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Engineering Circuit Analysis Transient Response of Second Order Circuits Perry Carlson, Nick Szapiro, Rio Akasaka Swarthmore College Engineering Performed October 25th Turned in November 13th, 2006 Abstract The purpose of this lab was to reinforce the understanding of the transient response of second order circuits. Multiple configurations of a second order RLC circuit were built, with constant capacitor and inductor values, of 0.01µF and 112mH respectively, and varying resistances values of 300Ω, 1000Ω, 6700Ω, and 10kΩ. The following equations were obtained for each resistor value: R300, V0(t) = -0.97e-1767.3t cos(29369t -0.1231) + 0.466, underdamped response; R1000, V0(t) = -0.97e-4824cos(29051t -0.1565) + 0.462, underdamped response; R6700, V0(t) = – 23098t e-27109t – 0.972e-27109t + 0.46, critically damped response; R10k, V0(t) = – 1.0759 e-10648t – 0.13163e-116100t + 0.46, overdamped response. The internal resistance or non-ideality of the 112mH inductor was then experimentally determined, using a LC circuit, to be 41.3Ω, given the internal resistance of 50Ω of the function generator. Finally, a Multisim model of the RLC circuit was assembled with a resistor value of 300Ω to compare the experimental data. Introduction Knowledge of the transient response, or the natural response of a circuit over time, is crucial in circuitry design and performance, and its importance is highlighted in the use of RLC circuits in common electrical appliances such as radio tuners to select particular wave frequencies. This lab reinforces several concepts of engineering studied thus far, including, but not limited to: the natural or homogenous response of resistorcapacitor-inductor circuits, the step and impulse response of first and second order circuits, the behavior at steady state of capacitors and inductors, as well as the more basic Ohm’s, Kirchoff’s current and voltage, and node voltage laws. The term “second order” comes from the fact that when the transient response for voltage or current is determined, using Kirchoff’s current or voltage laws, a differential equation in which the highest order derivative is two (2) is obtained. A second order circuit contains two energy storage devices – either an inductor and a capacitor, or a combination of only capacitors or only inductors that cannot be reduced to a singular passive element. Often, inductors and capacitors are combined in series or parallel with resistors, creating resistor-inductorcapacitor (RLC) circuits. Theory INDUCTOR AND CAPACITOR An inductor is an electrical component that opposes any change in electrical current (Nilsson and Riedel, pg 216). Inductors are measured in Henrys (H), noted graphically as a coiled wire, and represented as L (Nilsson and Riedel, p.216). Figure 1: Graphical representation of an inductor (www.wikipedia.org) A capacitor is an electrical component that opposes any change in voltage. Capacitance is measured in Farads (F), noted graphically by two short parallel plates, and represented by the symbol C (Nilsson and Riedel, p.216). Figure 2: Graphical representation of an capacitor (www.wikipedia.org) The definitions of the inductor and capacitor will help in determining the steady state behavior of these circuit elements. In the inductor there is a time-varying magnetic field (caused by charge in motion), which induces a voltage in any conductor in the field. Therefore, the inductance of an inductor is a relationship of the induced voltage to the change in current with respect to time. This relationship can be stated as Equation 1 (from Nilsson and Riedel, pg 218). V=L di dt (1) Since a capacitor, composed of two conductors separated by an insulating material can store electrical charge, the separation of charge, or voltage, varies with time, and creates an electric field that also