Section 24–3 Electrical transients

(Over-damped motion / Arbitrary constants / Critical value)

In this section, the three important points are how to solve an over-damped problem, determine arbitrary constants, and its critical value. To be specific, the section could be titled as “over-damped motion” instead of “electrical transients.”

1. Over-damped motion:

“If we increase the resistance still more, we get a curve like that of Fig. 24–6, which does not appear to have any oscillations, except perhaps one (Feynman et al., 1963, section 24–3 Electrical transients).”

Feynman solves an over-damped motion problem using the equation md²x/dt² + γmdx/dt + mω₀²x = 0. By assuming a solution is e^iαt, the equation can be simplified to −α² + iαγ + ω₀² = 0 that has two solutions: α₁ = iγ/2 + i√(γ²/4 − ω₀²) and α₂ = iγ/2 − i√(γ²/4 − ω₀²). However, one may prefer the equation md²x/dt² + 2γmdx/dt + mω₀²x = 0 by changing the resistive term γ to 2γ. In this approach, the two solutions can be re-expressed as α₁ = iγ + iÖ(γ² − ω₀²) and α₂ = iγ – iÖ(γ² − ω₀²). By substituting them into e^iαt, we have x = Ae^−(γ + ω_γ)t + Be^−(γ − ω_γ)t in which ω_γ = Ö(γ² − ω₀²). One advantage of this approach is to allow the two solutions to look simpler without the two fractions ½ and ¼. Another advantage is that we can easily distinguish under-damped oscillation (γ < ω₀) and over-damped motion (γ > ω₀).

To demonstrate electric transients, Feynman suggests using an oscilloscope to monitor the voltage across the inductance L in an electrical circuit and by suddenly closing the switch S to turn on a voltage. In this case, it is an oscillatory circuit that has a steady-state response and a transient response. Furthermore, he repeats the experiment by closing the switch 60 times a second and using the oscilloscope horizontal sweep. On the contrary, we can demonstrate an electric transient by first having the switch S in a closed position for a long time and this will result in a state steady response. An electrical transient will occur when we turn off the voltage by opening the switch S. The transient response is also dependent on the presence of energy storing elements such as an inductor or a capacitor.

2. Arbitrary constants:

“Now let us discuss how we can find the two coefficients A and B (or A and A*), if we know how the motion was started (Feynman et al., 1963, section 24–3 Electrical transients).”

Feynman discusses the two arbitrary constants A and B (or A and A*) that depends on how the motion was started. Firstly, we may assume the initial conditions at t = 0 are x = x₀ and dx/dt = v₀. Next, x(t) = e^−γt/2(Ae^iω_γt + Be^−iω_γt) must be real implies that Be^−iω_γt has to be the complex conjugate of Ae^iω_γt and thus, B = A*. Thus, we may write x = e^−γt/2(Ae^iω_γt + A*e^−iω_γt) and dx/dt = e^−γt/2[(−γ/2 + iω_γ)Ae^iω_γt  + (−γ/2 − iω_γ)A*e^−iω_γt ]. Alternatively, we may let x(t) = e^−γt/2[Ccos (ω_γt + q)] such that at t = 0, the arbitrary constants are related by the equations A + B = C cos q and A − B = iC sin q. (Using e^iq = cos q + isin q and by comparing the real parts and imaginary parts.)

In the first edition, the arbitrary constant A_R was correctly stated as x₀/2, but A_I was incorrectly stated as (v₀ + γx₀/2)/2ω_γ. This is later revised as A_I = −(v₀ + γx₀/2)/2ω_γ. To get the correct sign, we can use the initial conditions as well as A = A_R + iA_I and A* = A_R − iA_I. Firstly, we have x₀ = (A + A*) = 2A_R and A_R = x₀/2 because e⁰ = 1. On the other hand, v₀ = [(−γ/2+iω_γ)A + (−γ/2 − iω_γ)A*] = (−γ/2)(A + A*) + iω_γ(A − A*) = (−γ/2)(2A_R) + iω_γ(2iA_I) = −(γ/2)(x₀) − 2ω_γA_I. Thus, we have −(v₀ + γx₀/2)/2ω_γ instead of (v₀ + γx₀/2)/2ω_γ. The final equation (24.22) for the displacement x is still correct indicates that it could be a typo mistake.

3. Critical value:

“… often appears physically as a change from oscillatory to exponential behavior when some physical parameter (in this case resistance, γ) exceeds some critical value (Feynman et al., 1963, section 24–3 Electrical transients).”

Feynman ends the section by explaining the intimate mathematical relation of the sinusoidal and exponential function. In essence, we should expect a change from oscillatory to exponential behavior when the electrical resistance or friction (γ) exceeds a critical value. As a suggestion, we can use md²x/dt² + 2γmdx/dt + mω₀²x = 0 that has a simpler critical value. Firstly, when γ is smaller than ω₀, we have an under-damped oscillation. On the contrary, when γ is greater than ω₀, we have an over-damped motion. However, Feynman did not explain that if γ is equal to ω₀, then we will have a critically damped motion such that the oscillator is stopped in the shortest time. Thus, γ is a critical value that determines the nature of motion whether it slows down sinusoidally or exponentially.

In the last paragraph, Feynman adds that “all the behavior of such a system with no external force is expressed by a sum, or superposition, of pure exponentials in time” (which was written as e^iαt). However, in the case of critically damped oscillation (γ = ω₀), the general solution in Eq. (24.20) is no longer applicable because the roots α₁ and α₂ are both equal to −γ and this apparently implies that there is only one solution, e^−γt. Importantly, another possible solution can be written in the form te^−γt based on a mathematical result from the theory of linear differential equations. Thus, it is not exactly true to say that the behavior of a system is always a sum of pure exponentials in time.

Questions for discussion:

1. Would you use the equation md²x/dt² + γmdx/dt + mω₀²x = 0 or md²x/dt² + 2γmdx/dt + mω₀²x = 0 to solve an over-damped problem?

2. How would you determine the two arbitrary constants (A and B) of x(t) = Ae^iω_γt + Be^−iω_γt? 

3. Would you explain that the behavior of an oscillator with no external force can always be expressed by a sum of pure exponentials in time?

The moral of the lesson: an oscillator may change from oscillatory (under-damped) to exponential (over-damped) behavior when some physical parameter (in this case resistance, γ) exceeds some critical value.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.