and pink. Noise for audio op amps is specified with a graph of equivalent input noise versus frequency. These graphs usually show two distinct regions Lower frequencies where pink noise is the dominant effect Higher frequencies where white noise is the dominant effect Actual measurements for the TLV2772 show that the noise has both white and pink characteristics (Figure 10-6). Therefore, the noise equations for each...

## 1055 Noninverting Op Amp Circuit Noise

Taking the simplified equivalent op amp circuit from Paragraph 10.5.2 as the base, the noise equivalent of a noninverting op amp circuit is shown in Figure 10-11 Figure 10-11. Noninverting Equivalent Op Amp Circuit Noise Model Figure 10-11. Noninverting Equivalent Op Amp Circuit Noise Model

## 1056 Differential Op Amp Circuit Noise

Taking the simplified equivalent op amp circuit from Paragraph 10.5.2 as the base, the noise equivalent of a differential op amp circuit is shown in Figure 10-12 Figure 10-12. Differential Equivalent Op Amp Circuit Noise Model Assuming that R1 R3 and R2 R4, the gain of this circuit is Figure 10-12. Differential Equivalent Op Amp Circuit Noise Model Assuming that R1 R3 and R2 R4, the gain of this circuit is

## 1131 Input Offset Voltage

All op amps require a small voltage between their inverting and noninverting inputs to balance mismatches due to unavoidable process variations. The required voltage is known as the input offset voltage and is abbreviated VIO. VIO is normally modeled as a voltage source driving the noninverting input. Figure 11-1 shows two typical methods for measuring input offset voltage DUT stands for device under test. Test circuit (a) is simple, but since Vout is not at zero volts, it does not really meet...

## 11310Supply Voltage Rejection Ratio

Supply voltage rejection ratio, kSVR (AKA power supply rejection ratio, PSRR), is the ratio of power supply voltage change to output voltage change. The power voltage affects the bias point of the input differential pair. Because of the inherent mismatches in the input circuitry, changing the bias point changes the offset voltage, which, in turn, changes the output voltage. For a dual supply op amp, KSVR or KSVR 7 . The term AVCC means that the plus and minus power supplies are changed...

## 11313 Equivalent Input Noise

Noise is covered in more detail in Chapter 10. All op amps have parasitic internal noise sources. Noise is measured at the output of an op amp, and referenced back to the input. Therefore, it is called equivalent input noise. Equivalent input noise parameters are usually specified as voltage, Vn, (or current, In) per root Hertz. For audio frequency op amps, a graph is usually included to show the noise over the audio band. The spectral density of noise in op amps has a pink and a white noise...

## 11316 Settling Time

It takes a finite time for a signal to propagate through the internal circuitry of an op amp. Therefore, it takes a period of time for the output to react to a step change in the input. In addition, the output normally overshoots the target value, experiences damped oscillation, and settles to a final value. Settling time, ts, is the time required for the output voltage to settle to within a specified percentage of the final value given a step input. Figure 11-12 shows this graphically Figure...

## 1133Input Common Mode Voltage Range

The input common voltage is defined as the average voltage at the inverting and nonin-verting input pins. If the common mode voltage gets too high or too low, the inputs will shut down and proper operation ceases. The common mode input voltage range, V cr, specifies the range over which normal operation is guaranteed. Different input structures allow for different input common-mode voltage ranges The LM324 and LM358 use bipolar PNP inputs that have their collectors connected to the negative...

## 1135 Maximum Output Voltage Swing

The maximum output voltage, VOM , is defined as the maximum positive or negative peak output voltage that can be obtained without wave form clipping, when quiescent DC output voltage is zero. VOM is limited by the output impedance of the amplifier, the saturation voltage of the output transistors, and the power supply voltages. This is shown pictorially in Figure 11-4. Note that Vom depends on the output load. Voltage drop across R2 + Vsat of Q2 vcc Note that Vom depends on the output load....

## 1136 Large Signal Differential Voltage Amplification

Large signal differential voltage amplification, AVD, is similar to the open loop gain of the amplifier except open loop is usually measured without any load. This parameter is usually measured with an output load. Figure 11-11 shows a typical graph of AVD vs. frequency. Avd is a design issue when precise gain is required. The gain equation of a noninverting amplifier is a feedback factor, determined by the feedback resistors. The term tion is an error term. As long as AVD is large in...

## 1137 Input Parasitic Elements

Both inputs have parasitic impedance associated with them. Figure 11-5 shows a model of the resistance and capacitance between each input terminal and ground and between the two terminals. There is also parasitic inductance, but the effects are negligible at low frequency. Input impedance is a design issue when the source impedance is high. The input loads the source. Figure 11-5. Input Parasitic Elements Input capacitance, Ci, is measured between the input terminals with either input grounded....

## 1138 Output Impedance

Different data sheets list the output impedance under two different conditions. Some data sheets list closed-loop output impedance while others list open-loop output impedance, both designated by Zo. Zo is defined as the small signal impedance between the output terminal and ground. Data sheet values run from 50 Q to 200 Q. Common emitter (bipolar) and common source (CMOS) output stages used in rail-to-rail output op amps have higher output impedance than emitter follower output stages. Output...

