1• Linearity and superpositionECE 201: Lecture 9Borja PeleatoIntuition2Linear elements and circuits3• 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 properties1. Homogeneity: if the input is scaled by a constant k, then the output is also scaled by the same constant2. 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 domainPower is NOT linear4• 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.Properties5• 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 circuit1. Turn off ALL INDEPENDENT sources but ONE2. Solve the circuit for the chosen source (optionally, assume that it has value 1)3. Repeat for all independent sources considering them one by one4. 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)Example6Example (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 aswhere are the values of the independent current andvoltage sources7Linear 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
View Full Document
Unlocking...