There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

Origins of Newton's laws of motion

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

## Friday, June 28, 2013

### 16f. Differential equation solution for transient response and SPICE simulation

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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.

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

.     (1)

Differentiating and rearranging gives:

,     (2)

where Vres' is defined as the derivate of Vres 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 LC:

.     (3)

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

.     (4)

Finally, we have the definition of  Vres'   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 Vres' , IS and Vres. 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 Lc 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: τamplitude= 2QL/ω = 12.3, 7.70 and 3.47seconds, respectively. Comparing the above graphs, we see that higher loaded QL's result in longer rise and decay times. As indicated in Fig. 2, a coupling inductor Lc 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 Lc = 8H. One can see that the results are the same as in the middle graph in Fig. 2.

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