## 1139Common Mode Rejection Ratio

Common-mode rejection ratio, CMRR, is defined as the ratio of the differential voltage amplification to the common-mode voltage amplification, ADIF ACoM. Ideally this ratio would be infinite with common mode voltages being totally rejected. The common-mode input voltage affects the bias point of the input differential pair. Because of the inherent mismatches in the input circuitry, changing the bias point changes the offset voltage, which, in turn, changes the output voltage. The real mechanism...

## 121 Introduction

The typical transducer measurement system block diagram is shown in Figure 12-1. The transducer is the electronic system's interface with the real world, and it issues data about a variable. The transducer converts the data into an electrical signal adequate for processing by the circuitry that follows the transducer. Bias and excitation circuitry does the care and feeding of the transducer, thus this circuitry provides offset voltages, bias currents, excitation signals, external components,...

## 123 Design Procedure

A step-by-step design procedure that results in the proper op amp selection and circuit design is given below. This design procedure works best when the op amp has almost ideal performance, thus the ideal op amp equations are applicable. When nonideal op amps are used, parameters like input current affect the design, and they must be accounted for in the design process. The latest generation of rail-to-rail op amps makes the ideal op amp assumption more valid than it ever was. No design...

## 124Review of the System Specifications

The power supply has only one voltage available, and that voltage is 5V 5 5 V 250 mV. The power supply is connected with the negative terminal at ground and the positive terminal at VCC. This is not a portable application, thus the allowed current drain, 50 mA, is adequate for the job. No noise specifications are given, but the proposed power, ground, and signal traces are being done on high-quality circuit board material with planes and good size copper. A system of this quality should...

## 125Reference Voltage Characterization

A reference voltage is required to bias the transducer and act as a reference voltage for the analog interface amplifier (AIA). Selecting a reference with a total accuracy better than the accuracy specification (11 bits) does not guarantee meeting the system accuracy specification because other error sources exist in the design. Resistor tolerances, amplifier tolerances, and transducer tolerances all contribute to the inaccuracy, and the reference can't diminish these errors. The quandary here...

## 126 Transducer Characterization

The temperature transducer is a special silicon diode that is characterized for temperature measurement work. When this diode is forward biased at 2.0 mA 0.1 mA its forward voltage drop is 0.55 V 50 mV, and its temperature coefficient is -2 mV C. The wide acceptable variation in bias current makes this an easy device to work with. The circuit for the bias calculations is shown in Figure 12-12. Figure 12-12. Reference and Transducer Bias Circuit The current through RB1 is calculated in Equation...

## 128 Op Amp Selection

It is time to select the op amp, and the easiest way to do this is to list the known specifications or requirements, list a candidate op amp's specifications, and calculated the projected error that the candidate op amp yields. There should be almost no error from RIN because the transducer output impedance is very low. The high side of the op amp's output voltage swing (4.85 V) is much higher than the ADC input voltage (4 V). The low side of the op amp's output voltage swing (0.185 V) is less...

## 132 Wireless Systems

This chapter focuses on the requirements for the op amp and a number of techniques used in wireless communication systems to interface high-speed op amps to analog-to-digital converters (ADCs) and digital-to-analog converters (DACs). This section provides several examples of different op amp usage. Figure 13-1 shows an example of a dual-IF receiver. In this application, several stages with different IF frequencies are used to get the desired performance. The receiver converts the received radio...

## 133 Selection of ADCsDACs

In communication applications, the dc nonlinearity specifications that describe the converter's static performance are less important than the dynamic performance of the ADC. The receiver (overall system) specifications depend very much on the ADC dynamic performance parameters effective number of bits (ENOB), SFDR (spurious free dynamic range), THD (total harmonic distortion), and SNR (signal-to-noise ratio). Good dynamic performance and fast sampling rate are required for accurate conversion...

## 136 Communication DA Converter Reconstruction Filter

Modern communication DACs are, effectively, an array of matched current sources optimized for frequency domain performance. The most important dynamic specifications are SFDR, SNR, THD, IMD, ACPR and settling time. The dc parameters INL and DNL are considered important because of their influence on the SFDR parameter. Typical SFDR figures for 12-bit to 14-bit DACs, with a 5-MHz single-tone input at 50 MSPS, ranges from 75 dB to 80 dB. In order to prevent adjacent communication channels from...

## 137 External Vref Circuits for ADCsDACs

Figure 13-8 shows an op amp voltage follower circuit that is often used to interface the external precision voltage reference supplying the ADC DAC external reference voltage (see for example, Miller and Moore, 5 , 6 ,1999, 2000 for a more detailed discussion on voltage reference circuits used in ADC an DAC systems). Vin is the output from a precision voltage reference, such as the Thaler Corp. VRE3050. The low-pass filter (formed by C1R1) filters noise from the reference and op amp buffer. The...

