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16f. Differential equation solution for transient response and SPICE simulation
Keywords: differential equation, resonator, pulsed excitation, numerical solution, octave, SPICE simulation
Topics covered in this posting
 The response of the transmission line and resonator to a burst of excitation is analyzed.
 The governing differential equations are discussed and numerically solved.
 A standard electrical engineering tool, SPICE, is used to simulate the response to a burst of excitation. The result is the same as the differential equation solution.
Contents of this posting
1. Differential equations for pulsed excitation of the resonator  

Figure 1, at right, shows the circuit to be analyzed (it is the same as we've worked on in a number of previous postings). We start with the fact that the current flowing into the resonator (from the source and through R_{0} and L_{C}) equals the current flowing down through R, L and C: Differentiating and rearranging gives: where V_{res}' is defined as the derivate of V_{res} shown in (5) below. We next write an equation stating that the voltage across the resonator will equal the source voltage minus the voltage drop across R_{0} and L_{C}: This can be rearranged with the derivative on the left side: Finally, we have the definition of V_{res}' as noted above: The three boxed equations (2), (4) and (5) form a set of three linear firstorder differential equations governing the three coupled unknowns V_{res}' , I_{S} and V_{res}. These can be solved by various methods. Below we show the graphical results obtained using Octave's differential equation solver lsode. 
2. Numerical solutions to differential equations  graphs  


3. Graphical results of SPICE simulation of the electrical circuit  

Fig. 3. As a check, the circuit above was simulated in a standard SPICE program (a standard electrical engineering tool for analyzing circuits). See this tutorial for more on SPICE. The simulated circuit is shown at the left. Below we see the results of the SPICE simulation for the voltage across the resonator as a function of time. We used the same component and parameter values as given in the caption of Fig. 2 with L_{c} = 8H. One can see that the results are the same as in the middle graph in Fig. 2.  
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