1SuperpositionA direct result of the linearity property• Recall Linearity Theorem• The total response y is the sum of the responses aiui. Each aiuiis the response of the circuit to uiacting alone.mmuauauay+⋅⋅⋅++=2211SuperpositionPrinciple of Superposition:In a linear circuit containing multiple independent sources, any output (voltage or current) in the circuit may be calculated by adding together the contributions due to each independent source acting alone.Procedures:1. Determine contribution due to an independent source. Set all other independent source values to 0.2. Repeat for each independent source.3. Sum individual contributions to obtain desired output2Notes to Superposition• We consider one independent source at a time while all other independent sources are turned off (deactivated, killed).• Dependent sources are left intact .• Superposition cannot be used directly to find power.vs=0 ó short circuitis=0 ó open circuitSuperposition - Example3 V2 mA2 kΩ1 kΩ 6 kΩ+V0-2 mA2 k Ω1 kΩ 6 kΩ+V01-3 V2 k Ω1 kΩ6 kΩ+V02-V0=V01+V02Superposition - Example50 Ω 200 Ω50 Ωis1vs2+v1-i2Find v1and i2by superpositionV1=90is1+0.2vs2I2=-0.2is1+0.004vs23Superposition - ExampleFind v by superposition.1) Kill 18V and 2A sources à V12) Kill 6V and 2A sources à V23) Kill two voltage sources à V33 Ω6 Ω2 Ω6 V18 V2 A+ V -V=V1+ V2+ V3Superposition - ExampleFind the power delivered to the 3 Ω resistor.+-12 V1 Ω3 Ω6 A6 iiSource Transformation4Source Transformation• A source transformation is the process of replacing a voltage source in series with a resistor by a current source in parallel with a resistor, or vice versa.VsIsRVRiConcept of Equivalent Circuits• Two circuits are equivalent at a pair of terminals if and only ifthe v-I relation at the two terminals of one circuit is identical to the v-I relation at the corresponding two terminals of the other circuit. Circuit A Circuit B+v-ab.i...+v-iabEquivalent Circuitsv1v2v3v1+V2-V3i1i2i3i1-i2-i35Equivalent CircuitsR1R1R1R1+R2+R3G1G2G3G1+G2+G3Source TransformationVsIsRVRiConditions for equivalent circuits in source transformation:ivissRRRIV== and Notes to Source Transformation• Keep the polarity of the voltage source and the direction of the current source in agreement• Source transformation also applies to dependent sources.VsIsRR+-6Source Transformation - Example2 Ω2 Ω4 Ω6 V 18 V0.25Vx+Vx-Find VxSource Transformation - ExampleFind the power associated with the 6 V source6 V40 V4 Ω6 Ω 5 Ω20 Ω30 Ω10