## 1432The Resistor Ladder DA Converter

In this type of converter, a precision voltage reference is divided into 2N-1 parts in an internal voltage divider, where N is the number of bits specified for the converter. One switch at a time turns on, corresponding to the correct dc level (Figure 14-1). Figure 14 1. Resistor Ladder D A Converter Unfortunately, the number of resistors and switches doubles for each additional bit of resolution. This means that an 8-bit D A converter would have 255 resistors and 256 switches, and a 16-bit D A...

## 1435 The Sigma Delta DA Converter

The sigma delta D A converter takes advantage of the speed of advanced IC processes to do a conversion as a series of approximations summed together. A phase-locked loop-derived (PLL) sample clock operates at many times the overall conversion frequency in the case shown in Figure 14-5, it is 128X. The PLL is used to drive an interpolation filter, a digital modulator, and a 1-bit D A converter. The conversion is done by using the density ratio of the voltage out of the 1-bit D A as the analog...

## 1441Accuracy versus Resolution

It is important for the designer to understand the difference between converter accuracy and converter resolution. The number of bits determines resolution of a converter. Insufficient resolution is not error it is a design characteristic of the D A. If a given converter's resolution is insufficient, use a converter with better resolution (more bits). Accuracy is the error in the analog output from the theoretical value for a given digital input. Errors are described in the next paragraph. A...

## 1442DC Application Error Budget

DC applications will depend on the value of dc voltage coming out of the converter. THD and signal-to-noise will not be important because the frequency coming out of the converter is almost dc. The resolution of a converter is 1 2 LSB, where an LSB is defined as VFS Full-scale output voltage N Number of converter bits The number of bits in a dc system determines the dc step size that corresponds to a bit. Table 14-1 shows the number of bits, and the corresponding voltage step size for three...

## 145 DA Converter Errors and Parameters

The D A errors described in this section will add to the errors caused by the resolution of the converter. This section is divided into dc and ac sections, but many of the dc errors masquerade as ac errors. A given D A may or may not include either dc or ac error specifications. This should give the designer a clue that the device is optimized for dc or ac applications. Like any component, D A converters are designed with tradeoffs. It is possible to misapply a converter meant for...

## 1451 DC Errors and Parameters

The following paragraphs describe D A dc errors and parameters. The analog output voltage range for the complete range of input bits may be shifted linearly from the ideal 0 to full-scale value (Figure 14-7). The offset error is the A V from 0 V that results when a digital code is entered that is supposed to produce 0. Related to the offset error is the offset error temperature coefficient, which is the change in offset over temperature. This is usually specified in ppm C. Offset error is...

## 22 Laws of Physics

Ohm's law is stated as V IR, and it is fundamental to all electronics. Ohm's law can be applied to a single component, to any group of components, or to a complete circuit. When the current flowing through any portion of a circuit is known, the voltage dropped across that portion of the circuit is obtained by multiplying the current times the resistance (Equation 2-1). In Figure 2-1, Ohm's law is applied to the total circuit. The current, (I) flows through the total resistance (R), and the...

## 23 Voltage Divider Rule

When the output of a circuit is not loaded, the voltage divider rule can be used to calculate the circuit's output voltage. Assume that the same current flows through all circuit elements (Figure 2-5). Equation 2-6 is written using Ohm's law as V I (R- + R2). Equation 2-7 is written as Ohm's law across the output resistor. Substituting Equation 2-6 into Equation 2-7, and using algebraic manipulation yields Equation 2-8. A simple way to remember the voltage divider rule is that the output...

## 25 Thevenins Theorem

There are times when it is advantageous to isolate a part of the circuit to simplify the analysis of the isolated part of the circuit. Rather than write loop or node equations for the complete circuit, and solving them simultaneously, Thevenin's theorem enables us to isolate the part of the circuit we are interested in. We then replace the remaining circuit with a simple series equivalent circuit, thus Thevenin's theorem simplifies the analysis. There are two theorems that do similar functions....

## 26 Superposition

Superposition is a theorem that can be applied to any linear circuit. Essentially, when there are independent sources, the voltages and currents resulting from each source can be calculated separately, and the results are added algebraically. This simplifies the calculations because it eliminates the need to write a series of loop or node equations. An example is shown in Figure 2-11. When V1 is grounded, V2 forms a voltage divider with R3 and the parallel combination of R2 and R1. The output...

## 27 Calculation of a Saturated Transistor Circuit

The circuit specifications are when VIN 12 V, VOUT < 0.4 V at ISInK < 10 mA, and VIN < 0.05 V, VOUT > 10 V at IOUT 1 mA. The circuit diagram is shown in Figure 2-14. Figure 2-14. Saturated Transistor Circuit The collector resistor must be sized (Equation 2-26) when the transistor is off, because it has to be small enough to allow the output current to flow through it without dropping more than two volts to meet the specification for a 10-V output. Rc < V + - Vqut 12 J0 2k (2-26) When...

## 28 Transistor Amplifier

The amplifier is an analog circuit (Figure 2-15), and the calculations, plus the points that must be considered during the design, are more complicated than for a saturated circuit. This extra complication leads people to say that analog design is harder than digital design (the saturated transistor is digital i.e. on or off). Analog design is harder than digital design because the designer must account for all states in analog, whereas in digital only two states must be accounted for. The...