varies with time. This time-varying electric field produces a displacement current. The capacitance of a conductor is a relationship of the displaced current to the change in voltage with respect to time. This relationship can be given by Equation 2 (from Nilsson and Riedel, p.225). I=C dv dt (2) Employing Equations 1 and 2, one can conclude that at steady state an inductor acts like a short circuit and a capacitor acts like an open circuit. TRANSIENT RESPONSE OF SECOND ORDER CIRCUIT The circuit shown in Figure 3 is of second order due to the two energy storage devices (inductor and capacitor), and the second order differential equation one would derive to determine the natural response. d 2i ⎛ R ⎞ di i +⎜ ⎟ + =0 2 dt LC ⎝ L ⎠ dt (3) Figure 3: Electric circuit diagram for the second order series RLC circuit constructed during the lab. Courtesy of Lynne Molter. An alternative method, impedance, was used in finding the transient response of Vo (voltage of the capacitor) in the above RLC circuit, given a step input voltage, Vi (Figure 4). Figure 4 displays the input square wave from the function generator used in this experiment Since the capacitor (impedance, Z = 1 ) is in parallel with the inductor, L, (Z = sL), sC which is in series with the resistor, R, (Z = R), the impedance is: ⎛ 1 ⎞ ( R + sL) ⎜ ⎟ ⎝ sC ⎠ Zab ( s ) = 1 R + SL + sC (4) Simplifying, R s+ ( R + sL) R + sL L L Zab ( s ) = = = 2 R 1 ⎞ sCR + s LC + 1 LC s 2 + s + 1 ⎛ ⎜ R + sL + ⎟ sC L LC sC ⎠ ⎝ (5) Since Zab = ∞ as it is an open circuit, the denominator of Zab (s) is set to zero, yielding the characteristic equation: s2 + s R 1 + =0 L LC (6) Employing the quadratic equation to solve for s produces the characteristic equation: 2 −R 1 ⎛ R ⎞ ± ⎜ s= ⎟ − 2L ⎝ 2 L ⎠ LC (7) THE CHARACTERISTIC EQUATION AND CRITICAL DAMPING The characteristic equation can be rewritten as: s = −α ± α 2 − ω 2 where α, the damping factor or neper frequency, is ω0, the resonant radian frequency, is R and 2L 1 LC Consider the other suggested standard form of the characteristic equation: s2 + 2αs + ω02 = s2 + 2ζω0s + ω02 From this one can state: ζ. the damping coefficient, is α ω (8) Therefore when ζ = 1, α must equal ω0. Substituting in R for α, and 2L 1 LC for ω0, one obtains: R = 2L 1 LC It can therefore be said that when ζ = 1, the circuit is critically damped. From Equation 7 it becomes evident there are three categories for this response: overdamped, critically damped, and underdamped, each having two solutions. The complete transient response is not only dependent on the natural response of a circuit, but also the response due to the input, referred to as the particular response. Therefore the output can be described as follows: V0 complete = V0 homogenous + V0 particular 1) OVERDAMPED 2 1 ⎛ R ⎞ If ⎜ , s is two unique real numbers: the sum and difference of two real ⎟ > LC ⎝ 2L ⎠ numbers. Inserting the test solution (equation 8) into the differential equation (equation 3), yields a second order homogenous or natural test solution of V 0 = Ae s1t + Be s2t where: 2 s1= −R 1 ⎛ R ⎞ + ⎜ ⎟ − 2L ⎝ 2 L ⎠ LC 2 −R 1 ⎛ R ⎞ s2 = , − ⎜ ⎟ − 2L ⎝ 2 L ⎠ LC A and B are real constants, determined by the initial conditions. a) v0 (0 + ) = v0 (0 − ) = −0.5 V since instantaneous voltage change across a capacitor would require infinite current and is therefore impossible; dv0 + ic (0+ ) iL (0+ ) b) = = 0 since L and C are in parallel and the capacitor acts (0 ) = dt C C as an open circuit so there is no current flow. The output voltage, Vo, will decay to a final voltage value without oscillation, or is an overdamped response. Using R = 10 kΩ, L = 112mH, and C = 10−8 F, the solutions s = -11474 and -77810 were obtained. Substituting, V0(hom) (t ) = Ae−11474t + Be−77810t From the square input voltage, V 0( part ) = 0.5 V, as no current will flow when the capacitor has been charged (reached steady