00008 00001 t ime s 000004 rlc under damped response

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Unformatted text preview: .04 0 0.03 -0.05 0.02 -0.1 0.01 0 -0.15 0 0.0001Column U 0.0002 T ime (s) 0.0003 Column Q 0 0.00002 Column W Column R 0.00006 0.00008 0.0001 T ime (s) 0.00004 RLC Under Damped Response Parameters Parameter Calculated Deduced from Measurement -1 α 8759 (s ) -ln(V(t)RCωd/Vosin(ωd*t))/t = 6688.9 (s-1) ω0 91090 (rad/s) • ωd 2π/T=2π/(69μs ) = 91060 (rad/s) 90667 (rad/s) tA arctan(ωd/α)/ωd= 16.26 µsec 16 µsec RLC Critically Damped Response Parameters Parameter Calculated Deduced from Measurement Req R*R2/(R+R2)=(61.81)(554.2)/(61.81+554.2) =55.608Ω • α • 1/(2*R*C)=1/[2(55.608)(.103E-6)] = 87296 (s ) vmax @ t1[.92/(554.2*.103μF)](13.2μa)exp(-87296*13.2μa) = 67.2 mV 72 mV tA 1/α=1/87296 = 11.4µs 13.2µs ECE2100 Lab #3 - Transient Response: RC, RL, RLC Page 3 of 3 Fall 2012 Comment Block 1. Equation (4) Explanation. Equation (4) is a reduced form of the general case of a critically damped response. There is usally a second term in equation of the form Aexp(-αt) where A is a constant. Although it does not appear in equation (4) because at t=0 the initial voltage is 0V. The first term contains a outside any trignometric or exponential function and therefore for any initial conditions at t=0 will be dropped out. This means Aexp(-αt) at t=0 must equal zero. This only possible if the constant term (A) is zero itself. The first term's constant term is explained below under comment (4) but does not cover why α=1/[2(Req)(C)]. Req is present while in the amplitude it's only R=560Ω because α governs the time decay of the signal for t > 0 and must include Req because R2 has an effect on the circuit where at t=0+ it does not. 2. Amplitude in Equation (3). The amplitude in equation (3) is Vo/RCωd because to obtain the initial condition for Vout the derivative of the expression must be used. The initial current is Vo/R and using the relationship ic=C(dv/dt), dv/dt of time zero is Vo/RC. The derivative of Vout pulls a ωd out of the sine term and when the constant term is solved for, ωd is moved to the other side of the equation giving us Vo/RCωd. 3. RLC Underdamped Response. The predicted curve very closely modeled the under damped response of the RLC circuit. The negative peaks in the voltage are over compensated for by the predicted response. The positive peaks are under compensated for. This is most likely a result of not using ideal capacitors and inductors but overall equation (3) models the response well enough to use it to study its characteristics. On the graph two periods of oscillation is labeled as well as the voltage at peaks of oscillation. 4. Amplitude in Equation (4). The amplitude in equation (4) is Vo/RC where R is not Req but rather just the 560Ω resistor is a consequence of the inductors inability to instantaneously change current. The inductor acts as a short circuit when a steady state is reached. Therefore at the moment when the square wave reaches 1V the inductors current cannot change and still acts like a short circuit. If the inductor is pulling all the current initially, only th...
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