## 32 The Noninverting Op

The noninverting op amp has the input signal connected to its noninverting input (Figure 3-2), thus its input source sees an infinite impedance. There is no input offset voltage because VOS VE 0, hence the negative input must be at the same voltage as the positive input. The op amp output drives current into RF until the negative input is at the voltage, V N. This action causes V N to appear across RG. The voltage divider rule is used to calculate VIN VOUT is the input to the voltage divider,...

## 33 The Inverting Op

The noninverting input of the inverting op amp circuit is grounded. One assumption made is that the input error voltage is zero, so the feedback keeps inverting the input of the op amp at a virtual ground (not actual ground but acting like ground). The current flow in the input leads is assumed to be zero, hence the current flowing through RG equals the current flowing through RF. Using Kirchoff's law, we write Equation 3-4 and the minus sign is inserted because this is the inverting input....

## 35 The Differential Amplifier

The differential amplifier circuit amplifies the difference between signals applied to the inputs (Figure 3-5). Superposition is used to calculate the output voltage resulting from each input voltage, and then the two output voltages are added to arrive at the final output voltage. Figure 3-5. The Differential Amplifier The op amp input voltage resulting from the input source, V1, is calculated in Equations 3-10 and 3-11. The voltage divider rule is used to calculate the voltage, V+, and the...

## 36 Complex Feedback Networks

T Network in Feedback Loop Break the circuit at point X-Y, stand on the terminals looking into R4, and calculate the Thevenin equivalent voltage as shown in Equation 3-15. The Thevenin equivalent impedance is calculated in Equation 3-16. Replace the output circuit with the Thevenin equivalent circuit as shown in Figure 5-8, and calculate the gain with the aid of the inverting gain equation as shown in Equation 3-17. Figure 3-8. Thevenin's Theorem Applied to T Network Substituting...

## 37 Video Amplifiers

Video signals contain high frequencies, and they use coaxial cable to transmit and receive signals. The cable connecting these circuits has a characteristic impedance of 75 Q. To prevent reflections, which cause distortion and ghosting, the input and output circuit impedances must match the 75 Q cable. Matching the input impedance is simple for a noninverting amplifier because its input impedance is very high just make R N 75 Q. RF and RG can be selected as high values, in the hundreds of Ohms...

## 41 Single Supply versus Dual Supply

The previous chapter assumed that all op amps were powered from dual or split supplies, and this is not the case in today's world of portable, battery-powered equipment. When op amps are powered from dual supplies (see Figure 4-1), the supplies are normally equal in magnitude, opposing in polarity, and the center tap of the supplies is connected to ground. Any input sources connected to ground are automatically referenced to the center of the supply voltage, so the output voltage is...

## 42 Circuit Analysis

The complexities of single-supply op amp design are illustrated with the following example. Notice that the biasing requirement complicates the analysis by presenting several conditions that are not realizable. It is best to wade through this material to gain an understanding of the problem, especially since a cookbook solution is given later in this chapter. The previous chapter assumed that the op amps were ideal, and this chapter starts to deal with op amp deficiencies. The input and output...

## 51Why Study Feedback Theory

The gain of all op amps decreases as frequency increases, and the decreasing gain results in decreasing accuracy as the ideal op amp assumption (a breaks down. In most real op amps the open loop gain starts to decrease before 10 Hz, so an understanding of feedback is required to predict the closed loop performance of the op amp. The real world application of op amps is feedback controlled, and depends on op amp open loop gain at a given frequency. A designer must know theory to be able to...

## 52Block Diagram Math and Manipulations

Electronic systems and circuits are often represented by block diagrams, and block diagrams have a unique algebra and set of transformations . Block diagrams are used because they are a shorthand pictorial representation of the cause-and-effect relationship between the input and output in a real system. They are a convenient method for characterizing the functional relationships between components. It is not necessary to understand the functional details of a block to manipulate a block...

## 54 Bode Analysis of Feedback Circuits

Bode developed a quick, accurate, and easy method of analyzing feedback amplifiers, and he published a book about his techniques in 1945. 2 Operational amplifiers had not been developed when Bode published his book, but they fall under the general classification of feedback amplifiers, so they are easily analyzed with Bode techniques. The mathematical manipulations required to analyze a feedback circuit are complicated because they involve multiplication and division. Bode developed the...

## 55 Loop Gain Plots are the Key to Understanding Stability

Stability is determined by the loop gain, and when Ap -1 1 Z-180 instability or oscillation occurs. If the magnitude of the gain exceeds one, it is usually reduced to one by circuit nonlinearities, so oscillation generally results for situations where the gain magnitude exceeds one. Consider oscillator design, which depends on nonlinearities to decrease the gain magnitude if the engineer designed for a gain magnitude of one at nominal circuit conditions, the gain magnitude would fall below one...