state). The complete response is as follows: V0 (t ) = Ae −11474t + Be −77810t + 0.5 V for t > 0 until Vin switches to -0.5 V. Using the initial conditions one can solve for A and B. A + B + 0.5 = −0.5 → A + B = −1 0 = −11474 A − 77810 B Solving, A = -1.173 and B = 0.173. The complete solution is therefore: V 0(t ) = −1.173e −11474t + 0.173e−77810t + 0.5 V Graph 1: Plot of overdamped voltage vs. time using Graphing Calculator (9) 2) CRITICALLY DAMPED 2 −R −R 1 ⎛ R ⎞ If ⎜ , s= . Using the same procedure to find a test ± 0= ⎟ = 2L 2L LC ⎝ 2L ⎠ solution for the homogenous or natural response of a differential equation as in the over damped response, one yields: V 0 = Ate st + Be st where s = −R 2L a) v0 (0 + ) = v0 (0 − ) = −0.5 V as before; b) dv0 + i (0+ ) iL (0+ ) = = 0 as before. (0 ) = c dt C C The output voltage, V0, will decay until on the verge of oscillation about its final value, or is a critically damped response. Using a resistor with resistance equivalent constrained to α=ω0, or 2L LC or 6693Ω, the solution to s was -29881. Substituting s into the general homogenous test solution yields: V 0(hom)(t ) = Ate −29881t + Be−29881t From the square input voltage, V0 ( part ) = 0.5 V, as no current will flow when the capacitor has been charged (reached steady state). V 0(t ) = Ate −29881t + Be−29881t + 0.5 for t > 0 until Vin switches to -0.5 V. Using the initial conditions and solving for A and B: −0.5 = B + 0.5 → B = −1 0 = A − 29881B → A = −29881 Thus, the complete solution is V 0(t ) = −29881te −29881t − e −29881t + 0.5 V (10) Graph 2: Plot of critically damped voltage vs. time using Graphing Calculator 3) UNDERDAMPED 2 2 −R 1 1 ⎛ R ⎞ ⎛ R ⎞ + −1 ⎜ If ⎜ , we know s = . Using the same ⎟ − ⎟ < 2L LC ⎝ 2 L ⎠ LC ⎝ 2L ⎠ procedure to find a test solution for the homogenous or natural response for a differential equation as in the over and critically damped response, Euler’s Identity Me jθ = cos θ + j sin θ , and a trigonometric identity involving phase shifts, one yields: V 0 = Ae st cos(ωd t + φ ) where: −R s= 2L 2 ωd = 1 ⎛ R ⎞ ⎜ ⎟ − ⎝ 2 L ⎠ LC a) v0 (0 + ) = v0 (0 − ) = −0.5 V as before; b) dv0 + i (0+ ) iL (0+ ) = = 0 as before. (0 ) = c dt C C The output voltage, V0, will decay until it oscillates about its final value- the rate at which this oscillation takes places is fixed by the damped radian frequency, ωd, and the rate at which the oscillations subside is determined by α. (Nilsson, Riedel 369), or is an underdamped response. s, using the resistor value of 300Ω was then calculated to be equivalent to -1339 and ωd to be equal to 29851. Substituting into the homogenous test solution yields: V 0(hom)(t ) = Ae −446t cos(29851t + ϕ) From the square input voltage, V 0( part ) = 0.5 V, as no current will flow when the capacitor has been charged (reached steady state). V 0(t ) = Ae −446t cos(29851t + ϕ) + 0.5 V for t > 0 until Vin switches to -0.5 V. Using the initial conditions one can solve for A and φ . One then has two unknowns and the two following equations: 0.5 + A cos(ϕ ) = −0.5 → A cos(ϕ ) = −1 −1339 A cos(ϕ ) − 29851A sin(ϕ ) = 0 = 1339 + 29851tan(ϕ ) Solving, A = -1 and ϕ = -0.0448 radians and the complete solution is therefore: V0 (t) = -1e-1339t cos(29851t − 0.0448) + 0.5 V (11) Graph 3 (previous page): Plot of underdamped voltage vs. time using Graphing Calculator NON-IDEALITY OF THE INDUCTOR When determining the transient response in an RLC circuit it is assumed the inductor is ideal and has zero internal resistance. However, since inductors are usually composed of winded metal coils, the resistance encountered in the wires converts some of the electric current running through them to heat. Figure 5 shows the resistive qualities of a non-ideal inductor. Image courtesy of AllAboutCircuits.com NON-IDEALITY OF FUNCTION GENERATOR Though the internal resistance of electrical instruments is generally ignored, it is important to take them into consideration when taking measurements that involve small changes in resistance. The function generator used in this experiment had an internal resistance of 50Ω1. Figure 6 shows the internal resistance present in a function generator. Image courtesy of Erik Cheever. 