vp = vn ip + – io + vp – in + vo – Note: The resistances used in an **op**-**amp** **circuit** must be much larger than Ro and much smaller than Ri in order for the ideal **op**-**amp** **equations** to be accurate. + vn – EE 42/100 Fall 2005 Week 8, Prof. White Unity-Gain Voltage-Follower **Circuit** + VIN vn V0(V) VIN(V) 1 2 IIN vp vp = vn V0 = VIN/

discharge **equation** Electronic **circuits** - Overview Semiconductors such as Si and Ge Diode: current in one direction Come up with one or two applications of diodes Transistor: amplifies current (current in the collector-emitter path is amplified version of the current in the base) Use of a transistor to amplify audio sound: interface a microphone and a loud speaker Operational amplifier Ideal **op** **amp**, Real **op** **amp**/

.A linear system can be described by (a) state transition cavation (b) differential **equation** (c) dynamic **equation** (d) none of the above 37.At which of the following frequency, the gain of **op**-**amp** will be zero? (a) β cut-off frequency (b) α cut-off frequency/ all of the above 55.Thermal drift of **op**-**amp** parameters (a) forced air cooling only (b) careful printed **circuit** board layout (c) both a and b (d) keep **op**-**amp** away from source of heat 56.AC characteristics of **op**-**amp** includes (a) Frequency response (b) slew /

Multiple-Choice Quiz 24 What are some possible signs of an unstable **op** **amp** **circuit**? a)oscillations b)large overshoot and ringing c)unpredictable or unexpected response d)all of the above Many common **circuits** inadvertently cause delay in the feedback network, resulting in stability issues. a/& C LOAD Pole: R O, R ISO, and C LOAD Z1Z1 Z2Z2 Vin Method 1: R ISO – Theory 53 Transfer Function: Zero **Equation**: Pole **Equation**: Method 1: R ISO – Theory X A OL (from data sheet)A OL Load Loaded A OL = 54 Method 1: R ISO/

**Op**-**Amp** **Circuits** +V 2 : Non-inverting input -V 1 : Inverting input +V s : Positive source PS -V s : Negative source PS V out : Output voltage ON: Offset Null NC: Not Connected **Op**-**Amp** Ideal, Equivalent **Circuit**, Characteristics and Features Ideal **Op**-**Amp** **Op**-**Amp** Equivalent **Circuit** **Op**-**Amp** Symbol **Op**-**Amp** Characteristics 741 **Op**-**Amp**/ the dc output voltage is zero when the input signals are zero. (b). The output voltage **equation** is valid for both ac and dc input signals. The output voltage is given by Thus the /

unity-gain low-pass filters. (a) The block diagram (b) The **circuit** © 2008 Pearson Education 15.4 Higher Order **Op** **Amp** Filters The transfer function of an nth–order Butterworth low-pass filter with a cutoff frequency of 1 rad/s can be determined from the **equation**: © 2008 Pearson Education 15.4 Higher Order **Op** **Amp** Filters By Finding the roots of the denominator polynomial. Assigning/

, and infinite input resistance. These conditions lead to two assumptions useful in analyzing ideal **op** **amp** **circuits**: 1. The voltage difference across the input terminals is zero. 2. The input currents are zero. Jaeger/Blalock 7/1/03 Microelectronic **Circuit** Design McGraw-Hill Microelectronic **Circuit** Design Ideal **Op** **Amp** Example Writing a loop **equation**: From assumption 2, we know that i- = 0. Assumption 1 requires v- = v+ = 0/

Frequency synthesizer FM detector Active Filters Active filters use **op**-**amp**(s) and RC components. Advantages over passive filters: **op**-**amp**(s) provide gain and overcome **circuit** losses increase input impedance to minimize **circuit** loading higher output power sharp cutoff characteristics can be /is 150 mA Dropout voltage is 3 V (i.e. VCC > Vo(max) + 3) LM723 in High-Voltage Configuration Design **equations**: Choose R1 + R2 = 10 kW, and Cc = 100 pF. External pass transistor and current sensing added. To make Vo /