## 61 Introduction

There are two types of error sources in op amps, and they fall under the general classification of dc and ac errors. Examples of dc errors are input offset voltage and input bias current. The dc errors stay constant over the usable op amp frequency range therefore, the input bias current is 10 pA at 1 kHz and it is 10 pA at 10 kHz. Because of their constant and controlled behavior, dc errors are not considered until later chapters. AC errors are flighty, so we address them here by developing a...

## 62 Review of the Canonical Equations

A block diagram for a generalized feedback system is repeated in Figure 6-1. This simple block diagram is sufficient to determine the stability of any system. Figure 6-1. Feedback System Block Diagram The output and error equation development is repeated below. Combining Equations 6-1 and 6-2 yields Equation 6-3 Collecting terms yields Equation 6-4 Rearranging terms yields the classic form of the feedback equation. Notice that Equation 6-5 reduces to Equation 6-6 when the quantity Ap in...

## 63 Noninverting Op Amps

A noninverting op amp is shown in Figure 6-3. The dummy variable, VB, is inserted to make the calculations easier and a is the op amp gain. Equation 6-8 is the amplifier transfer equation. The output equation is developed with the aid of the voltage divider rule. Using the voltage divider rule assumes that the op amp impedance is low. Combining Equations 6-8 and 6-9 yields Equation 6-10. VOUT aVIN - aZZG + ZUT (6-10) Rearranging terms in Equation 6-10 yields Equation 6-11, which describes the...

## 64 Inverting Op Amps

The inverting op amp circuit is shown in Figure 6-5. The dummy variable (VA) is inserted to make the calculations easier, and a is the op amp open loop gain. The transfer equation is given in Equation 6-16 The node voltage (Equation 6-17) is obtained with the aid of superposition and the voltage divider rule. Equation 6-18 is obtained by combining Equations 6-16 and 6-17. Equation 6-16 is the transfer function of the inverting op amp. By virtue of the comparison between Equations 6-18 and 6-14,...

## 65 Differential Op Amps

The differential amplifier circuit is shown in Figure 6-7. The dummy variable, VE, is inserted to make the calculations easier, and a is the open loop gain. Figure 6-7. Differential Amplifier Circuit Equation 6-20 is the circuit transfer equation. VGUT aVE v+- V- The positive input voltage, V+, is written in Equation 6-21 with the aid of superposition and the voltage divider rule. The negative input voltage, V-, is written in Equation 6-22 with the aid of superposition and the voltage divider...

## 71 Introduction

Voltage-feedback amplifiers (VFA) have been with us for about 60 years, and they have been problems for circuit designers since the first day. You see, the feedback that makes them versatile and accurate also has a tendency to make them unstable. The operational amplifier (op amp) circuit configuration uses a high-gain amplifier whose parameters are determined by external feedback components. The amplifier gain is so high that without these external feedback components, the slightest input...

## 73 External Compensation Stability and Performance

Nobody compensates an op amp just because it is there they have a reason to compensate the op amp, and that reason is usually stability. They want the op amp to perform a function in a circuit where it is potentially unstable. Internally and noninternally compensated op amps are compensated externally because certain circuit configurations do cause oscillations. Several potentially unstable circuit configurations are analyzed in this section, and the reader can extend the external compensation...

## 79 Comparison of Compensation Schemes

Internally compensated op amps can, and often do, oscillate under some circuit conditions. Internally compensated op amps need an external pole to get the oscillation or ringing started, and circuit stray capacitances often supply the phase shift required for instability. Loads, such as cables, often cause internally compensated op amps to ring severely. Dominant pole compensation is often used in IC design because it is easy to implement. It rolls off the closed-loop gain early thus, it is...

## 810 Compensation of CF and CG

When CF and CG both are present in the circuit they may be adjusted to cancel each other out. The stability equation for a circuit with CF and CG is Equation 8-33. Rr( 1 + -A- I(Rb Rp Rg (Cp + Cg) S + 11 (8 33) If the zero and pole in Equation 8-33 are made to cancel each other, the only poles remaining are in Z. Setting the pole and zero in Equation 8-33 equal yields Equation 8-34 after some algebraic manipulation. Rb dominates the parallel combination of RB and RG, so Equation 8-34 is reduced...

## 82 CFA Model

The CFA model is shown in Figure 8-1. The noninverting input of a CFA connects to the input of the input buffer, so it has very high impedance similar to that of a bipolar transistor noninverting VFA input. The inverting input connects to the input buffer's output, so the inverting input impedance is equivalent to a buffer's output impedance, which is very low. ZB models the input buffer's output impedance, and it is usually less than 50 Q. The input buffer gain, GB, is as close to one as IC...

## 83 Development of the Stability Equation

Vqut Becomes Vto The Test Signal Output Figure 8-2. Stability Analysis Circuit output impedance have been deleted from the circuit to simplify calculations. This approximation is valid for almost all applications. Figure 8-3. Stability Analysis Circuit The transfer equation is given in Equation 8-1, and the Kirchoff's law is used to write Equations 8-2 and 8-3. Equations 8-2 and 8-3 are combined to yield Equation 8-4. VTI l1(ZF + ZG II ZB) 1 + 5 1 UZC( 1 + Dividing Equation 8-1 by Equation 8-4...