1 Ybarra. Gary. “Lab 2: Operation of the Digital Instruments and Basic Measurements” http://www.educatorscorner.com/media/Exp90b.doc Accessed November 9th, 2006. KALEIDAGRAPH FITTING This lab relied extensively on the use of Kaleidagraph to fit exponential curves to the data extracted from the lab. In order to obtain the most accurate best-fit curve, data points that were not of concern to the particular segment (the half-period of the square wave during which the capacitor voltage decayed) were masked using the Mask function. The best fit equations (see Appendix) were modeled after the calculated (expected) equations, since though Kaleidagraph’s best-fit is capable of finding most best-fit curves given the correct parameters, it was nonetheless necessary to provide good estimates for the constants in the equations. Particular attention was made in minimizing the error of the constants obtained using the initial conditions, while errors with the solutions to the characteristic equation did not necessarily have to have small error since they could be explained by the non-ideality of the inductor. Procedure The circuit shown in Figure X was assembled, where Vi was connected to the function generator, V0 was connected to the oscilloscope, and another channel of the oscilloscope was connected across Vi. The function generator was set to a 1-volt peak to peak square wave with a frequency of approximately 100Hz. The circuit also required a variable resistor box, a 0.01µF capacitor and a 112mH inductor. Figure 3: Electric circuit diagram for the second order series RLC circuit constructed during the lab. Courtesy of Lynne Molter. After using the Agilent function in Excel to extract the output voltage provided through the oscilloscope, the resistor value was varied by twisting the knob on the resistor box to 300, 1000, 3000 and 10kΩ, as well as RC (the resistance calculated for critical damping) and for each value, the outputs were then printed. Rc was calculated by employing the definition for critical damping and equating α2 to ω02. To investigate the non-ideality of the inductor, the resistor was then replaced by a short circuit to create an LC circuit, and the data points were extracted through Agilent. Lastly, the DC resistance of the inductor was measured using an ohmmeter, and a RLC measuring tool was used to find the value of resistance and inductance of the same inductor. Figure 7 shows the layout of the circuit modeled using MultiSim. Note the oscilloscope is reading both the voltage source and the voltage across the capacitor. Results The following table outlines the expected values for the damping factor, α, and the resonant radian frequency, ω, and compares them in order to determine the nature of their response. Table 1 Expected values for frequencies and damping for varying resistors: Expected values Resistor α (rad/s) 300Ω 1000Ω 6700Ω 10000Ω 1339 4464 29910 44643 ω0 (rad/s) ωd (rad/s) 29881 29851 29545 N/A N/A α compared to ω02 α2 < ω02 α2 < ω02 α2 ≈ ω02 α2 > ω02 Damping Underdamped Underdamped Critically damped† Overdamped † We suggest later in this lab that this value, due to non-idealties in the inductor and the internal impedance in the function generator, provides a overdamped response. Table 2: Comparison of best-fit to expected equations Resistor Equation obtained using best fit in Kaleidagraph Expected equation 300Ω y = -0.97e-1767.3x cos (29369x -0.1231) + 0.466 V0 (t) = -1e-1339t cos(29851t − 0.0448) + 0.5 1000Ω y = -0.97e-4824x cos (29051t -0.1565) + 0.462 V0 (t) = -1e-4464t cos(29545t − 0.1499) + 0.5 6700Ω y = – 23098x e-27109x – 0.972e-27109x + 0.46 V 0(t ) = −29881te −29881t − e−29881t + 0.5 10000Ω y = − 1.0759 e -10648x + 0.13163 e -116100x + 0.46 V 0(t ) = −1.173e −11474t + 0.173e−77810t + 0.5 The above table compares the equation obtained using Kaleidagraph’s best-fit curve equation with the expected equation. 