Section 23–3 Electrical resonance

(Circuit elements / Mechanical-electrical analogies / Electrical symbols)

In this section, Feynman discusses circuit elements, mechanical-electrical analogies, and electrical symbols that are related to the concept of electrical resonance.

1. Circuit elements:

“The three main kinds of circuit elements are the following. The first is called a capacitor… (Feynman et al., 1963, section 23–3 Electrical resonance).”

Feynman discusses an example of a capacitor that has two plane metallic plates spaced a very short distance apart by an insulating material. He says that all we need to know is that the potential difference across a capacitor is proportional to the charge: V = q/C; where C is the capacitance of the object. However, in The Art of Electronics, Horowitz and Hill (1989) write: “there’s a ‘memory’ effect (known as dielectric absorption), which is rarely discussed in polite society: if you charge a capacitor up to some voltage V₀ and hold it there for a while, and then discharge it to 0 V, then when you remove the short across its terminals it will tend to drift back a bit toward V₀ (p. 28).” In addition, real capacitors behave like they have resistance, inductance, and some frequency-dependent parallel resistance.

Feynman explains that an inductor is a circuit element that is analogous to the mass of an object. It is a coil which builds up a magnetic field within itself when there is a current in it and the coil’s changing magnetic field results in a voltage that is proportional to dI/dt. On the other hand, Horowitz and Hill (1989) explain that “inductors are magnetic devices, in which two things are going on: current flowing through the coil creates a magnetic field aligned along the coil’s axis; and then changes in that field produce a voltage (sometimes called ‘back EMF’) in a way that tries to cancel out those changes (an effect known as Lenz’s law) (p. 28).” Importantly, an ideal inductor has no resistance and no power is dissipated within the coil, whereas a real inductor has resistance and capacitance (due to a separation between the wires of the coil).

2. Mechanical-electrical analogies:

“If we think of the charge q on a capacitor as being analogous to the displacement x of a mechanical system, we see that the current, I = dq/dt, is analogous to velocity… (Feynman et al., 1963, 23–3 Electrical resonance).”

Feynman suggests that the charge q on a capacitor is analogous to the displacement x of a mechanical system and the current dq/dt is analogous to velocity. This is also known as Maxwell’s analogy or “force-voltage, velocity-current” analogy. In essence, the force on a mechanical element is analogous to the voltage (or electromotive force) on a circuit element, and the velocity of the mechanical element is analogous to the electric current through the circuit element. The analogy is derived from the similarity of the equations z = force/velocity and Z = voltage/current where z is the mechanical impedance and Z is the electrical impedance. However, Feynman could have clarified a limitation of this analogy: circuit elements connected in series are analogous to the corresponding mechanical elements connected in parallel, or vice versa. (Thus, he has inappropriately used the phrase exact analogy in chapter 25.)

Feynman mentions that the quantity R+iωL+1/iωC is a complex number, and it is called the complex impedance Z that is used in electrical engineering. One may explain that the concept of electrical impedance is seen as a ratio of the cause (voltage) to its effect (current). To have a better understanding of mechanical-electrical analogies, one should distinguish Maxwell’s analogy from Firestone’s analogy. Firestone (1933) suggests another analogy such that the force and velocity of a mechanical system are analogous to the Kirchhoff’s laws of current and voltage. The analogy is derived from the similarity of the equations ẑ = velocity/force and Z = voltage/current where ẑ is the reciprocal of the mechanical impedance.

3. Electrical symbols:

“The difficulties of science are to a large extent the difficulties of notations, the units, and all the other artificialities which are invented by man, not by nature (Feynman et al., 1963, section 23–3 Electrical resonance).”

In electrical engineering, the symbol j is commonly used to denote √−1 (instead of using i). Feynman elaborates that i must be the current for electrical engineers, but they get into trouble when they use j to denote current density. However, this is not true because electrical engineers use J (instead of j) to denote current density. In 1893, Charles Proteus Steinmetz introduces the concept of phasor and proposes the use of the symbol j because he found it useful in doing efficient calculations on alternating currents (Araújo & Tonidandel, 2013). One may add that the imaginary number j corresponds to a rotation of 90^o or a phase difference between the electric current and the electric potential difference.

Feynman explains that the difficulties of science are to a large extent the difficulties of notations, the units, and all the other artificialities, which are invented by man, not by nature. In his biography, Feynman (1997) writes that “I thought my symbols were just as good, if not better, than the regular symbols--it doesn’t make any difference what symbols you use-but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, ‘What the hell are those?’ I realized then that if I’m going to talk to anybody else, I’ll have to use the standard symbols, so I eventually gave up my own symbols (p. 24).” In a sense, the difficulties of notations are worsened because Feynman also likes to invent his own symbols.