## 84 The Noninverting CFA

The closed-loop gain equation for the noninverting CFA is developed with the aid of Figure 8-4, where external gain setting resistors have been added to the circuit. The buffers are shown in Figure 8-4, but because their gains equal one and they are included within the feedback loop, the buffer gain does not enter into the calculations. Equation 8-6 is the transfer equation, Equation 8-7 is the current equation at the inverting node, and Equation 8-8 is the input loop equation. These equations...

## 86 Stability Analysis

The stability equation is repeated as Equation 8-18. Comparing Equations 8-9 and 8-15 to Equation 8-18 reveals that the inverting and non-inverting CFA op amps have identical stability equations. This is the expected result because stability of any feedback circuit is a function of the loop gain, and the input signals have no affect on stability. The two op amp parameters affecting stability are the transimpedance, Z, and the input buffer's output impedance, ZB. The external components...

## 87 Selection of the Feedback Resistor

Equation 8-27 leads one to believe that a new value for ZF can easily be chosen for each new gain. This is not the case in the real world the assumptions don't hold up well enough to rely on them. When you change to a new gain not specified on the data sheet, Equation 8-27, at best, supplies a starting point for Rf, but you must test to determine the final value of Rf. When the Rf value recommended on the data sheet can't be used, an alternate method of selecting a starting value for Rf is to...

## 88 Stability and Input Capacitance

When designer lets the circuit board introduce stray capacitance on the inverting input node to ground, it causes the impedance ZG to become reactive. The new impedance, ZG, is given in Equation 8-28, and Equation 8-29 is the stability equation that describes the situation. Equation 8-29 is the stability equation when ZG consists of a resistor in parallel with stray capacitance between the inverting input node and ground. The stray capacitance, CG, is a fixed value because it is dependent on...

## 89 Stability and Feedback Capacitance

When a stray capacitor is formed across the feedback resistor, the feedback impedance is given by Equation 8-31. Equation 8-32 gives the loop gain when a feedback capacitor has been added to the circuit. This loop gain transfer function contains a pole and zero, thus, depending on the pole zero placement, oscillation can result. The Bode plot for this case is shown in Figure 8-9. The original and composite curves cross the 0-dB axis with a slope of -40 dB decade, so either curve can indicate...

## 96 Equation Comparison

The pertinent VFA and CFA equations are repeated in Table 9-1. Notice that the ideal closed-loop gain equations for the inverting and noninverting circuits are identical. The ideal equations for the VFA depend on the op amp gain, a, being very large thus making Ap large compared to one. The CFA needs two assumptions to be valid to obtain the ideal equations. First, the ideal equations for the CFA depend on the op amp transimpedance, Z, being very large thus making Ap large compared to one....

## As

Comparing the variables of Equation 16-22 with Equation 16-20 provides the equations that determine the filter parameters To calculate the individual component values, establish the following design procedure 1) Define fm and C and calculate R with 1 2) Specify Q and determine a via 3) Specify Aq and determine p via 4) Define R2 and calculate R3 and R4 with In comparison to the twin-T circuit, the Wien-Robinson filter allows modification of the passband gain, Aq, without affecting the quality...

## Design Reference

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## Jr

First-Order Noninverting Low-Pass Filter with Unity Gain There are two topologies for a second-order low-pass filter, the Sallen-Key and the Multiple Feedback (MFB) topology. The general Sallen-Key topology in Figure 16-15 allows for separate gain setting via A0 1+R4 R3. However, the unity-gain topology in Figure 16-16 is usually applied in filter designs with high gain accuracy, unity gain, and low Qs (Q < 3). Figure 16-15. General Sallen-Key Low-Pass Filter Figure 16-16....

## Q At

With Am being the gain at mid frequency, fm, of the overall filter. Example 16-6. Fourth-Order Butterworth Band-Pass Filter The task is to design a fourth-order Butterworth band-pass with the following parameters From Table 16-2 the following values are obtained In accordance with Equations 16-14 and 16-15, the mid frequencies for the partial filters are The overall Q is defined as Q fm B , and for this example results in Q 10. Using Equation 16-16, the Qi of both filters is (1 + 1.0362)-1 With...

## R2

The Sallen-Key circuit has the advantage that the quality factor (Q) can be varied via the inner gain (G) without modifying the mid frequency (fm). A drawback is, however, that Q and Am cannot be adjusted independently. Care must be taken when G approaches the value of 3, because then Am becomes infinite and causes the circuit to oscillate. To set the mid frequency of the band-pass, specify fm and C and then solve for R 1 Because of the dependency between Q and Am, there are two options to...