300 ohms Graph 4 shows the sinusoidal damping that can be seen in an underdamped RLC circuit. The best-fit equation that was derived using Kaleidagraph was y = 0.466 - 0.97e-1767.3x cos(29369x -0.1231) Graph 5 shows Kaleidagraph’s best fit curve equation applied to the data for MultiSim’s output of a second-order RLC circuit. Note the abundance of data points- MultiSim’s output consisted of over 1400 data points, of which approximately 500 were used. Graph 6 shows the square wave input and output voltage as seen by the oscilloscope for the 300Ω and the 1 volt peak-to-peak square wave with a frequency of about 100Hz. 1000 ohms Graph 7 shows the sinusoidal damping that can be seen in an underdamped RLC circuit. The best-fit equation that was derived using Kaleidagraph was y = 0.462 - 0.97e-4824xcos (29051x -0.1565) Graph 8 shows the square wave input and output voltage as seen by the oscilloscope for the 1000Ω and the 1 volt peak-to-peak square wave with a frequency of about 100Hz. 6700 ohms, Resistor value at critical damping Graph 9 shows the sinusoidal damping that can be seen in a RLC circuit close to critical damping. Due to non-idealities and the internal impedance of the function generator, however, we note that the graph represents an overdamped respons. The best-fit equation that was derived using Kaleidagraph was y = 0.46 – 23098x e-27109x – 0.972e-27109x Graph 10 shows the square wave input and output voltage as seen by the oscilloscope for the 6700Ω and the 1 volt peak-to-peak square wave with a frequency of about 100Hz. The graph obtained using the 6700Ω resistor looks similar to the graph of the overdamped response using the 10kΩ because even though the resistor value at critical damping was found using the equivalence R = 2L 1 (that is, with a resistor value of LC 6693.3Ω), the resistor value at critical damping should be less because of the resistance in the inductor and the function generator. 10k ohms Graph 11 shows the sinusoidal damping that can be seen in an over damped RLC circuit. The best-fit equation that was derived using Kaleidagraph was y = − 1.0759 e -10648x + 0.13163 e -116100x + 0.46 Figure 12 shows the square wave input and output voltage as seen by the oscilloscope for the 10kΩ and the 1 volt peak-to-peak square wave with a frequency of about 100Hz. Graph 13: This output and best-fit curve from Kaleidagraph shows the damping in a LC circuit. The best fit equation is y = -0.865e-431.93x cos(29384x -0.17144) + 0.47 Graph 14 shows the sinusoidal damping that can be seen in an LC circuit with a 1 volt peak-topeak square wave. Discussion We can examine the non-ideality of the circuits by comparing the following values: Circuit parameters for 300Ω s ω0 A φ Expected Measured MultiSim Error 1339 29851 -1 -0.0448 1767.3 29369 -0.97 -0.1231 1310 29633 -1.0019 -0.0397 32% 1.6% 3% 175% where V0 (t) = Ae-st cos( w0t + ϕ ) + V p in the underdamped response Circuit parameters for 1000Ω s ω0 A φ Expected Measured Error 4464 29545 -1 -0.1499 4824 29051 -0.97 -0.1565 8.1% 1.7% 3% 4.4% where V0 (t) = Ae-st cos( w0t + ϕ ) + V p in the underdamped response Circuit parameters for 6700Ω s A B Expected Measured Error 29881 -29881 -1 27109 -23098 -0.972 9.3% 23% 2.8% where V 0(t ) = − Ate− st − Be− st + Vp in the critically damped response Circuit parameters for 10000Ω s1 s2 A B Expected Measured Error 11474 77810 -1.173 -0.173 10648 116100 -1.0759 -0.13163 7.2% 49% 8.3% 24% where V 0(t ) = Ae− s1t − Be− s2t + Vp in the overdamped response Table 3: Inductance and Resistance of Inductor