Lecture 7 Last Time The Circuit Abstraction Circuits represent systems as connections of elements through which currents through variables ow across which voltages across variables develop Last Time Analyzing Circuits Circuits are analyzed by combining three types of equations KVL sum of voltages around any closed path is zero KCL sum of currents out of any closed surface is zero Element constitutive equations Resistor V IR Voltage Source V V Current Source I I Many KVL and KCL equations are redundant We looked at three methods to systematically identify a linearly independent set Check yourself Analyzing Circuits Correct answer 2 and 5 How many of the following are true V i v i e Interaction of Circuit Elements Circuit design is complicated by interactions among the elements Adding an element changes voltages and currents throughout the circuit Example closing a switch is equivalent to adding a new element Concept Question Closing a switch Correct answer 3 By closing the switch V decreased by more than a factor of 1000 Buffering with Op Amps Dependent Sources Interactions between elements can be reduced or eliminated by using an op amp as a buffer Opening and closing the switch has no effect on I or V When the switch is closed the voltage across the motor is the same as the voltage at the input to the op amp To analyze op amps we must introduce a new kind of element a dependent source A dependent source generates a voltage or current whose value depends on another voltage or current Example current controlled current source Check yourself Dependent Sources Correct answer 4 Dependent sources are two ports characterized by two equations By contrast one ports resistors voltage sources current sources are characterized by a single equation An op amp operational ampli er can be represented by a voltage controlled voltage sources A voltage controlled voltage source is a two port Example Find V V for the following circuit Op Amp Non Inverting Ampli er This circuit implements a gain For large K the gain is always greater than or equal to one We call this a non inverting ampli er Check yourself Non Inverting Ampli er Correct answer 4 V V when R is in nity and or R is zero The Ideal Op Amp The approximation that V V is referred to as the idea op amp approximation Check yourself Determine Output Correct answer 3 V Check yourself Determine Resistance Correct answer 3 V 2 V V when R 2 Paradox If quantity is conserved then the difference between what comes in and what goes out must accumulate Example One Example Two These circuits seem to have identical responses if K is large The reasoning is wrong because it ignores a critical property of circuits For a voltage to change charged particles must ow To understand ow we need to understand continuity Flow and Continuity Charge accumulates in a capacitor Charge Accumulation in an Op Amp We can add a resistor and capacitor to model the accumulation of charge in an op amp This is not an accurate representation of what is inside an op amp We can add a resistor and capacitor to model the accumulation of change in an op amp This is a model of how the op amp works Dynamic Analysis of Op Amps If the input voltage to this circuit suddenly increases then current will ow into the capacitor and gradually increases voltages As V increases the difference V V decreases less current ows and V approaches a nal value to V If the input voltage to this circuit suddenly decreases then current will ow out of the capacitor and decreases V As V decreases the V V decreases the magnitude of the current decreases and V approaches a nal value equal to V Switching the plus and minus inputs ips these relations If the input increases current will ow out of the capacitor and decrease V This makes the difference between the input and output even bigger If the input decreases current will ow into the capacitor and increase V As the output diverges from the input the magnitude of the capacitor current increases and the rate of divergence increases Check yourself Brightness of a Bulb correct answer 3 Closing the switch will make bulb 1 brighter and bulb 2 dimmer Superposition Summary If a circuit contains only linear parts resistors current and voltage sources then any voltage or current can be computed as the sum of those that result when each source is turned on one at a time Short circuit guarantees the voltage is zero Combining the two halves of the circuit we get the whole picture An op amp can be represented as a voltage dependent voltage source The ideal op amp approximation is V V The ideal op amp approximation only makes sense when the op amp is connected with negative feedback If a circuit contains only linear parts then any voltage or current can be computed as the sum of those that result when each source is turned on one at a time
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