Questions for discussion:

1. How would you define a capacitor and an inductor?

2. What are the limitations of the Maxwell’s analogy and Firestone’s analogy?

3. Do electrical engineers really get into trouble because of the electrical symbol j?

The moral of the lesson: the difficulties of science are to a certain extent due to difficulties of notations, the units, and all the other artificialities, which are invented by man.

References:

1. Araújo, A. E. A. D., & Tonidandel, D. A. V. (2013). Steinmetz and the Concept of Phasor: A Forgotten Story. Journal of Control, Automation and Electrical Systems, 24(3), 388-395.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

3. 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.

4. Firestone, F. A. (1933). A new analogy between mechanical and electrical systems. The Journal of the Acoustical Society of America, 4(3), 249-267.

5. Horowitz, P. & Hill, W. (1989). The Art of Electronics (Second ed.). New York: Cambridge University Press.

Section 23–2 The forced oscillator with damping

(Complex factor R / Phase angle θ / Quality factor Q)

In this section, Feynman discusses a complex factor R, phase angle θ, and quality factor Q of a forced oscillation.

1. Complex factor R:

“There is no technical name for this factor, no particular letter for it, but we may call it R for discussion purposes: R = 1/m(ω₀²−ω²+iγω)… (Feynman et al., 1963, section 23–2 The forced oscillator with damping).”

Feynman says that there is no technical name for the factor R = 1/m(ω₀²−ω²+iγω), but we may call it R for discussion purposes. By using this factor R, the amplitude of the response becomes F^R in which F^ is a complex force. However, in a mechanics textbook, Barger and Olsson (1973) call R a complex factor and clarify that it is a time-independent factor. In the first edition of this textbook, the equation of motion is dz²/dt² + 2g(dz/dt) + w₀²z = (F₀/m)e^iwt and they let R = 1/m(w₀²-w²+2igw) = re^-iq. This allows them to have z = RF₀e^iwt that is similar to Feynman’s approach. In the second edition, they let R = w₀²-w²+2igw and this allows them to have z = (f/R)e^iwt in which f = F₀/m. In a sense, this revised approach is better because the R factor is simpler and it also helps to understand the phase angle better (to be discussed later).

Feynman defines R = re^iq and calls r the magnification factor because it represents the size of the response. Furthermore, ρ² = 1/m²[(ω²−ω₀²)²+γ²ω²] is proportional to the square of the amplitude or the energy of the oscillator. Initially, Barger and Olsson (1973) defines R = re^-iq and thus, the magnification factor r² = |R|² = 1/m²[(w₀² - w²)² + 4g²w²]. In the second edition, Barger and Olsson (1995) defines R = re^iq and thus, r² = |R|² = (w₀²-w²)²+4g²w². The magnification factor r depends mainly on the relative magnitude of the natural frequency w₀ and the driving frequency w. However, it is not absolutely necessary to use the symbol r or r, but it is a shorthand that saves time, instead of writing (w₀² - w²)²+g²w² or (w₀²-w²)²+4g²w² again and again.

2. Phase angle θ:

“Therefore ρ and θ represent the size of the response and the phase shift of the response (Feynman et al., 1963, section 23–2 The forced oscillator with damping).”

According to Feynman, if we write 1/R = 1/ρe^iθ = (1/ρ)e^−iθ = m(ω₀²−ω²+iγω), then tan θ = −γω/(ω₀²−ω²). However, it should be easier to understand the derivation of phase angle θ if we choose ρe^iθ = m(ω₀²−ω²+iγω). We may explain that ρe^iθ = ρ(cos θ + isin θ) = m(ω₀²−ω²+iγω) or equivalently, ρ(cos θ) = m(ω₀²−ω²) and ρ(isin θ) = imγω by comparing the real and imaginary part. This allows us to easily deduce that tan θ = mγω/m(ω₀²−ω²) = γω/(ω₀²−ω²). Note that some textbook authors may prefer tan θ = 2γω/(ω₀²−ω²) because their resistive force is 2γv instead of γv. In addition, due to different definitions of θ, we should carefully distinguish tan θ = γω/(ω₀²−ω²) and tan θ = γω/(ω²−ω₀²) that involves a difference in sign.

Feynman explains that the phase angle θ is minus because tan (−θ) = −tan θ. He adds that a negative value for θ results for all ω and this corresponds to the displacement x lagging the force F. However, one may change tan θ = −γω/(ω₀²−ω²) into tan θ = γω/(ω²−ω₀²) without mentioning the minus sign and still explain that the displacement is lagging the force. Whether the phase angle is positive or negative is also dependent on your choice, e.g., let x = x₀e^i(ωt+θ) or x = x₀e^i(ωt-θ). More important, one has a choice to either say system A leads system B by q or system A lags system B by (2p - q). We choose to say that the force leads the displacement by the phase angle θ because we consider the force is the cause and the displacement is the effect that is lagging. The “delay” of the displacement is dependent on the magnitude of the resistive force.