d = v 2 - v 1 v o = Av d = A(v 2 – v 1 ) Typical **Op**- **Amp** **circuits** Non-inverting Summing **Amp** From this frequency response curve we can see that the product of the gain against frequency is constant at any point along the curve. / error of an ideal ADC is half of the step size. Number of bits N: The higher is the number of bits, the more precise is the digital output. ADC **EQUATIONS** V in = input voltage, V ref+ = ref voltage, V ref- = 0 V, (Note: V ref = V FS ) N = number of bits of precision output_code = V/

voltage gain, and infinite input resistance. These conditions lead to two assumptions useful in analyzing ideal **op**-**amp** **circuits**: 1. The voltage difference across the input terminals is zero. 2. The input currents are zero. Microelectronic **Circuit** Design, 4E McGraw-Hill Chap 1 - 46 Ideal **Op** **Amp** Example Writing a loop **equation**: From assumption 2, we know that i - = 0. Assumption 1 requires v - = v + = 0. Combining/

Inverting and Noise Gain Non-Inverting (Output Cload) Original **Circuit**: Transient Response 59 Noise-Gain Compensation **Circuit** Configurations Non-Inverting: Inverting: Noise-Gain **Circuit** **Equations** 61 1/β Transfer Function: DC 1/β Magnitude: 1/Beta Pole Frequency: 1/Beta Zero Frequency: => Solve for CN The **op** **amp** equivalent input capacitance, CI, has been left out of these **equations** to simplify them. If CI and RF||RI are causing/

Inverting and Noise Gain Non-Inverting (Output Cload) Original **Circuit**: Transient Response 15 Noise-Gain Compensation **Circuit** Configurations Non-Inverting: Inverting: Noise-Gain **Circuit** **Equations** 17 1/β Transfer Function: DC 1/β Magnitude: 1/Beta Pole Frequency: 1/Beta Zero Frequency: => Solve for CN The **op** **amp** equivalent input capacitance, CI, has been left out of these **equations** to simplify them. If CI and RF||RI are causing/

45 degrees phase margin at unity gain Control Loop - Second Order System The traditional second order system control loop block diagram and characteristic **equation** are shown above. G(s) for us represents our closed loop, 2-pole dominant, **op** **amp** **circuit**. There are well established, documented, and derived behaviors for such a 2-pole dominant system that we can use to help us assess/

these designs. The internationally acknowledged authority on electronic amplifiers wrote five very popular books about **op** **amps**, the latest being Photodiode Amplifiers: **Op** **Amp** Solutions and Optimizing **Op** **Amp** Performance. The latter, subtitled "A new approach for maximizing **op** **amp** behavior in **circuit** designs without extensive mathematical analysis," offers design **equations** and models that reflect real-world **op** **amp** behavior and makes analysis of difficult-looking configurations easy. Graemes earlier books are/

Effect Transistor (FET) **circuit**. 1.1.21 Describe Integrated **Circuit** (**Op** **Amp**) device design and application. 1.1.22 Build/test/analyze integrated (**Op** **Amp**) **circuits**. 1.1.23 Describe Digital Logic symbols and truth tables. 1.1.24 Describe Fiber Optic **circuit** device design and / By assuming that the two input terminals are at the same potential and using Kirchhoff Current law for a series **circuit**. 2. We can write the **equation** for current through Rf and Ra as follows: Vout -Vin / Ra =Vin- 0/Rf We can therefore/

**Amps**.ppt 1 Bruce Mayer, PE Engineering-43: Engineering **Circuit** Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 43 Chp 4 **Op** **Amp** **Circuits** BMayer@ChabotCollege.edu ENGR-43_Lec-04_Op-**Amps**.ppt 2 Bruce Mayer, PE Engineering-43: Engineering **Circuit**/BMayer@ChabotCollege.edu ENGR-43_Lec-04_Op-**Amps**.ppt 38 Bruce Mayer, PE Engineering-43: Engineering **Circuit** Analysis Example Required Draw The Linear Equivalent **Circuit** Write The Loop **Equations** 1.Locate Nodes + - o/

