Electrical Principles: Steady-State AC Circuits - Resistors, Inductors & Capacitors - Prof, Study notes of Electrical and Electronics Engineering

Download Electrical Principles: Steady-State AC Circuits - Resistors, Inductors & Capacitors - Prof and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! 1 ECET 3000 Electrical Principles Steady-State AC Circuits Resistors, Inductors & Capacitors in 2 Resistors Resistors are devices that resist or oppose the flow of current in an electric circuit. A potential force (voltage) must exist across a resistor for a current to flow through the resistor. Electrical energy is consumed by the resistor during this process and is converted into thermal energy, causing the resistor to heat-up. + vR(t) - iR(t) Resistor V-I Relationship Whether supplied by a DC or an AC source, the voltage/current relationship for a resistor remains the same, as defined by Ohm’s Law: + vR(t) - iR(t) R Rtitv RIV RR RR DCDC ⋅= ⋅= )()( )()( 5 Inductors An inductor is basically a coil of wire in which a time-varying magnetic field is created proportional to the magnitude of the inductor’s current. If also time-varying, the magnetic field will induce a counter-e.m.f. (voltage) across the coil proportional to the field’s rate of change. The induced voltage will oppose the source voltage, thus limiting the rate of change in the current. + vL(t) - iL(t) L Inductors Energy is required to initially create the magnetic field in the inductor, but once it is created, the field requires no energy to maintain it. The electrical energy is basically converted into a magnetic form. If the field strength is decreased due to a decrease in current, the energy is converted back to electrical and released back into the circuit. + vL(t) - iL(t) L 6 Inductor V-I Relationship Unlike resistors which have a linear V-I relationship, inductors have a non-linear V-I relationship where the voltage is proportional to the rate of change of the current: + vL(t) - iL(t) L o t L t LL L L Idttv L dttv L ti dt tdiLtv +== ⋅= ∫∫ ∞− 0 )(1)(1)( )()( Inductor V-I Relationship If the current is sinusoidal in nature: Then the resultant inductor voltage can be defined by: + vL(t) - iL(t) L )90sin(2 )cos(2)()( °+⋅⋅⋅⋅= ⋅⋅⋅⋅⋅=⋅= tLI tIL dt tdiLtv RMS RMS L L L L ωω ωω )sin(2)( tIti RMSLL ⋅⋅⋅= ω 7 Inductor V-I Relationship Thus, given the inductor’s current and induced voltage: It can be seen that the inductor’s voltage is phase shifted ahead of the current by 90º and has an RMS magnitude defined by: + vL(t) - iL(t) L )90sin(2)( )sin(2)( °+⋅⋅⋅⋅= ⋅⋅⋅= tLItv tIti RMS RMS LL LL ωω ω LIV RMSRMS LL ω⋅= Inductor V-I Relationship If the inductor’s voltage and current are expressed by their phasor values: A phasor V-I relationship may be expressed using complex numbers as: + vL(t) - iL(t) L °∠= °∠= 0~ 90~ RMS RMS LL LL II VV )(~90)(~~ LjILIV LLL ωω ⋅=°∠⋅= 10 Capacitors A capacitor is basically two plates of conductive material that are placed parallel to each other and separated by a thin layer of insulation. When subjected to an externally supplied voltage, current will flow into the capacitor to charge the plates, one positive and the other negative, to create the electric field between the plates required by: + vC(t) - iC(t) C ∫ ⋅= a b C dlEV Capacitors The rate at which charge is stored in the capacitor (i.e. – current flow) is proportional to the rate of change of the applied voltage. Since current must flow into the capacitor to build-up the voltage, the created voltage will lag behind the applied current. Thus, a capacitor tends to oppose a change in voltage. + vC(t) - iC(t) C 11 Capacitor V-I Relationship Similar to inductors, capacitors also have a non-linear V-I relationship, but with the voltage and current terms reversed compared to the inductor: o t C t CC C C Vdtti C dtti C tv dt tdvCti +== ⋅= ∫∫ ∞− 0 )(1)(1)( )()( + vC(t) - iC(t) C Capacitor V-I Relationship As with the inductor, we can solve the V-I relationship. Given voltage: We can solve for the current: + vC(t) - iC(t) C )sin(2)( tVtv CRMSC ⋅⋅⋅= ω )90sin(2 )cos(2)()( °+⋅⋅⋅⋅⋅= ⋅⋅⋅⋅⋅=⋅= tCV tVC dt tdvCti RMS RMS C C C C ωω ωω 12 Capacitor V-I Relationship + vC(t) - iC(t) C Thus, given the capacitor’s voltage and resultant current: It can be seen that the capacitor’s current is phase shifted ahead of the voltage by 90º and has an RMS magnitude defined by: )90sin(2)( )sin(2)( °+⋅⋅⋅⋅= ⋅⋅⋅= tCVti tVtv CRMS CRMS C C ωω ω CVI RMSRMS CC ω⋅= Capacitor V-I Relationship + vC(t) - iC(t) C Thus, similar to inductors, if the capacitor’s voltage and current are expressed by their phasor values: A phasor V-I relationship may be expressed using complex numbers as: °∠= °∠= 90~ 0~ CRMS CRMS II VV C C Cj I C jI C IV CCCC ωωω 1~~90)1(~~ ⋅=−⋅=°−∠⋅=