Common Reference Example

Transcription

1 Operational Amplifiers Overview Common reference circuit diagrams Real models of operational amplifiers Ideal models operational amplifiers Inverting amplifiers Noninverting amplifiers Summing amplifiers Difference amplifiers Cascaded op amp circuits Portland State University ECE 221 Operational Amplifiers Ver Common Reference Example 5kΩ ip + 35 V 7 ma 20 kω - 35 V 5kΩ ip 7 ma 20 kω Portland State University ECE 221 Operational Amplifiers Ver Common Reference Circuit Diagrams All circuits we have seen show all connections explicitly In many circuits, there is a common node that interconnects many circuit elements For nodal analysis, we usually chose this as the reference node In many practical circuits, all voltages are measured relative to a common reference (like nodal analysis In these cases, the connections to the reference are often replaced with the reference symbol + This makes cleaner circuit diagrams Makes analysis more difficult conceptually because not all of the wires are drawn Cannot perform mesh analysis on these circuits Portland State University ECE 221 Operational Amplifiers Ver Variable versus Numerical Analysis We have mostly analyzed circuits that contain elements with known values Resistor and independent source values have been specified For this chapter, will mostly use variables Requires more algebra Can not solve numerically by inverting matrices + Is more useful for design How do you design a circuit that has this mathematical relationship between variables, e.g. vo =12v1 3v2 Start by designing a circuit that has vo = αv1 βv2 Can always pick values to meet design requirements after you know the relationships You should be proficient at both types of circuit analysis Portland State University ECE 221 Operational Amplifiers Ver

2 Operational Amplifier Symbol Inverting input Noninverting input Upper Power Supply V ss Lower Power Supply V dd v Output - v + V ss V dd Operational amplifiers amplify voltage Operational amplifiers are often called op amps Note symbols for upper and lower power supplies To function properly Vss >Vdd Often called rails Portland State University ECE 221 Operational Amplifiers Ver Real Operational Amplifier A ( - Like transistors, Op amps are nonlinear devices We will only use them in their linear range We can model the op amp behavior by the equivalent circuit shown above Note that this is just a model, it is a good approximation Amplifies the difference of voltages on the input terminals Portland State University ECE 221 Operational Amplifiers Ver Operational Amplifier Symbol Continued V ss Linear Region Saturation (v+ - Saturation V dd Often the supply rails are omitted from circuit diagrams Often Vdd =-Vss Donotforgettheyarepresent They produce the power for amplification The output voltage can not exceed the rails For our analysis, must be in linear region Portland State University ECE 221 Operational Amplifiers Ver Real Operational Amplifier Continued A ( - Typical values: Ro =75Ω, Ri =5MΩ, A = 100 k In most low-power analog circuits, resistors have a range of 1 kω to 100 kω Op amps have extreme values Input resistance Ri is very large Output resistance Ro is very small The gain A is very large Portland State University ECE 221 Operational Amplifiers Ver

3 Example 1: Real Op Amp Voltage Follower v i A ( - Use the real model of an op amp to solve for vo in terms of vi. Step 1: replace op amp with model of a real op amp Step 2: nodal analysis Portland State University ECE 221 Operational Amplifiers Ver Example 1: Real Op Amp Voltage Follower vo = vi 1+ 1 A +1+ ARL + Note that variables are more difficult to work with than real values But are also more flexible Note the relationship of vo to vi is complicated Difficult to understand the circuit How is vo related to vi? What is the sensitivity of vo to RL? This is actually the simplest possible circuit with an op amp! If we can make certain approximations, there is an easier way to analyze and understand op amp circuits Portland State University ECE 221 Operational Amplifiers Ver Example 1: Real Op Amp Voltage Follower A ( - vo A(vi vo + vo Ro vo ( 1 ( 1+A Ro RL + vi Ri RL Ri A +1+ ARL + = 0 ( A = vi + 1 Ro Ri ( = vi 1+ R o Portland State University ECE 221 Operational Amplifiers Ver Ideal Operational Amplifier The ideal model of an op amp is easier to work with than the real model Ideal op amp model has same symbol as a real op amp model Problem must state which model to use Ideal op amps make approximations that dramatically simplify analysis In most cases these approximations are reasonable In most engineering contexts, we are satisfied if analysis and experimental results are within 1% of each other Portland State University ECE 221 Operational Amplifiers Ver

4 Ideal Op Amp Approximations Recall for practical real op amps Input resistance Ri is very large Output resistance Ro is very small The gain A is very large Ideal op amp approximations Input resistance Ri is infinite Output resistance Ro is zero The gain A is infinite Portland State University ECE 221 Operational Amplifiers Ver Example 2: Continued (1 From our previous analysis using the real op amp model, vo = vi 1+ 1 A +1+ ARL + Approximations: Ri, Ro 0, anda. vo = vi vo = vi Portland State University ECE 221 Operational Amplifiers Ver Example 2: Ideal Op Amp Models Use the ideal model of an op amp to solve for vo as a function of vi. We can start with the solution derived from the real model and take the limit as Ri, Ro 0, anda. Portland State University ECE 221 Operational Amplifiers Ver Voltage Followers This is a common op amp circuit This is called a voltage follower Also called a current buffer Useful for driving circuits with an output voltage without loading the circuit Portland State University ECE 221 Operational Amplifiers Ver