3. Quality factor Q:

“As another measure of the width, some people use a quantity Q which is defined as Q = ω₀/γ (Feynman et al., 1963, section 23–2 The forced oscillator with damping).”

Feynman mentions that some people use a quantity Q which is defined as a measure of the frequency width (ω₀/γ). This quantity Q is commonly known as the “quality factor,” but Johnson who originates the term, says that it did not stand for quality (Green, 1955). He proposes the symbol Q to represent the ratio of reactance to effective resistance in a coil or anything else because the other letters of the alphabet had been used. In chapter 24, Feynman adds that many have tried to define Q in the simplest and most useful way such that various definitions differ a bit from one another, but they are in agreement if Q is very large. Interestingly, the term Q can be used to describe a characteristic of many physical systems such as a resonant circuit, a spectral line, a mechanical vibrator, and a bouncing ball.

Feynman explains that the narrower the resonance, the higher the Q, e.g., Q = 1000 means a resonance whose width is only 1000th of the frequency scale. In chapter 24, he provides another definition of Q: the mean stored energy is multiplied by 2π and is divided by the work done per cycle. One may clarify that we can define the Q factor in terms of energy of instead of frequency and this is why it is also known as the “storage factor.” This factor is dependent on the presence of resistive forces that affect the energy stored during an oscillation. Alternatively, Kleppner and Kolenkow (1973) explain that the degree of damping of an oscillator is specified by the quality factor and defines Q = (energy stored in the oscillator)/(energy dissipated per radian).

Questions for discussion:

1. Would you prefer to define R = 1/m(ω₀²−ω²+iγω) or simply R = (ω₀²−ω²+iγω)?

2. Would you prefer to have tan θ = γω/(ω₀²−ω²), tan θ = γω/(ω²−ω₀²), or tan θ = 2γω/(ω₀²−ω²)? 

3. Would you describe Q as a quality factor or storage factor?

The moral of the lesson: we would really appreciate the magic of the “complex” method that can obtain the response x to a given force F by comparing with the working needed if we use the old straightforward way, the “trignometric” method.

References:

1. Barger, V. D., Olsson, M. G. (1973). Classical mechanics: a modern perspective. New York: McGraw-Hill.

2. 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.

3. Green, E. I. (1955). The story of Q. American Scientist, 43(4), 584-594.

4. Kleppner, D., & Kolenkow, R. (1973). An Introduction to Mechanics. Singapore: McGraw-Hill.

Section 23–1 Complex numbers and harmonic motion

(Real force / Real part of x / Right frequency)

In this section, Feynman discusses the real force, the real part of x, and the right frequency of a forced oscillation.

1. Real force:

“The complex number F that we have so defined is not a real physical force, because no force in physics is really complex; actual forces have no imaginary part, only a real part (Feynman et al., 1963, section 23–1 Complex numbers and harmonic motion).”

Feynman says that the real force, F = F₀cos ωt, can be written as the real part of a complex number F = F₀e^iωt. Physicists use complex functions to represent oscillatory functions because it is easier to work with an exponential function than with a cosine. In other words, the use of complex functions is based on pragmatic grounds or a matter of convenience. One may confess that this trick involves guessing x = e^iωt or x = te^ωt. On the other hand, Schrödinger struggled in determining between a real and complex wave function for his quantum theory. However, Wigner (1990) argues that the use of complex numbers in quantum mechanics is not a calculational trick. On the use of complex numbers, Wilczek (2015) explains that some follow “the advice of Father Jim Malley — ‘It is more blessed to ask forgiveness than permission’—and used them (p. 348).”

Feynman suggests a mathematical trick to represent a force that is out of phase by a delayed phase Δ. It refers to the real part of F₀e^i(ωt−Δ), but we may write e^i(ωt−Δ) = e^iωte^−iΔ. An advantage of the trick is that we can simplify the general solution of x using xe^iωt instead of xe^i(ωt−Δ). Some students may have difficulties understanding the term F₀e^i(ωt−Δ) because of the symbol F_0. It is worth mentioning that the complex force F is assumed to be out of phase to the displacement x by a phase difference Δ. One should realize that F₀ is the complex force at time t = 0 and it is not necessarily the maximum force. Importantly, a multiplication of F₀ by e^−iΔ can increase the magnitude of force to its maximum value. On the contrary, some physicists may prefer to let the factor e^−iΔ to appear in the displacement as xe^i(ωt−Δ).

