ircuits onsider the R and series circuits shown: ++++ ---- R ++++ ---- Suppose that the circuits are formed at t with the capacitor charged to value. There is a qualitative difference in the time development of the currents produced in these two cases. Why?? onsider from point of view of energy! In the R circuit, any current developed will cause energy to be dissipated in the resistor. In the circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!
Energy in the Electric and Magnetic Fields Energy stored in a capacitor... U V energy density... u +++ +++ E --- --- ε E electric Energy stored in an inductor. U I energy density... u B magnetic B μ
+++ --- I R R/ ircuits +++ --- I R: current decays exponentially I I : current oscillates t t
Oscillations (qualitative) + + - - I + I I I +I - - + + I
ecture 8, Act At t, the capacitor in the circuit shown has a total charge. At t t, the capacitor is uncharged. A + + t tt - - What is the value of V ab V b -V a, the voltage across the inductor at time t? (a) V ab < (b) V ab (c) V ab > a b B What is the relation between U, the energy stored in the inductor at tt, and U, the energy stored in the capacitor at tt? (a) U < U (b) U U (c) U > U
ecture 8, Act At t, the capacitor in the circuit shown has a total charge. At t t, the capacitor is uncharged. A + + t tt - - What is the value of V ab V b -V a, the voltage across the inductor at time t? (a) V ab < (b) V ab (c) V ab > a b V ab is the voltage across the inductor, but it is also (minus) the voltage across the capacitor! Since the charge on the capacitor is zero, the voltage across the capacitor is zero!
ecture 8, Act At t, the capacitor in the circuit shown has a total charge. At t t, the capacitor is uncharged. What is the relation between U, B the energy stored in the inductor at tt, and U, the energy stored in the capacitor at tt? + + - - t tt a b (a) U < U (b) U U (c) U > U At tt, the charge on the capacitor is zero. At tt, the current is a maximum. U U I >
Oscillations (quantitative, but only for R) What is the oscillation frequency ω? Begin with the loop rule: d + dt + + - - I Guess solution: (just harmonic oscillator!) cos( ω t + φ) remember: where φ, determined from initial conditions m dx dt + kx Procedure: differentiate above form for and substitute into loop equation to find ω. Note: Dimensional analysis ω
Oscillations (quantitative) General solution: cos( ω t + φ) + + - - Differentiate: d dt ω sin( ω t + φ) Substitute into loop eqn: d dt ω cos( ωt + φ) d + dt ( ) ω cos( ) ( cos( )) ω t + φ + ω t + φ ω + Therefore, ω which we could have determined from the mass on a spring result: k / ω m
ecture 8, Act 3 t At t the capacitor has charge ; the resulting oscillations have frequency ω. The maximum current in the circuit during these oscillations has value I. 3A + + - - What is the relation between ω and ω, the frequency of oscillations when the initial charge? (a) ω / ω (b) ω ω (c) ω ω 3B What is the relation between I and I, the maximum current in the circuit when the initial charge? (a) I I (b) I I (c) I 4I
ecture 8, Act 3 t At t the capacitor has charge ; the resulting oscillations have frequency ω. The maximum current in the circuit during these oscillations has value I. + + - - 3A What is the relation between ω and ω, the frequency of oscillations when the initial charge? (a) ω / ω (b) ω ω (c) ω ω determines the amplitude of the oscillations (initial condition) The frequency of the oscillations is determined by the circuit parameters (, ), just as the frequency of oscillations of a mass on a spring was determined by the physical parameters (k, m)!
ecture 8, Act 3 t At t the capacitor has charge ; the resulting oscillations have frequency ω. The maximum current in the circuit during these oscillations has value I. 3B What is the relation between I and I, the maximum current in the circuit when the initial charge? + + - - (a) I I (b) I I (c) I 4I The initial charge determines the total energy in the circuit: U / The maximum current occurs when! At this time, all the energy is in the inductor: U / I o Therefore, doubling the initial charge quadruples the total energy. To quadruple the total energy, the max current must double!
Preflight 8: The current in a circuit is a sinusoidal oscillation, with frequency ω. 5) If the inductance of the circuit is increased, what will happen to the frequency ω? a) increase b) decrease c) doesn t change 6) If the capacitance of the circuit is increased, what will happen to the frequency? a) increase b) decrease c) doesn t change
Oscillations Energy heck Oscillation frequency ω has been found from the loop equation. The other unknowns (, φ ) are found from the initial conditions. E.g., in our original example we assumed initial values for the charge ( i ) and current (). For these values: i, φ. uestion: Does this solution conserve energy? U E U B ( t) ( t) cos ( ω t + φ) ( t) i ( t) ω sin ( ω t + φ)
Energy heck Energy in apacitor U E ( t) cos ( ω t + φ) Energy in Inductor U B U B ( t) ω sin ( ω t + φ) ω ( t) sin ( ω t + φ ) U E U B t Therefore, U E ( t) + U B ( t) t
Inductor-apacitor ircuits Solving a circuit problem; Suppose ω/sqrt()3 and given the initial conditions, t 5 Solve find and φ, to get complete solution using, I and we find, I ( ) ( t ) 5A ( ) 5 cos( + φ ) ( t ) 5 ω sin( + φ ) 3 sin( + φ ) t 5 3 () 5 + sin ( φ ) + cos ( φ ) 5 φ inv.tan, φ 5 3 ( t) cos( ωt + φ ) [ ] 45 I( t) o ω, sin( ωt + φ ) 5
Mathematical Insert The following are all equally valid solutions t t t t () ( ) () ( ) () ( ( ) ( ) ( ) ( )) cos ωt + φ sin Acos ωt ωt + φ cos ωt + cos Bsin φ ωt ( ) ( ) ( ) sin ωt sin The circuit eqn is the analog of the spring force eqn d dt ω + m d dt x ω k m Kx φ
Inductor-apacitor-Resistor ircuit + RI + d dt d dt + R d dt + Second order homogeneous differential equation. Solution can give rather different behaviors depending on values of circuit components. + - + -
Inductor-apacitor-Resistor ircuit 3 solutions, depending on,r, values R > 4 R R 4 < 4
Inductor-apacitor-Resistor ircuit Solving for all the terms Solution for underdamped circuit; t () Ae cos( ω' t + φ) Ae α R t R αt cos andω' R 4 R 4 > t R + φ 4 For other solutions, use starting form, solve for λ and λ, t λt λ' t ( ) Ae + Be
Alternating urrents (hap 3.) In this chapter we study circuits where the battery is replaced by a sinusoidal voltage or current source. V ( t) V cos( ωt) or I( t) I cos( ωt) The circuit symbol is, An example of an R circuit connected to sinusoidal source is,
Alternating urrents (hap 3.) Since the currents & voltages are sinusoidal, their values change over time and the averages are zero. A more useful description of sinusoidal currents and voltages are given by considering the average of the square of this quantities. We define the RMS (root mean square), which is the I ( t) I cos ωt square root of the average of, ( ( )) I ( ) cos ω ω I t ( I ( t) ) I + ( t) I I ( t) RMS I ( cos )
Alternating urrents ; Phasors A convenient method to describe currents and voltages in A circuits is Phasors. Since currents and voltages in circuits with capacitors & inductors have different phase relations, we introduce a phasor diagram. For a current, We can represented this by a vector rotating about the origin. The angle of the vector is given by ωt and the magnitude of the current is its projection on the X-axis. If we plot simultaneously currents & voltages of different components we can display different phases. i I cos ( ωt) Note this method is equivalent to imaginary numbers approach where we take the real part (x-axis projection) for the magnitude