Resonances, waves and fields: 16f. Differential equation solution for transient response and SPICE simulation

Friday, June 28, 2013

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

Differential equations for pulsed excitation of the resonator
Numerical solution of differential equations - graphs
SPICE simulation of pulsed response

1. Differential equations for pulsed excitation of the resonator

Fig. 1. Circuit being analyzed.

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 R₀ and L_C) equals the current flowing down through R, L and C:

. (1)

Differentiating and rearranging gives:

, (2)

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 R₀ and L_C:

. (3)

This can be rearranged with the derivative on the left side:

. (4)

Finally, we have the definition of V_res' as noted above:

. (5)

The three boxed equations (2), (4) and (5) form a set of three linear first-order 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

Fig. 2. Responses of the above resonant circuit to bursts of a sinusoidal driver. Parameters used were the same as in Fig. 2 in posting 3.16c (ω₀ = 2.5rad/sec, Q₀ = 20 , Z₀ = R₀ = 5Ω, C = 0.08F, L = 2H, and R = 100Ω). The coupling inductance L_c was set at 16, 8 and 4 Henries for the three graphs. The drive frequencies used were 2.65, 2.78 and 3.0rad/sec. Fig. 2 indicates that these frequencies provided the peak responses to the coupling inductance used in each graph. The time constants for the rise and decay of the envelope (in green) in the three cases are: τ_amplitude= 2Q_L/ω = 12.3, 7.70 and 3.47seconds, respectively.

Comparing the above graphs, we see that higher loaded Q_L's result in longer rise and decay times. As indicated in Fig. 2, a coupling inductor L_c equal to 8H results in the largest eventual amplitude (and is almost unity coupling).

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.

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.