2. Real part of x:

“… we deduce that the real part of x satisfies the equation with the real part of the force (Feynman et al., 1963, section 23–1 Complex numbers and harmonic motion).”

Feynman mentions that the real part of x satisfies the equation with the real part of the force, but the method is valid only for equations which are linear. That is, he refers to equations in which x appears in every term only in the first power or zeroth power. Feynman’s description of linear differential equation is potentially misleading. For example, Lorenz equations dx/dt = –sx + sy, dy/dt = – xz + rx – y, dz/dt = xy – bz are three ordinary differential equations that do not have any term that is in the second power or higher. After Feynman delivered this lecture, Lorenz (1963) discovered this system that is non-linear, non-periodic, and deterministic, but it leads to chaotic results. One feature of non-linearity is: we cannot combine two independent solutions to get a general solution (i.e., the whole is not equal to the sum of the parts).

Feynman elaborates that if there were a term λx² in the equation, then when we substitute x_r+ix_i, we would get λ(x_r+ix_i)² and it would yield λ(x_r²−x_i²) as the real part and 2iλx_rx_i as the imaginary part. In short, it introduces an artificial thing (imaginary part) in the analysis. Perhaps the following simplification may help some students: e.g., let the differential equation be dx/dt + x² = 0. If the two possible solutions are x₁ and x₂, this means that dx₁/dt + x₁² = 0 and dx₂/dt + x₂² = 0. To check the linear property of the equation, we need to find out whether d(x₁+x₂)/dt + (x₁+x₂)² = 0. By expanding the equation, d(x₁+x₂)/dt + (x₁+x₂)² = (dx₁/dt + dx₂/dt) + (x₁² + 2x₁x₂ + x₂²) = (dx₁/dt + x₁²) + (dx₂/dt + x₂²) + 2x₁x₂ = 2x₁x₂. It introduces a cross-term 2x₁x₂ that prevents the equation to be definitely equal to zero, and thus, x₁ + x₂ is not a general solution.

3. Right frequency:

“So we get a very strong response when we apply the right frequency ω (Feynman et al., 1963, section 23–1 Complex numbers and harmonic motion).”

Feynman explains that we get a very strong response when we apply the right frequency ω. He provides a simple example: if we hold a pendulum on the end of a string and shake it at just the right frequency, then it will swing very high. Alternatively, the explanations could include basic physical principles. Firstly, if the forcing frequency ω is exactly equal to natural frequency ω₀, the swing would oscillate at an infinite amplitude is an idealization. Specifically, it is about shaking at the right time (or right phase) instead of only right frequency. If the force and the motion of the pendulum are in the same direction (or same phase), then there is a maximum energy transfer in accordance with the work-energy theorem. However, one should realize that the phase difference between the force and displacement is p/2 radians (see Fig. 23-3).

According to Feynman, the magnitude of x is related to the size of the F by the factor 1/m(ω₀²−ω²). In section 21–5, he uses a trigonometric method to get ω₀² = k/m. To use an exponential method, we may first guess x = Ce^wt. By substituting it into md²x/dt² + kx = 0, we have m(w²)(Ce^wt) + k(Ce^wt) = mw²+ k = 0 whose solution is w = ±Ö(-k/m). Feynman has explained in section 21–5 that w₀ = Ö(k/m) is the natural frequency and this allows us to write w = ±iw₀. Thus, the general solution is x = Ce^(iw₀t) + C*e^(-iw₀t), whereby the two arbitrary constants C and C* should be complex numbers such that we have real solution (Symon, 1971). By solving the equations, we will get x = (C/2)e^(iw₀t + q) + (C/2)e^(-iw₀t + q) = C (cos w₀t + q).

Questions for discussion:

1. Would you prefer using a mathematical trick by letting F = F₀e^i(ωt−Δ) or x = x₀e^i(ωt−Δ)?

2. Is the imaginary part of the equation definitely an artificial thing without meaning?

3. Why do we get a very strong response if we apply the right frequency ω?

The moral of the lesson: The complex number F that we have defined is not a real physical force, because no force in physics is really complex; actual forces have no imaginary part, but only a real part.

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.

2. Lorenz, E. (1963). Deterministic Non-Periodic Flow. Journal of the Atmospheric Sciences, 20(2), 130–141.

3. Symon, K. R. (1971). Mechanics (3rd ed.). Reading, MA: Addison-Wesley.

4. Wigner, E.P. (1990). The unreasonable effectiveness of mathematics in the natural sciences. In R. E. Mickens (ed.). Mathematics and Science (pp. 291–306). Singapore: World Scientific.

5. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.