Differential Amplifier A basic differential amplifier **circuit** and its symbol are shown in Figure 12-4. The diff-**amp** stages that make up part of the **op**-**amp** provide high voltage gain and common-mode rejection. Notice that the differential amplifier has two outputs where an **op**-**amp** has only one output. Modes/0 to 50 A. The interbase resistance RBB is the resistance of the device between terminals B1 and B2 when IE is 0. In **equation** form; R BB = (R B1 + R B2 ) IE = 0 (RBB is typically within the range of 4 to 10 /

shown explicitly But they MUST physically be there to actually Power the Operational Amplifier OpAmp **Circuit** Model DRIVING **CIRCUIT** LOAD **OP**-**AMP** vi→vo Transfer Characteristics Linear Region vo/vi = Const Saturation The OUTPUT Voltage Level /**Circuit** Using Linear Model For **Op**-**Amp** Example Required Draw The Linear Equivalent **Circuit** Locate Nodes Write The Loop **Equations** + - o v R i O Locate Nodes Place the nodes in linear **circuit** model Example cont Add Remaining Components to Complete Linear Model Examine **Circuit**/

CONFIGURATION INVERTING CONFIGURATION BUFFER The above **equations** are valid only if the gain A 0 of the **op** **amp** is very high! ELEC 2005Giovanni Anelli - CERN22 Single-stage **Op** **Amp** V DD I SS T1T1 T2T2 /SMALL PMLARGE PM ELEC 2005Giovanni Anelli - CERN33 Frequency Compensation Single-pole **op**-**amps** would always be stable (the phase does not go below - 90 ). But a typical **op**-**amp** **circuit** always contains several poles (and zeros!). These **op**- **amps** can easily be unstable, and they need therefore to be compensated./

computers Basis for analog computers **OP** **Amp** Questions What is V o equal to in the **circuit** below? What is V o equal to in the **circuit** below? AC **Circuits** Introduction Sine waves, frequency, period Sine waves, frequency, period RMS values of voltage RMS values of voltage Angular measurement of sine waves Angular measurement of sine waves **Equation** for sine waves **Equation** for sine waves Non sinusoidal/

A infinity then, V + - V 0 V + = V I + = I = 0 Seems strange, but the input terminals to an **op**-**amp** act as a short and open at the same time Summary of **op**-**amp** behavior To analyze an **op**-**amp** **circuit** for linear operation Write node **equations** at + and - terminals (I i =I + = I - = 0) Set V + = V - Solve for V o Inverting amplifier gain One of/

Q2 (a) + (c), Q3 (b) + (d), Q4 Differential Mode When the **op**-**amp** is in differential mode, both inputs are used. **Circuit** + - R1R1 VoVo V1V1 RfRf 0 V R2R2 R3R3 V2V2 Resistor R 3 is usually chosen/**op**-**amp** connected to a Wheatstone bridge **circuit** is shown. 3000 Ω Calculate the output voltage of the **op**-**amp**. Transistor Output TURD – temperature up, resistance of thermistor goes down. Voltage across thermistor goes down. V 1 goes down. V1V1 0 V +V S V2V2 + - 230 V motor for cooling fan M VoVo Consider the **equation**/

R so that V o = 2 (V 1 − V 2 ). 8 No current in +ve or -ve inputs : Ideal **op**-**amp**: 9 **Op** **Amp** Configurations (II) Fill in the values of R1 and R2 required to satisfy the **equations** in the left column of the following table. The values must be non-negative (i.e., in the range [0,∞]) 10 R1R1/ (a number) for R 2. If R 3 = R 4 then the right motor input is 5V. If α i = α o then the gain of the left **op**-**amp** **circuit** must be 5 so that the motor voltage is 0. The gain is R 1 + R 2 /R 1, so R 2 must be 4000Ω. 27 5 55 1 0/

is present. Assume that the differential input voltage and the input current of the **op** **amp** are forced to zero. (This is the summing-point constraint.) Apply standard **circuit**-analysis principles, such as Kirchhoff’s laws and Ohm’s law, to solve for/ resistor: Charging a Capacitance from a DC Source through a Resistance Rearranging: This is a linear first-order differential **equation** with contant coefficients. Charging a Capacitance from a DC Source through a Resistance The boundary conditions are given by the/

practice, the fabricated value of K (which depends on emitter area ratio, current ratio, and resistance ratio) may not satisfy the given **equation**. This will lead to Vref value at testing temp to differ from the therretically given value. A resistance value (typically R3) can be/in Vref, so that Vref temp co matches R4 temp co. CMOS version in subthreshold With a good **op** **amp**, ID1=ID2 Characterization of a bandgap **circuit** Assuming an ideal **op** **amp** with an infinite gain, we have V A = V B and I 1 = I 2. /