## Rg R1 R2

The specifications for an example design are Vqut 15 V V N 0.2 V, Vqut 4.5 V VIN 0.5 V, VREF VCC 5 V, RL 10 kQ, and 5 resistor tolerances. The simultaneous equations, (Equations 4-40 and 4-41), are written below. From these equations we find that b -0.5 and m 10. Making the assumption that R1 R2< < RG simplifies the calculations of the resistor values. Let RG 20 kQ, and then RF 180 kQ. Vcc(RG )(rt) 5( 20 )(RTT2R2) (4-44)

## Zi f

000 001 010 011 100 101 Digital Input Code Figure 14-10. Integral Nonlinearity Error Both the INL and DNL errors affect ac applications as distortion and spectral harmonics (spurs). In dc applications, they will result in an error in the dc output voltage. The mechanical steps of a positioning table, for instance, may not be exact increments. 14.5.1.5 Power Supply Rejection Ratio The power supply rejection ratio is sometimes called the power supply sensitivity. It is the ability of the...

## 129 Amplifier Circuit Design

Enough information exists for the AIA to be designed. The TLV247X op amp is selected because it meets all the system requirements. The first step in the design is to determine the AIA input and output voltages, and this has already been done. These voltages are taken from Tables 12-1 and 12-2, and repeated here as Table 12-4. Table 12-4. AIA Input and Output Voltages Table 12-4. AIA Input and Output Voltages The equation of an op amp is the equation of a straight line as given in Equation...

## 11314 Total Harmonic Distortion Plus Noise

Total harmonic distortion plus noise, THD + N, compares the frequency content of the output signal to the frequency content of the input. Ideally, if the input signal is a pure sine wave, the output signal is a pure sine wave. Due to nonlinearity and noise sources within the op amp, the output is never pure. THD + N is the ratio of all other frequency components to the fundamental and is usually specified as a percentage (2 Harmonic voltages + Noise Voltages) Fundamental Figure 11-10 shows a...

## Active Filter Design Techniques

A filter is a device that passes electric signals at certain frequencies or frequency ranges while preventing the passage of others. Webster. Filter circuits are used in a wide variety of applications. In the field of telecommunication, band-pass filters are used in the audio frequency range (0 kHz to 20 kHz) for modems and speech processing. High-frequency band-pass filters (several hundred MHz) are used for channel selection in telephone central offices. Data acquisition systems usually...

## 106 Putting It All Together

This example is provided for analysis only actual results depend on a number of other factors. Expanding on the techniques of Paragraph 10.2.5 A low-noise op amp is needed over an audio frequency range of 20 Hz to 20 kHz, with a gain of 40 dB. The output voltage is 0 dBV (1V). The schematic is shown in Figure 10-13 Figure 10-13. Split Supply Op Amp Circuit It would be nice to use a TLE2027 with a noise figure of 2.5 nV vHz . The data sheet, however, reveals that this is a 15-V part, and that...

## 134 Factors Influencing the Choice of Op Amps

IF amplifiers and filters can be built from discrete components, though most modern applications use integrated circuits. High-speed wideband op amps are employed as buffer amplifiers in the LO circuit, at the front end of ADCs, at the output of the DAC, in the external voltage reference circuits for ADCs and DACs, and in the AGC amplifier and anti-aliasing stage. Op amps operating at IF frequencies, such as the AGC amplifier in Figure 13-1, must attain a large gain control range. How well the...

## 122 Transducer Types

This is not a treatise on transducers, but an appreciation for the many different types of transducers gives a feel for the extent and complexity of the transducer characterization problem following section . The variety of electrical output that transducers offer loosely groups transducers. Various types of transducer outputs are resistive, optical, ac-excited, junction voltage, and magnetic, and each of these outputs must be converted to an electrical signal that can be amplified to fit the...

## 53 Feedback Equation and Stability

Figure 5-7 shows the canonical form of a feedback loop with control system and electronic system terms. The terms make no difference except that they have meaning to the system engineers, but the math does have meaning, and it is identical for both types of terms. The electronic terms and negative feedback sign are used in this analysis, because subsequent chapters deal with electronic applications. The output equation is written in Equation 5-1. Figure 5-7 shows the canonical form of a...

## 11315 Unity Gain Bandwidth and Phase Margin

There are five parameters relating to the frequency characteristics of the op amp that are likely to be encountered in Texas Instruments data sheets. These are unity-gain bandwidth B- , gain bandwidth product GBW , phase margin at unity gain m , gain margin Am , and maximum output-swing bandwidth BOM . Unity-gain bandwidth Bi and gain bandwidth product GBW are very similar. Bi specifies the frequency at which AVD of the op amp is 1 GBW specifies the gain-bandwidth product of the op amp in an...

## 72 Internal Compensation

Miller Effect Compensation Figure 7-2 shows the gain phase diagram for an older op amp TL03X . When the gain crosses the 0-dB axis gain equal to one the phase shift is approximately 108 , thus the op amp must be modeled as a second-order system because the phase shift is more than 90 . LARGE-SIGNAL DIFFERENTIAL VOLTAGE AMPLIFICATION AND PHASE SHIFT vs LARGE-SIGNAL DIFFERENTIAL VOLTAGE AMPLIFICATION AND PHASE SHIFT vs 10 100 1 k 10 k 100 k 1 M 10 M f - Frequency - Hz 10 100 1 k 10 k...

