1 Wednesday, March 26, 2014 RC Circuits . Going around the circuit in a counterclockwise direction we can write . We can rewrite this equation remembering that i = dq/dt . The solution is The term Vc is negative since the top plate of the capacitor is connected to the positive - higher potential - terminal of the battery. Thus analyzing . where q = CV and τ = RC 0 emf counter-clockwise leads to a drop in voltage across the capacitor! 31 Wednesday, March 26, 2014 RL Circuits . Thus we can write the sum of the potential drops around the circuit as . The solution to this differential equation is . We can see that the time constant of this circuit is τL = L/R University Physics, Chapter 29 52 Wednesday, March 26, 2014 Summary: LC Circuit (1) . 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 . The current in the inductor as a function of time is given by . where φ is the phase and ω0 is the angular frequency 21 Wednesday, March 26, 2014 5 Wednesday, March 26, 2014 RLC Circuit (1) . Now let’s consider a single loop circuit that has a capacitor C and an inductance L with an added resistance R . We observed that the energy of a circuit with a capacitor and an inductor remains constant and that the energy translated from electric to magnetic and back gain with no losses . If there is a resistance in the circuit, the current flow in the circuit will produce ohmic losses to heat . Thus the energy of the circuit will decrease because of these losses 23 Wednesday, March 26, 2014 RLC Circuit (2) . The rate of energy loss is given by . We can rewrite the change in energy of the circuit as a function time as . Remembering that i = dq/dt and di/dt = d 2q/dt2 we can write 24 Wednesday, March 26, 2014 RLC Circuit (3) . We can then write the differential equation . The solution of this differential equation is (damped harmonic oscillation!), where 25 Wednesday, March 26, 2014 RLC Circuit (4) . 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 . This behavior with time is illustrated below 26 Wednesday, March 26, 2014 RLC Circuit (5) .Observations: •The charge varies sinusoidally with but the amplitude is damped out with time •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 . We can see that the energy stored in the capacitor decreases exponentially and oscillates in time 27 Wednesday, March 26, 2014 Alternating currents 11 Wednesday, March 26, 2014 DC and AC Motors and Generators 23 Wednesday, March 26, 2014 Direct and Alternating Current Generators . In a direct current generator the rotating coil is connected to an external circuit using a split commutator ring . As the coil turns, the connection is reversed such that the induced voltage always has the same sign . In alternating current generator, each end of the loop is connected to the external circuit through a slip ring • Thus this generator produces an induced voltage that varies from positive to negative and back, and is called an alternator . The voltages and currents produced by these generators are illustrated below Direct voltage/ Alternating voltage/ current current 22 Wednesday, March 26, 2014 Alternating Current (1) . Now we consider a single loop circuit containing a capacitor, an inductor, a resistor, and a source of emf . This source of emf is capable of producing a time varying voltage as opposed to the sources of emf we have studied in previous chapters . We will assume that this source of emf provides a sinusoidal voltage as a function of time given by . where ω is that angular frequency of the emf and Vmax is the amplitude or maximum value of the emf 28 Wednesday, March 26, 2014 Alternating Current (2) . The current induced in the circuit will also vary sinusoidally with time . This time-varying current is called alternating current . However, this current may not always remain in phase with the time- varying emf • The sinusoidal wave may crest earlier or later than that for emf . We can express the induced current as where the angular frequency of the time-varying current is the same as the driving emf but the phase φ is not zero 29 Wednesday, March 26, 2014 Alternating Current (3) . Note that traditionally the phase enters here with a negative sign . Thus the voltage and the current in the circuit are not necessarily in phase . Notation: instantaneous values are denoted by small letters (v, i ), amplitudes by capital letters (V, I) 30 Wednesday, March 26, 2014 Circuit with Resistor (1) . 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 . Applying Kirchhoff’s loop rule to this circuit we get . where vR is the voltage drop across the resistor . Substituting into our expression for the emf as a function of time we get . Remembering Ohm’s Law, V = iR, we get 31 Wednesday, March 26, 2014 Circuit with Resistor (2) . Thus we can relate the current amplitude and the voltage amplitude by . We can represent the time varying current by a phasor IR and the time-varying voltage by a phasor VR as shown below . Phase difference is 0 32 Wednesday, March 26, 2014 Circuit with Capacitor (1) . Now let’s address a circuit that contains a capacitor and a time varying emf as shown to the right . The voltage across the capacitor is given by Kirchhoff’s loop rule . Remembering that q = CV for a capacitor we can write . We would like to know the current as a function of time rather than the charge so we can write 33 Wednesday, March 26, 2014 Circuit with Capacitor (2): Capacitive Reactance . We can rewrite the last equation by defining a quantity that is similar to resistance and is called the capacitive reactance . Which allows us to write 1 Effective resistivity of a capacitor ωC . We can now express the current in the circuit as . We can see that the current and the time varying emf are out of phase by 90° 34 Wednesday, March 26, 2014 Circuit with Capacitor (2): Capacitive Reactance . We can rewrite the last equation by defining a quantity that is similar to resistance and is called the capacitive reactance compare with . Which allows us to write V i = 1 R Effective resistivity of a capacitor ωC . We can now express the current in the circuit as . We can see that the current and the time varying emf are out of phase by 90° 34 Wednesday, March 26, 2014 Circuit with Capacitor (3): Phasor . We can represent the time varying current by a phasor IC and the time-varying voltage by a phasor VC as shown below . The current flowing this circuit with only a capacitor is similar to the expression for the current flowing in a circuit with only a resistor except that the current is out of phase with the emf by 90° 35 Wednesday, March 26, 2014 Circuit with Capacitor (4) . We can also see that the amplitude of voltage across the capacitor and the amplitude of current in the capacitor are related by . This equation resembles Ohm’s Law with the capacitive reactance replacing the resistance . One major difference between the capacitive reactance and the resistance is that the capacitive reactance depends on the angular frequency of the time-varying emf 36 Wednesday, March 26, 2014 Circuit with Inductor (1) . Now let’s consider a circuit with a source of time-varying emf and an inductor as shown to the right . We can again apply Kirchhoff’s Loop Rule to this circuit to obtain the voltage across the inductor as . A changing current in an inductor will induce an emf given by . So we can write 37 Wednesday, March 26, 2014 Circuit with Inductor (2) . We are interested in the current rather than its time derivative so we integrate . We define inductive reactance as which, like the capacitive reactance, is similar to a resistance . We can then write which again resembles Ohm’s Law except that the inductive reactance depends on the angular frequency of the time-varying emf 38 Wednesday, March 26, 2014 Circuit with Inductor (3) . The current in the inductor can then be written as . Thus the current flowing in a circuit with an inductor and a source time-varying emf will be -90° out of phase with the emf . We can write the relationship between the amplitude of the current and the amplitude of the voltage as 39 Wednesday, March 26, 2014 Summary: RLC Circuit . If we have a single loop RLC circuit, the charge in the circuit as a function of time is given by where . The energy stored in the capacitor as a function of time is given by 40 Wednesday, March 26, 2014 Summary: Resistance and Reactance . Time-varying emf . Time-varying emf VR with resistor Resistance R . Time-varying emf V with capacitor C Capacitive Reactance XC . Time-varying emf VL with inductor Inductive Reactance XL 41 Wednesday, March 26, 2014 Summary: Phase and Phasors 42 Wednesday, March 26, 2014 Average powers IV <P >= IV < sin ωt sin ωt >= 2 <P >= IV < sin ωt cos ωt >=0 <P >= IV < sin ωt cos ωt >=0 42 Wednesday, March 26, 2014 Series RLC Circuit (1) .