Section 25–5 Series and parallel impedances

 (Series impedance / Parallel impedance / Kirchhoff’s laws)

 

In this section, Feynman discusses how to calculate the series impedance and parallel impedance of alternating current (AC) circuits as well as Kirchhoff’s circuital laws.

 

1. Series impedance:

“This means that the voltage on the complete circuit can be written V = IZ_s, where the Z_s of the combined system in series is the sum of the two Z’s of the separate pieces: Zs = Z₁ + Z₂ (Feynman et al., 1963, section 25–5 Series and parallel impedances).”

 

Feynman asks what happens if we put two circuit elements in series that have impedance Z₁ and Z₂ by applying a voltage. Then, he shows that the total voltage is the sum of the voltages across the two circuit elements and it is equal to V = V₁ + V₂ = (Z₁ + Z₂)I. However, there is a problem of sequence because Feynman could have first proved the Kirchhoff’s circuital laws. Alternatively, he could first explain why the current is the same in the circuit elements and apply the law of conservation of energy for circuit elements that are connected in series. To have a better understanding, one may distinguish reactance and impedance as follows: reactance of a capacitor or inductor is like the resistance that opposes the AC current, whereas impedance of an AC circuit is the vector sum of the net reactance and the resistance in the circuit.

 

Feynman explains that the voltage on the complete circuit can be written as V = IZ_s, where Z_s is the effective impedance of the combined system in series. That is, the effective impedance of two circuit elements is the sum of the two Z’s: Z_s = Z₁ + Z₂. Perhaps one may provide a definition of the impedance of an AC circuit as a measure of the total opposition to the current that is analogous to the resistance by using the equation Z_s = V/I. It is beneficial to clarify that although impedance is a complex number, but it is not the same as V or I that is a phasor. One may represent the phasor using a complex number and explain how it behaves like a vector that can rotate continuously. In short, all phasors are complex numbers (or a kind of sinusoidal vectors), but not all complex numbers are phasors.

 

2. Parallel impedance:

“This can be written as V = I/[(1/Z₁) + (1/Z₂)] = I/Z_p. Thus, 1/Z_p = 1/Z₁ + 1/Z₂ (Feynman et al., 1963, section 25–5 Series and parallel impedances).”

 

According to Feynman, if the connecting wires are perfect conductors and a constant voltage is effectively applied to both of the impedances, it will cause currents in each independently. He shows that the current in Z₁ is equal to I₁ = V/Z₁ and the current in Z₂ is I₂ = V/Z₂. Alternatively, one may explain why the voltage is the same and apply the law of conservation of charge for circuit elements that are connected in parallel. Note that Kirchhoff’s current law is not really a fundamental principle, but it is a consequence of the law of conservation of charge. When we apply this law in DC circuit, we assume the ideal condition of steady-state current (or constant current).

 

Feynman states that the total current which is supplied to the terminals is the sum of the currents in the two sections: I = V/Z₁ + V/Z₂. In addition, he says that “it is the same voltage” without further explanations. The phrase “same voltage” could be unclear because it is about the “same AC voltage” that is varying sinusoidally. Kirchhoff’s circuit laws are intuitively correct and applicable to DC voltage, but it can be generalized to AC voltage provided both AC voltage and AC current are of the same frequency. Essentially, we have idealized the circuit elements only interact with each other through connecting wires. Specifically, we have assumed that there is no induced current and the wires are not capacitively coupled when the circuit devices are connected in parallel.

 

3. Kirchhoff’s laws:

“These rules are called Kirchhoff’s laws for electrical circuits. Their systematic application to complicated circuits often simplifies the analysis of such circuits (Feynman et al., 1963, section 25–5 Series and parallel impedances).”

 

Feynman ends the chapter by stating Kirchhoff’s laws for electrical circuits. Kirchhoff’s current law and voltage law are also known as Kirchhoff’s junction rule (or nodal rule) and loop rule (or mesh rule) respectively. That is, they are rules that involve idealizations and approximations. In Volume II, Feynman elaborates that “one of our idealizations has been that negligible electrical charges accumulate on the terminals of the impedances. We now assume further that any electrical charges on the wires joining terminals can also be neglected… (Feynman et al., 1964, section 22–3 Networks of ideal elements; Kirchhoff’s rules).”

 

Feynman states a condition for Kirchhoff’s law: the voltages in little generators have no impedance. This implies that the voltage of a generator is approximately equal to the electromotive force provided its internal resistance is negligible and thus, there is no loss of electrical energy in the generator. Feynman adds that the systematic application of Kirchhoff’s laws to complicated circuits can simplify the analysis of such circuits. Interestingly, we can apply Kirchhoff’s laws for mechanical elements: 1. Kirchhoff’s force law means that “the algebraic sum of all the forces acting on any junction of mechanical elements is zero (Firestone, 1933, pp. 254-255).” 2. Kirchhoff’s velocity law means that “the algebraic sum of the velocity differences around any closed mechanical circuit is zero (Firestone, 1933, p. 255).”

 

Questions for discussion:

1. Would you derive the formula for two circuit elements in series without using Kirchhoff’s voltage law?

2. Would you derive the formula for two circuit elements in parallel without using Kirchhoff’s current law? 

3. How would you state Kirchhoff’s laws for electrical circuits?

 

The moral of the lesson: we can apply Kirchhoff’s circuital laws to determine the effective impedance of any circuit, however, we have essentially used Kirchhoff’s laws to derive the formula for series impedance and parallel impedance.

 

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

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