5 Example 2: Continued (2 A ( - vo = vi: Notes on ideal op amp model approximations RL has no effect on the output Ri, Ro, anda do not affect the relationship of vo to vi Output is linearly related to the input Input current is zero Much simpler equation than for real op amp model Portland State University ECE 221 Operational Amplifiers Ver Negative Feedback Continued A ( - In both cases The output is fed back to the inverting input of the op amp The error is proportional to the negative of the output This feedback minimizes the error What if we connected the output to the positive terminal of the op amp? Portland State University ECE 221 Operational Amplifiers Ver Negative Feedback A ( - Op amp circuits require negative feedback For this circuit, define the error as ɛ = vi vo If the error becomes positive due to a sudden increase in vi, The output increases: vo Aɛ The error becomes smaller If the error becomes negative due to a sudden decrease in vi, The output decreases: vo Aɛ The error becomes smaller Portland State University ECE 221 Operational Amplifiers Ver Positive Feedback Again, define the error as ɛ = vi vo If the error becomes positive due to a sudden increase in vi, The output decreases: vo Aɛ The error becomes larger If the error becomes negative due to a sudden decrease in vi, The output increases: vo Aɛ The error becomes larger In both cases, the op amp will quickly saturate This is sometimes called hitting the rails Portland State University ECE 221 Operational Amplifiers Ver

6 Negative Feedback Importance Linear Circuit The assumption (v+ v =0is only true if there is a non-inverting path from the output vo to the negative input terminal vi This negative feedback is essential If this condition is not satisfied, the op amp will be unstable and the output will saturate at one of the rails With negative feedback v+ = v (almost always If uncertain about negative feedback, (1 solve in terms of A and (2 take the limit as A Portland State University ECE 221 Operational Amplifiers Ver Example 3: Inverting Amplifier Use the ideal model of an op amp to find vo as a function of vi. Portland State University ECE 221 Operational Amplifiers Ver Ideal Op Amp Analysis i - i - i + (v+ v i + Ri = implies i and i+ are zero Ro =0implies the op amp output voltage is not affected by what it is connected to A = implies either vo = ± or (v+ v =0 Since we are only interested in linear circuits with finite voltages and currents, A = implies v+ = v Portland State University ECE 221 Operational Amplifiers Ver Example 3: Workspace Portland State University ECE 221 Operational Amplifiers Ver

7 Example 4: Voltage Divider 50 V + 50 V v R R 2 L o - Solve for vo in both circuits. (If type of model is not specified, use the ideal model. Portland State University ECE 221 Operational Amplifiers Ver Example 5: Current-to-Voltage Converter R I S Solve for vo as a function of Is. Portland State University ECE 221 Operational Amplifiers Ver Example 4: Workspace Portland State University ECE 221 Operational Amplifiers Ver Example 6: Noninverting Amplifier Solve for vo as a function of vi. Portland State University ECE 221 Operational Amplifiers Ver

8 Example 7: Summing Amplifier v 1 R1 R f R2 v 2 R3 v 3 Solve for vo as a function of v1, v2, and v3. Portland State University ECE 221 Operational Amplifiers Ver Example 8: Difference Amplifier v 1 v 2 R 3 R 4 Solve for vo as a function of v1 and v2. Portland State University ECE 221 Operational Amplifiers Ver Example 7: Workspace Portland State University ECE 221 Operational Amplifiers Ver Example 8: Workspace Portland State University ECE 221 Operational Amplifiers Ver

9 Example 9: Instrumentation Amplifier v 1 R 3 R 4 R 5 v 2 Solve for vo as a function of v1 and v2. Portland State University ECE 221 Operational Amplifiers Ver Difference & Instrumentation Amplifier Tradeoffs The following questions should be considered when choosing between two designs (e.g. difference amplifier versus instrumentation amplifier How much does each cost? How many op amps does each require? How flexible is the design of each? How much does each load down the inputs v1 and v2? How much tuning is necessary in each? Portland State University ECE 221 Operational Amplifiers Ver Example 9: Workspace Portland State University ECE 221 Operational Amplifiers Ver Example 10: General Op Amp Circuit Analysis 24 kω 18 kω 3kΩ 1.1 V i o 3kΩ 4.5 kω 27 kω Solve for vo and io. Portland State University ECE 221 Operational Amplifiers Ver

10 Example 10: Workspace Portland State University ECE 221 Operational Amplifiers Ver Example 11: Workspace Portland State University ECE 221 Operational Amplifiers Ver Example 11: General Op Amp Circuit Analysis R f i 1 3kΩ R i 2 2 Select the values of R2 and Rf so that vo = 2000(i2 i1. Portland State University ECE 221 Operational Amplifiers Ver Example 12: General Op Amp Circuit Analysis 20 kω 240 kω i a + v a 60 kω - 9 V 6kΩ i o 120 mv -9 V 80 kω Solve for va, vo, ia, andio. Portland State University ECE 221 Operational Amplifiers Ver

11 Example 12: Workspace Summary Op amps dramatically expand the operations we can perform with electric circuits Buffer current Summation Difference Integration Differentiation Amplification In ECE 222 we will see that op amps enable us to solve and model differential equations Portland State University ECE 221 Operational Amplifiers Ver Portland State University ECE 221 Operational Amplifiers Ver