82-3 Linearity and superposition Let us first define a linear element as a passive element that has a linear voltage-current relationship linear-resistor D=Ri linear-inductor w=Li linear-capacitor q=Cu We must also define a linear dependent source as a dependent current or voltage source where output current or voltage is proportional only to first power (-) of some current or voltage variable in the circuit or to the sum of such quantities. U,=061+24U2(Yes)U,=0.61(No)
§2-3 Linearity and superposition Let us first define a linear element as a passive element that has a linear voltage-current relationship. q 0 C linear − capacitor q = C i 0 L linear − inductor = Li We must also define a linear dependent source as a dependent current or voltage source where output current or voltage is proportional only to first power(幂) of some current or voltage variable in the circuit or to the sum of such quantities. 0.6 2.4 ( ) 0.6 ( ) 2 s = i 1 + 2 Yes s = i 1 No 0 i R linear − resistor = Ri
a linear circuit - independent sources, linear dependent sources, linear elements Linearity theorem: The response is proportional to the source or that multiplication of all independent source voltages and currents by a constant k increases all the current and voltage responses by the same factor k(including the dependent source voltage or current outputs)
A linear circuit -- independent sources; linear dependent sources; linear elements. The response is proportional to the source or that multiplication of all independent source voltages and currents by a constant k increases all the current and voltage responses by the same factor k (including the dependent source voltage or current outputs). Linearity theorem:
Superposition theorem In any linear resistive network containing several sources. The voltage across or the current through any resistor or source may be calculated by adding algebraically all the individual voltage or current caused by each independent source acting alone, with all other independent voltage sources replaced by short circuit and all other independent current sources replaced by open circuit. Note that dependent sources are in general active in every experiment
In any linear resistive network containing several sources. The voltage across or the current through any resistor or source may be calculated by adding algebraically all the individual voltage or current caused by each independent source acting alone, Note that dependent sources are in general active in every experiment. Superposition theorem: with all other independent voltage sources replaced by short circuit and all other independent current sources replaced by open circuit
It usually turns out that little if any time is saved in analyzing a circuit containing one or more dependent sources by use of the superposition principle, for there must always be at least two sources in operation: one independent source and all the dependent sources
It usually turns out that little if any time is saved in analyzing a circuit containing one or more dependent sources by use of the superposition principle, for there must always be at least two sources in operation: one independent source and all the dependent sources
A group of independent sources may be made active and inactive collectively, if we wish. For example, suppose there are three independent sources. The theorem above staes that we may find a given response by considering each of the three sources acting alone and adding the three results. Altenatively, we may find the response due to the first and second sources operating with the third inactive, and then add to this the response caused by the third source acting alone
A group of independent sources may be made active and inactive collectively, if we wish. For example, suppose there are three independent sources. Altenatively, we may find the response due to the first and second sources operating with the third inactive, and then add to this the response caused by the third source acting alone. The theorem above staes that we may find a given response by considering each of the three sources acting alone and adding the three results
A voltage source in parallel with a current source ls i'=b./Ri'=0 ①D
A voltage source in parallel with a current source: i'= s / R i''= 0 i − + s s i R " i s i R ' i − + s = R + i − + s s i i − + s