Engineering College of Engineering, University of North Texas **Op**-**Amp** **Circuit** Analysis General rule for **op**-**amp** **circuit** analysis Use the ideal **op**-**amp** model conditions Write nodal **equations** at the **op**-**amp** input terminals Example 4.5: Determine v o Example 4.6: Determine v o This is a precision differential voltage gain device 1 2 Comparator Comparator is a variant of **op**-**amp** Ideal comparator and its transfer curve Comparator is/

Jose Schutt-Aine 31 Example - II (d) The incremental differential voltage gain of the **circuit** is defined as: Calculate r e and ECE 342 – Jose Schutt-Aine 32 Example - II Applying the gain **equation** and assuming r out >> 1.6 k gives (e) The voltage at the /3 0.40.6 r o (k ) 222 111 ECE 342 – Jose Schutt-Aine 74 2-Stage **Op** **Amp** – Frequency Response Incremental **Circuit** ECE 342 – Jose Schutt-Aine 75 2-Stage **Op** **Amp** – Frequency Response ECE 342 – Jose Schutt-Aine 76 Transmission zero at s = s Z with Two poles/

also be expressed in logarithmic terms EXAMPLE Calculate the CMRR for the **circuit** measurements Inverting **Op**-**Amp** Non-inverting **Op**-**Amp** Summation Difference If all resistors are equal: Differentiating **Op**-**Amp** Integrating **Op**-**Amp** Differentiating **Op**-**Amp** (where Vin and Vout are functions of time) SEMICONDUCTOR Si/ and zener tunneling. PN Junction Diode V-A Characteristic PN Junction Diode V-A Characteristic Current **Equations** • The forward bias current is closely approximated by where VT =kT/q is the thermal /

function, which may be a voltage or a current source. 30 7.4 The Step-Response of a RC **Circuit** (2) Integrating both sides and considering the initial conditions, the solution of the **equation** is: Final value at t -> ∞ Initial value at t = 0 Source-free Response Complete Response = / t > 0. Calculate i for t = 2 s and t = 5 s. 7.5 The Step-Response of a RL **Circuit** (6) 40 7.6 First-Order **Op** **Amp** **Circuits** (1) Example 21 For the **op** **amp** **circuit**, find v o for t > 0, given that that v (0) = 3 V. Let R f = 80 k,/

equivalent capacitance at terminals A-B. Series Inductors Parallel Inductors RC Operational Amplifier **Circuits** - Differentiator General rules for **op**-**amp** **circuit** analysis Use the ideal **op**-**amp** model conditions: Write nodal **equations** at the **op**-**amp** input terminals RC Operational Amplifier **Circuits** - Integrator General rules for **op**-**amp** **circuit** analysis Use the ideal **op**-**amp** model conditions: Write nodal **equations** at the **op**-**amp** input terminals Example 6.17:The waveform below is applied to the input of/

Generators Study example 14.7-page786 for this pulse generator with diode EE3601-13 Electronics **Circuit** Design 3 Summary of Design **Equations** **Op**. **Amp**. Pulse Generators EE3601-13 Electronics **Circuit** Design 4 Example: Given the **Op**. **Amp**. Pulse Generator **circuit** below, (a) calculate and sketch the output waveform (b) frequency output and (c) duty cycle of the output waveform if C = 0.1 F, R = 20k , R 1/

C IRCUITS Filters, frequency response, time domain connection, bode plots, resonant **circuits**. O UTLINE AND TOPICS Low-pass filters High-pass filters Other filters Resonance (Ch 20) Ideal **op**-**amps** and active filters Decibels & log scales Linear systems and transfer functions Bode plots/ scale referenced to 1 mW, 600Ω and a 3 V rms voltage scale. L INEAR SYSTEMS RLC **circuits**, **op**-**amps** are linear **circuit** elements i.e. a differential **equation** can describe them. You can add solutions at a given ω i.e. if exp(jωt) /

Lecture Discuss analog computing and the application of 1 st order operational amplifier **circuits**. Derive the **equations** that relate the output voltage to the input voltage for a differentiator and /and Subtractors Summing and difference amplifiers Differentiators Integrators 1 st order **op** **amp** **circuits** Capacitors Differentiator Ideal **Op** **Amp** Model Virtual ground **Op** **Amp** Model Virtual ground Analysis Since current is not allowed to enter the input terminals of an ideal **op** **amp**. Example #1 Suppose v S (t) = 3V u/