## VN R l 1 RCs 1 xs58

The magnitude of this transfer function is VOUT VIN 1 tm 2 . This magnitude, Vout V N 1 when ra 0.1 t, it equals 0.707 when ra 1 t, and it is approximately 0.1 when ra 10 t. These points are plotted in Figure 5-9 using straight line approximations. The negative slope is -20 dB decade or -6 dB octave. The magnitude curve is plotted as a horizontal line until it intersects the breakpoint where ra 1 t. The negative slope begins at the breakpoint because the magnitude starts decreasing at that...

## R

Second-Order MFB Low-Pass Filter The transfer function of the circuit in Figure 16-19 is 1 cC1 2 R3 -Rt3 s c2 C1 C2R2R3s2 Through coefficient comparison with Equation 16-2 one obtains the relation Given C1 and C2, and solving for the resistors R1-R3 In order to obtain real values for R2, C2 must satisfy the following condition 16.3.3 Higher-Order Low-Pass Filters Higher-order low-pass filters are required to sharpen a desired filter characteristic. For that purpose, first-order...

## 138 High Speed Analog Input Drive Circuits

Communication ADCs, for the most part, have differential inputs and require differential input signals to properly drive the device. Drive circuits are implemented with either RF transformers or high-speed differential amplifiers with large bandwidth, fast settling time, low output impedance, good output drive capabilities, and a slew rate of the order of 1500 V S. The differential amplifier is usually configured for a gain of 1 or 2 and is used primarily for buffering and converting the...

## 43 Simultaneous Equations

Taking an orderly path to developing a circuit that works the first time starts here follow these steps until the equation of the op amp is determined. Use the specifications given for the circuit coupled with simultaneous equations to determine what form the op amp equation must have. Go to the section that illustrates that equation form called a case , solve the equation to determine the resistor values, and you have a working solution. A linear op amp transfer function is limited to the...

## 85 The Inverting CFA

The current equation for the input node is written as Equation 8-12. Equation 8-13 defines the dummy variable, VA, and Equation 8-14 is the transfer equation for the CFA. These equations are combined and simplified leading to Equation 8-15, which is the closed-loop gain equation for the inverting CFA. When ZB approaches zero, Equation 8-15 reduces to Equation 8-16. When Z is very large, Equation 8-16 becomes Equation 8-17, which is the ideal closed-loop gain equation for the inverting CFA. The...

## 75 Gain Compensation

When the closed-loop gain of an op amp circuit is related to the loop gain, as it is in voltage-feedback op amps, the closed-loop gain can be used to stabilize the circuit. This type of compensation can not be used in current-feedback op amps because the mathematical relationship between the loop gain and ideal closed-loop gain does not exist. The loop gain equation is repeated as Equation 7-11. Notice that the closed-loop gain parameters ZG and ZF are contained in Equation 7-11, hence the...

## 1053 Op Amp Circuit Noise Model

Texas Instruments measures the noise characteristics of a large sampling of devices. This information is compiled and used to determine the typical noise performance of the device. These noise specifications refer the input noise of the op amp. Some noise portions can be represented better by a voltage source, and some by a current source. Input voltage noise is always represented by a voltage source in series with the noninverting input. Input current noise is always represented by current...

## 1443 AC Application Error Budget

The error budget for an ac application will most likely be specified as total harmonic distortion, dynamic range, or signal-to-noise ratio. Assuming no internal noise, and no noise in the buffer op amp circuitry, the inverse of the dynamic range is the signal-to-noise ratio of the converter D A. Of course, noise is always present, and is measured with all input data set to zero. Noise will make the S N ratio decrease. The number of converter bits, however, is the overwhelming factor determining...

## 1433 The Weighted Resistor DA Converter

This type of converter is very similar to the Resistor Ladder D A converter. In this case, however, each resistor in the string is given a value proportional to the binary value of the bit it represents. Currents are then summed from each active bit to achieve the output Figure 14-2 . Figure 14 2. Binary Weighted D A Converter Figure 14 2. Binary Weighted D A Converter The number of resistors and switches reduced to one per bit, but the range of the resistors is extremely wide for...

## 76 Lead Compensation

Lead-Compensation Circuit transfer function The equation for the inverting op amp closed-loop gain is repeated below. Figure 7-14. Lead-Compensation Bode Plot When a approaches infinity, Equation 7-13 reduces to Equation 7-14. Substituting RF C for ZF and RG for ZG in Equation 7-14 yields Equation 7-15, which is the ideal closed-loop gain equation for the lead compensation circuit. The forward gain for the inverting amplifier is given by Equation 7-16. Compare Equation 7-13 with...

## 135 Anti Aliasing Filters

Spurious effects in the receiver channel Figure 13-1 appear as high frequency noise in the baseband signal present at the ADC. The spurious signals gt 2 must be blocked from getting to the ADC sampling at Nyquist rate, s where they will cause aliasing errors in the ADC output. A suitable anti-aliasing low-pass analog filter placed immediately before the ADC can block all frequency components capable of causing aliasing from reaching the ADC. The anti-aliasing filter cutoff frequency fc is set...