ECE 201 Lecture 9 Borja Peleato Linearity and superposition 1 Intuition 2 Linear elements and circuits From now on we will refer to the voltages or currents provided to a circuit by sources other circuits etc as inputs and the voltages or currents that we want to measure as outputs A circuit element is linear if it has two properties 1 2 Homogeneity if the input is scaled by a constant k then the output is also scaled by the same constant Additivity The response to a sum of inputs is the sum of the corresponding responses Resistors are linear obvious from Ohm s law A linear circuit is one whose output is directly proportional to its input A circuit built with linear components is linear The circuits studied in 201 are linear although later on we will see capacitors and inductors which are only linear in the complex domain 3 Power is NOT linear The power consumed by a resistor is not directly proportional to its inputs If I double the current or voltage the power quadruples Superposition does not apply to power calculations If two current sources are injecting current into a resistor the amount of power consumed is not the sum of the power consumed with each source alone 4 Properties Superposition The output voltage across some element or current through it of a linear circuit is the algebraic sum of the outputs voltages or currents in that element due to each INDEPENDENT source acting alone This means that you can solve the circuit by considering each independent source in turn with all the other INDEPENDENT sources turned off and adding all the corresponding contributions Turning off a voltage source implies V 0 i e replacing it by a short circuit Turning off a current source implies I 0 i e replacing it by an open circuit Linearity when you are considering a single INDEPENDENT source you can solve the circuit assuming that it has value 1 and then scale the contribution by the corresponding factor be very careful if you do this Summarizing to solve a circuit 1 2 3 4 Turn off ALL INDEPENDENT sources but ONE Solve the circuit for the chosen source optionally assume that it has value 1 Repeat for all independent sources considering them one by one Find total contribution by adding all the contributions if for any source you assumed that it had value 1 scale the corresponding contribution by the correct value 5 Example 6 Example Cont Once you have solved the circuit once you can easily update the solution for any changes in the sources For example If I now asked you to solve the circuit again with the current source having a value of 3A and the voltage source having a value of 25V you could do the following In general any voltage or current in the circuit can always be expressed as where voltage sources are the values of the independent current and 7 Linear circuit characterization You can completely characterize a linear circuit by knowing its output to a few inputs For example assume that you have an unknown linear circuit with two inputs and one output 8
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