Inductor Inductance (mH) 116mH Resistance (Ω) Ohmmeter RLC meter 7.0 6.4 The above table shows the inductance and resistance values obtained using the RLC meter and the ohmmeter. As the capacitor value is increased, the resonant frequency of the circuit decreases, a result of the equation 1 LC . When shorting the resistor in the RLC circuit, the resulting circuit is still an RLC circuit since the non-ideal inductor has an internal resistance, as does the function generator, as previously ascertained. The assumption was made that the combined internal resistance would be much less than the 6693Ω characteristic of critical damping, since the internal resistance of circuit elements and instruments are kept at a minimum to lessen their influence on the rest of the circuit. Therefore, using the relationship determined in the theory that α = R (note that R is the equivalent resistance of the non-ideal elements) and 2L with the values for α from the curve fitting of the data points using Kaleidagraph, a measure for the non-ideality of the internal resistances of the circuit elements can be obtained using s = α = R that is, Requivalent = 2sL 2L Considering the data from the 1000Ω resistor, the measured value for the roots of the characteristic equation is s = 4824. Hence, Requivalent = 1081Ω. So, Rint ernal = 1081 − 1000 = 81 Ω. Considering the data from the 300 Ω resistor, s = 1767.3 → Rint ernal = 96 Ω. For the 0Ω (LC circuit with internal resistance) case, s = 431.93 → Rint ernal = 97 Ω. The other two resistor values, 6700Ω and 10kΩ, were not taken into consideration because they did not lead to coherent results: both overdamped cases gave a measured resistor value that was less than the expected value, suggesting consistent errors due to the relative negligible effect of the internal resistances, which are considerably smaller. The average value of the root of the characteristic equation was 91.3Ω. The values for Rint ernal included the internal resistances of all of the circuit elements. Noting the negligible effect of the capacitor on the value of Requivalent , Rint ernal for the inductor was calculated to be 41.3Ω since the function generator has an internal resistance of 50Ω 2. ERROR CONSIDERATIONS The value of 41.3Ω, however, is altogether inconsistent with the measured resistance of 7Ω, and the error can therefore be attributed to lack of sufficient data points or a higher internal impedance of the function generator. Noting the general trend that the error in this lab is considerably high, which, considering the data extraction using Agilent, may suggest the possibility that insufficient data points did not allow for proper curve-fitting using Kaleidagraph’s function. Furthermore, in choosing the data range on the oscilloscope screen, a sufficient range might not have been chosen, leading to the lack of necessary or pertinent data points. It is also important to note that the assumption was made that, because the voltage across a capacitor cannot change instantaneously, the capacitor voltage before the step change of the source from -0.5V to 0.5V was equal to the capacitor voltage immediately thereafter, that is vc(0-) = vc(0-) = 0.5 However, it is also assumed that the capacitor has discharged to its steady state in the time of half a period, which introduces an error. 