College, Department of ECE & EEE, Pavoorchatram. Introduction An integrator **op** **amp** (operational amplifier) is one type of **op** **amp** **circuit**. The Integrator operational amplifier **circuit** performs the mathematical operation of Integration. The magnitude of its output is/capacitor C is given as: Assuming that the input impedance of the **op**-**amp** is infinite (ideal **op**-**amp**), no current flows into the **op**-**amp** terminal. Therefore, the nodal **equation** at the inverting input terminal is given as: From which we have/

2) Compensate with 1/Beta Pole (CF) 37 3) 1/Beta Pole/Zero **Equations** 38 1/Beta Transfer Function: DC 1/Beta: 1/Beta Pole Frequency: 1/Beta Zero Frequency: => Solve for CF CI is the equivalent input capacitance of the **op** **amp**. (See Appendix #7) 3) Select CF to Compensate **Circuit** Calculate CF(min) from fp3 < *fcl : Calculate CF(max) from fz1 < f/

> ƒc, Just like the low pass filter, the operation of a high pass active filter can be verified from the frequency gain **equation** above as: Frequency response curve Cut-off frequency of High-pass filter: Second-order (Sallen-Key) High-pass filter - A first-/ Band Pass Filter is to connect the basic passive high and low pass filters we look at previously to an amplifying **op**- **amp** **circuit** as shown. Active Band Pass Filter - this cascading together of the individual low and high pass passive filters produces a/

gain bandwidth” (UGB) or the “gain-bandwidth product” (GBW) Bandwidth: A CL & A OL The closed-loop gain (A CL ) of an **op**-**amp** **circuit** is typically much less than the open-loop gain (A OL ). On a Bode plot, A CL is a horizontal line. At some frequency, call /output. The **equation** says we can use higher frequencies if we keep the amplitude low, or we can have higher amplitudes if we keep the frequency low. Troubleshooting As always, check to see if DC voltages are within the correct range. If an **op**-**amp** needs to /

Solve **Equations** I 1 = 239.9 + j0.23 A I 2 = -12.36 + j5.98 A I 3 = -12.54 + j3.46 A ECE201 Lect-1312 Solve for V out ECE201 Lect-1313 |V out | as a function of ECE201 Lect-1314 Class Example Learning Extension E7.14(a) ECE201 Lect-1315 **Op** **Amps** **Op** **Amp** is / the input(s). ECE201 Lect-1328 Where is the Feedback? – + V in + – V out R1R1 R2R2 +–+– ECE201 Lect-1329 Review To solve an **op**-**amp** **circuit**, we usually apply KCL at one or both of the inputs. We then invoke the consequences of the ideal model. –The/

frequency components will be amplified significantly over the signal of interest (look at the **equation** for the output voltage after you have taken the derivative of the input voltage). ◦ The **circuit** may become unstable – and certainly the shape of the output signal will not / pass filter. If the difference in the voltage between the negative input terminal on the **op** **amp** and Vo is relatively constant, C1 acts like an open **circuit** and all of the current through R2 and C1 will flow through R1. If the difference/

Natural solution Forced solution (steady state) Unknowns from initial conditions Example 6-13: **Op**-**Amp** **Circuit** Substitute v out into KCL expression, rearrange for diff. **equation** in terms of i L Example 6-13: **Op**-**Amp** **Circuit** (cont.) Cont. Example 6-13: **Op**-**Amp** **Circuit** (cont.) Cont. Example 6-13: **Op**-**Amp** **Circuit** (cont.) Multisim Example of RLC **Circuit** RFID **Circuit** Tech Brief 12: Micromechanical Sensors and Actuators Tech Brief 13: Touchscreens and Active Digitizers/

active mode and entering saturation. **Equations** (8.66) and (8.67) define the minimum and maximum common-mode input voltages. Oxford University Publishing Microelectronic **Circuits** by Adel S. Sedra and Kenneth C. Smith (0195323033) Summary The differential-pair or differential-amplifier configuration is most widely used building block in analog IC designs. The input stage of every **op**-**amp** is a differential amplifier. There/

