Review Consider a circuit consisting of an inductor L and a capacitor C The charge on the capacitor as a function of time is given by Physics for Scientists Engineers 2 q qmax cos 0t The current in the inductor as a function of time is given by Spring Semester 2005 i imax sin 0t Lecture 29 where is the phase and 0 is the angular frequency 0 March 10 2005 Physics for Scientists Engineers 2 1 March 10 2005 Review 2 2 qmax cos 2 0t 2C If there is a resistance in the circuit the current flow in the circuit will produce ohmic losses to heat The total energy stored in the circuit is given by March 10 2005 2 We observed that the energy of a circuit with a capacitor and and an inductor remains constant and that the energy translated from electric to magnetic and back gain with no losses L 2 imax sin 2 0t 2 U UE UB Physics for Scientists Engineers 2 Now let s consider a single loop circuit that has a capacitor C and an inductance L with an added resistance R The energy stored in the magnetic field of the inductor L as a function of time is UB 1 LC RLC Circuit The energy stored in the electric field of the capacitor C as a function of time is UE 1 LC 2 qmax 2C Thus the energy of the circuit will decrease because of these losses Physics for Scientists Engineers 2 3 March 10 2005 Physics for Scientists Engineers 2 4 RLC Circuit 2 RLC Circuit 3 The rate of energy loss is given by We can then write the differential equation d 2 q dq q L 2 R 0 dt dt C dU i 2 R dt We can rewrite the change in energy of the circuit as a function time as dU d q dq di U E U B Li i 2 R dt dt C dt dt The solution of this differential equation is q qmax e Remembering that i dq dt and di dt we can write 2 q dq di q dq dq d 2 q dq Li i 2 R L R 0 C dt dt C dt dt dt 2 dt Physics for Scientists Engineers 2 Rt 2L cos t where d2q dt2 March 10 2005 R 02 2L 5 March 10 2005 2 0 1 LC Physics for Scientists Engineers 2 RLC Circuit 4 6 RLC Circuit 5 Now consider a single loop circuit that contains a capacitor an inductor and a resistor The charge varies sinusoidally with but the amplitude is damped out with time If we charge the capacitor then hook it up to the circuit we will observe a charge in the circuit that varies sinusoidally with time and while at the same time decreasing in amplitude After some time no charge remains in the circuit We can study the energy in the circuit as a function of time by calculating the energy stored in the electric field of the capacitor This behavior with time is illustrated below UE Rt 2L q e cos t max 1 1 q2 2C 2 C 2 Rt 2 qmax e L cos 2 t 2C We can see that the energy stored in the capacitor decreases exponentially and oscillates in time March 10 2005 Physics for Scientists Engineers 2 7 March 10 2005 Physics for Scientists Engineers 2 8 Alternating Current Alternating Current 2 Now we consider a single loop circuit containing a capacitor an inductor a resistor and a source of emf The current induced in the circuit will also vary sinusoidally with time This source of emf is capable producing a time varying voltage as opposed the sources of emf we have studied in previous chapters However this current may not always remain in phase with the time varying emf This time varying current is called alternating current We can express the induced current as i I sin t We will assume that this source of emf provides a sinusoidal voltage as a function of time given by where the angular frequency of the time varying current is the same as the driving emf but the phase is not zero Vemf Vmax sin t Note that traditionally the phase enters here with a negative sign where is that angular frequency of the emf and Vmax is the amplitude or maximum value of the emf March 10 2005 Physics for Scientists Engineers 2 Thus the voltage and the current in the circuit are not necessarily in phase 9 March 10 2005 Circuit with Resistor 10 Circuit with Resistor 2 To begin our analysis of RLC circuits let s start with a circuit containing only a resistor and a source of time varying emf as shown to the right Thus we can relate the current amplitude and the voltage amplitude by VR I R R We can represent the time varying current by a phasor IR and the timevarying voltage by a phasor VR as shown below Applying Kirchhof s loop rule to this circuit we get Vemf vR 0 Physics for Scientists Engineers 2 Vemf vR where vR is the voltage drop across the resistor Substituting into our expression for the emf as a function of time we get vR VR sin t Remembering Ohm s Law V iR we get The current flowing through the resistor and the voltage across the resistor are in phase which means that the phase difference between the current and the voltage is zero v iR R I R sin t R March 10 2005 Physics for Scientists Engineers 2 11 March 10 2005 Physics for Scientists Engineers 2 12 Circuit with Capacitor Circuit with Capacitor 2 Now let s address a circuit that contains a capacitor and a time varying emf as shown to the right We can rewrite the last equation by defining a quantity that is similar to resistance and is called the capacitive reactance The voltage across the capacitor is given by Kirchhof s loop rule XC vC VC sin t Which allows us to write Remembering that q CV for a capacitor we can write iC q CvC CVC sin t We would like to know the current as a function of time rather than the charge so we can write Physics for Scientists Engineers 2 VC cos t XC We can now express the current in the circuit as iC I C cos t I C sin t 90 dq d CVC sin t iC CVC cos t dt dt March 10 2005 1 C We can see that the current and the time varying emf are out of phase by 90 13 March 10 2005 Circuit with Capacitor 3 Physics for Scientists Engineers 2 14 Circuit with Capacitor 4 We can represent the time varying current by a phasor IC and the time varying voltage by a phasor VC as shown below We can also see that the amplitude of voltage across the capacitor and the amplitude of current in the capacitor are related by VC I C XC This equation resembles Ohm s …
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