2 Cheever, Erik. Personal communication, Ybarra. Gary. “Lab 2: Operation of the Digital Instruments and Basic Measurements” http://www.educatorscorner.com/media/Exp90b.doc Accessed November 9th, 2006 The reading from the RLC meter suggested that the accurate value for the inductance used should have been 116mH rather than the labeled value of 112mH. The other discrepancies that were present in this experiment could have been prevented if more data points were retrieved in Agilent, as only 250 were taken, where 500 or 1000 points could have been taken. The internal resistance of the function generator could have been confirmed by connecting the input and output of the function generator with a variable resistor close to 50Ω, and measuring the output- if the function generator is set to a low voltage, the oscilloscope should measure half the initial voltage due to voltage divider3. Figure X: Using a voltmeter, the function generator set to a sinusoidal frequency of 50Hz, a 1.6V output, and a load resistor of 50Ω, the resistance of the function generator can be easily determined.4 Conclusion and Future Work It was determine from this lab that the non-ideality of the circuit lent itself to an average additional resistance of 91.3Ω. With the RLC meter, the resistance of the inductor was found to be 6.3Ω, and therefore it could be concluded that other circuit elements, including the function generator and possibly the oscilloscope, provided the remaining unaccounted resistances. It was also possible to approximate the non-ideality of the inductor by assembling a LC circuit with the knowledge that the non-idealities in the form of resistances contribute to what can be then considered to be a RLC circuit. Furthermore, in using several different resistor values, it was possible to qualitatively examine the effect of their values on the overall transient response of the circuit, with 3 Sannuti, P. “Step Response of a RLC series circuit” The State University of New Jersey Rutgers. http://www.ece.rutgers.edu/~panayot/224_06/PEEII_Expt_5_06.pdf Accessed November 9th, 2006 4 “Equivalent Equipment Circuits”, University of Minnesota Duluth, <http://www.d.umn.edu/~snorr/ece2006s5/Lab7b.pdf> Accessed November 11th, 2006 sinusoidal (under)damping occurring for resistor values less than 6693Ω. Τhe value of the damping factor, α, contributed to determining whether damping took place and what type of response could be expected when comparing the value to the neper frequency, ω0. Though this lab was well conducted and successfully completed, more in-lab time devoted to the discussion of the calculated decay rates of the output voltage for the circuit would have helped in a more thorough understanding of the circuit. Consideration for the number of data points extracted through Agilent would also have been helpful in fitting a more precise curve using Kaleidagraph. Acknowledgements Grateful acknowledgement is made to Lynne Molter and Ed Jaoudi, who contributed to the completion of this lab, as well as to Erik Cheever, for answering queries regarding the content of this report. Appendix The following are the best-fit equations that were used in Kaleidagraph’s curve-fit function. The equations were modeled after the expected equations calculated previously. Note for both the critically damped and the overdamped curve-fit equations it would have been better to not include the 0.46V value in the fit. Underdamped curve-fit equation m1 + m2*exp(-m3*M0)*cos(m4*M0 + m5); m1=0.5; m2=-1;m3=4464;m4=30000;m5=-0.1483 Critically damped curve-fit equation M1*x*exp(-m2*x) + m3*exp(-m2*x) + 0.46;m1=-29881;m2=29881;m3=-1; Overdamped curve-fit equation m1*exp(-m2*x) + m3*exp(-m4*x) + 0.46;m1=1.173;m2=11474;m3=0.173;m4=77810 Bibliography Nilsson and Riedel, Electrical Circuits, 7th ed. Chapters 6 and 7 (Nilsson and Riedel) Molter, Lynne. Engineering 11 2004 Electric Circuit Analysis Lab 4 – Background Second Order Systems, <http://www.swarthmore.edu/NatSci/lmolter1/courses/e112005/Labs/E11L4/E11-02%20Lab4_background.html> Accessed November 10th, 2006. Sannuti, P. “Step Response of a RLC series circuit” The State University of New Jersey Rutgers. http://www.ece.rutgers.edu/~panayot/224_06/PEEII_Expt_5_06.pdf Accessed November 9th, 2006 Ybarra. Gary. “Lab 2: Operation of the Digital Instruments and Basic Measurements” http://www.educatorscorner.com/media/Exp90b.doc Accessed November 9th, 2006