”. Filter can be passive or active filter. Passive filters Passive filters: The **circuits** built using RC, RL, or RLC **circuits**. Active filters Active filters : The **circuits** that employ one or more **op**-**amps** in the design an addition to resistors and capacitors Advantages of Active Filters over Passive/ it a high Q. The Q is set by the feedback resistors R 5 and R 6 according to the following **equations** : The configuration is similar to the band-pass version BUT R 3 has been moved and R 4 has been added/

. It can operate as an integrator over a short frequency. **Op**-**amp** parameters affect the output. Gain reduces with increase in frequency. In a triangular wave or ramp generators. In the ADC. In the integral type controllers in a closed loop control system. In analog computers to solve differential **equation**. As a low pass filter. In the communication **circuits** for recovering the modulating signal.

transfer function of Eq. (16.68) is realized by feeding the input signal through appropriate components to the inputs of the three **op** **amps**. This **circuit** can realize all special second-order functions. The design **equations** are given in Table 16.2. Microelectronic **Circuits**, Sixth Edition Sedra/Smith Copyright © 2010 by Oxford University Press, Inc. Figure 16.27 (a) Feedback loop obtained by placing a/

, values R and 2R. R-2R Ladder DAC R-2R DAC **Equation** b3, b2, b1, and b0 are binary values either ‘1’ or ‘0’. MC1408 Integrated **Circuit** DAC Popular, inexpensive 8-bit multiplying DAC. Also designated DAC0808. Output is proportional to the reference voltage. Operation of the MC1408 Requires an external **op**-**amp** to increase the output voltage and current. Can be wired to/

theory: complex notation, phasor diagrams, RC, RL, LCR **circuits**, resonance, bridges… **Op** **amps**: ideal operational amplifier **circuits**… **Op**-**amps** are now on the exam syllabus Stored energy, RC and RL transient **circuits** Reading List Electronics: **Circuits**, Amplifiers and Gates, D V Bugg, Taylor and Francis Chapters/R 1 =3kΩ R 2 =2kΩ R 3 =6kΩ VXVX 0V USE PASSIVE SIGN CONVENTION!!! - - + - Only one **equation**, Mesh analysis would give two. All currents leave all labeled nodes And apply V/R to each current. 2V 9V + + /

the highest quality voltage regulation 3-pin types make regulator **circuit** design simple Multipin IC Voltage Regulator LM 723C Schematic The LM723 has an equivalent **circuit** that contains most of the parts of the **op**-**amp** voltage regulator discussed earlier. It has an internal voltage reference/ (i.e. V CC > V o(max) + 3) LM723 in High-Voltage Configuration External pass transistor and current sensing added. Design **equations**: Choose R 1 + R 2 = 10 k , and C c = 100 pF. To make V o variable, replace R 1 /

= 87.13 o Cc=1 pF, Phase Margin = 56.99 o Lower bandwidth BG Simulation for different diode current id=13uA Characterization of a bandgap **circuit** Assuming an ideal **op** **amp** with an infinite gain, we have V A = V B and I 1 = I 2. Schematic of the current-mode bandgap/,pVB,pVC,pVD,pID1,pID2); pVCpVos= -(Vt+ID1*R1)*(ID2*R0+Vt+ID2*R2) /(-Vt*R2*ID2+ID1*R0*R1*ID2+ID1*Vt*R1) Schematic and Nodal **Equations** Derivative wrpt to 1/A: VC*(1/A)=VA-VB eq1=(pVA-pVC)/R1+pID1=0; eq2=(pVB-pVC)/R2+pID2=0; eq3=VC+pVC/A=pVA-pVB; /

-order **op** **amp** **circuit**. Example 14 (p.344) In the **op** **amp** **circuit**, f ind v o (t) for t > 0 when v s = 10u(t) mV. Let R 1 = R 2 = 10 k , C 1 = 20 μF, and C 2 = 100 μF 35 8.8 Duality (1) Two **circuits** are said to be duals of one another if they are described by the same charactering **equations** with/

: (1)The voltage between V + and V is zero V + = V (2)The current into both V + and V termainals is zero For ideal **Op**-**Amp** **circuit**: (1)Write the kirchhoff node **equation** at the noninverting terminal V + (2)Write the kirchhoff node eqaution at the inverting terminal V (3)Set V + = V and solve for the desired closed-loop gain Noninverting Amplifier (/

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