PES 0 Spring 04, Spenier ecture 36/Page Toay: chapter 3 - R circuits: Dampe Oscillation - Driven series R circuit - HW 9 ue Wenesay - FQs Wenesay ast time you stuie the circuit (no resistance) The total energy of the system is conserve an oscillates between magentic an electric potential energy. We notice that this circuit is analogous to a spring-mass system (simple harmonic oscillator) without friction. However, we know that typically there is resistance in the circuit (wires, resistors, internal resistance). So, what o you think happens if we put a resistor in series with the capacitor an inuctor? Remember that with the resistor, the total energy will ecrease ue to the power issipate in the resistor. R circuit: Dampe Oscillation A circuit containing resistance, inuctance, an capacitance is calle an R circuit. Figure shows a series R circuit. As the charge containe in the circuit oscillates back an forth through the resistance, electromagnetic energy is issipate as thermal energy, amping (ecreasing the amplitue of) the oscillations. the capacitor is initially charge to Q 0. After the switch is close current will begin to flow. However, unlike the circuit energy will be issipate through the resistor. The rate at which energy is issipate is U t i( t) R where the negative sign on the right-han sie implies that the total energy q( t) i( t) U U E U B
PES 0 Spring 04, Spenier ecture 36/Page is ecreasing. After substituting for the left-han sie of the above equation, we obtain the following ifferential equation: q( t) q i( t) i i( t) R t t q use i( t) t q( t) i ( ) ( ) ( ) i t i t i t R t q( t) q q t t q t R q t R 0 q( t) 0 i not show these steps This expression shoul remin you of the equation for ampe simple harmonic oscillations. The general solution is: initial conition: q(t=0) = Q 0 t R q( t) Q0e cos t 4 R R ' ( angular frequency) ( amping factor) 4 - The amping effect is ue to the presence of resistance R. - The amping factor γ etermines the rate at which the response is ampe. - If R=0, the circuit is sai to be lossless an the oscillatory response will continue. How o these oscillations look like?
PES 0 Spring 04, Spenier ecture 36/Page 3 a) Unerampe circuit (small resistance R) With a relatively small resistor, 4 R then there are oscillations whose amplitue ecreases exponentially in time. The ampe oscillation exhibite by the unerampe response is known as ringing. It stems from the ability of the an to transfer energy back an forth between them. b) ritically ampe circuit (larger resistance R) 4 R then the system no longer oscillates, but instea amps own as quickly as is possible. ' R 0 4 All the other properties (i(t), E(t), B(t), U E (t), U B (t)) will also ecay in amplitue over time until all electromagnetic energy has been lost to heat. A an D Until now, we have ealt with circuits where the source of EMF (e.g., the battery) has a constant value. This is known as a irect current (D) source. For many reasons however, much of the worl s power is not elivere as a uniirectional EMF. For commercial an resiential electricity, A current is use. The main reason for this is the with a changing current, Faraay's aw of inuction can be use to inuce EMFs/currents in other circuits. This is the main iea behin transformers, which we will iscuss next lecture.
PES 0 Spring 04, Spenier ecture 36/Page 4 We just saw that for an R circuit, energy is taken away from the circuit as heat in the resistor. Therefore the oscillating current ies away over time. Not very useful for commercial electricity. - So how can we stop this ecay over time? We coul put energy at the same rate as it is being lost! Alternating EMF So, now we want to examine how the circuit elements we have behave when they are riven with an alternating current (A) source. An A source supplies an EMF which follows a cosine epenency: ( t) sin( t)... imum emf amplitue... riving angular frequency After an initial transient time, an A current will flow in the circuit as a response to the riving voltage source. The current, written as i( t) I sin( t ) will oscillate with the same frequency as the voltage source an may be out of phase with the emf. Driven series R circuit We are now reay to apply the alternating emf ( t) sin( t) to the full R circuit. Because R,, an are in series, the same current i( t) I sin( t ) is riven in all three of them. We wish to fin the current amplitue I an the phase constant.
PES 0 Spring 04, Spenier ecture 36/Page 5 Applying Kirchhoff s loop rule, we obtain i q( t) ( t) VR ( t) V ( t) V ( t) ( t) i( t) R 0 t which leas to the following ifferential equation: i q( t) i( t) R sin( t) t Assuming that the capacitor is initially uncharge so that i(t)=+q/t is proportional to the increase of charge in the capacitor, the above equation can be rewritten as ( ) R sin( t) q q q t t t One possible solution to this equation is q( t) Q cos( t ) where the imum charge amplitue is Q R /( ). The corresponing current is q i( t) Q0 sin t t with an imum current amplitue I Q0 R /( ) an phase
PES 0 Spring 04, Spenier ecture 36/Page 6 tan R (Note book uses a ifferent way of eriving these results) We can see that the quantities X an X must have the same units as resistance. They are calle "inuctive reactance (X )" an the "capacitive reactance (X )". Resonance in an R circuit I Q 0 R /( ) gives the current amplitue I in an R circuit as a function of the riving angular frequency ω of the external alternating emf.for a given resistance R, that amplitue is a imum when the quantity 0 that is when: ( X X ) Because the natural angular frequency ω of the circuit is also equal to /, the imum value of I occurs when the riving angular frequency matches the natural angular frequency that is, at resonance. Thus, in an R circuit, resonance an imum current amplitue I occur when (resonance)
PES 0 Spring 04, Spenier ecture 36/Page 7 Usefulness of Reactances X an X Notice that the reactances are epenent on the angular frequency, (the resistance is not). As ω 0 (D), there is no inuctive effect X goes to zero an current is passe through the inuctor, while no current is passe through the capacitor X iverges. As ω gets large, X goes to zero an current is passe through the capacitor, while no current is passe through the inuctor X gets large because of the quickly changing current. We can use these properties to create frequency filters. - Inuctors are use as low-pass filters. - apacitors are use as high-pass filters. - In combination, you can create a cross-over circuit. The inuctor an capacitor fee low frequencies mainly fee low frequencies mainly to the woofer an high frequencies mainly to the tweeter.