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Circuit Theory Based on New Concepts and Its Application to Quantum Theory

For commensurate transmission line circuits constructed using a unit element, transient responses as well as resonance as a steady-state response can be obtained. Such circuits can be analyzed using the z-transform, and transient responses that vary with time can be obtained. In addition, unit elements are symmetrical and lossless circuit elements. If the phase of a unit element is nπ and the internal resistance of the power source is equal to the load resistance, resonance occurs. Resonance also occurs if the phase of a unit element is nπ + π/2 and the product of the internal resistance and the load resistance is equal to the square of the characteristic resistance of the unit element.


Introduction
In general, there are two types of circuit response: transient and steady-state. When external factors such as the input voltage to the circuit change, the response does not instantaneously transit to another steady-state; a transient period is required for the transition. The response that occurs during the transient period is called the transient response. The transient phenomena of a lumped constant circuit can be determined by the Laplace transform only when the function of the circuit has a very particular form. Because of this limitation, the determination of the steadystate response alone is considered to be sufficient in circuit theory. The transition response that occurs and the transition to the steady-state are not clearly described by theory.
The z-transform is used in signal processing and can also be used to clarify the phenomena that occur in transient responses. The z-transform can be used to analyze commensurate transmission line circuits [1]. In this session, we first examine the transient responses of a circuit constructed using a unit element. Then, we demonstrate that resonance can be obtained for a circuit constructed using a unit element, which is a lossless circuit element, and we determine the active and reactive powers at resonance.

Deriving Unit Elements
The unit elements used in commensurate transmission line circuits are circuit elements obtained from lossless telegrapher's equations. The procedure for deriving unit elements was explained in Session 1 of this lecture series. However, we review the procedure because the operations used in the procedure are extremely important.

Solution of lossless telegrapher's equations
The lossless telegrapher's equations describe transmission lines such as coaxial cables. Let us denote the round-trip inductance per unit length as L, the capacitance between the lines as C, and the instantaneous values of voltage and current as v(x,t) and i(x,t), respectively. Then, the lossless telegrapher's equations are given by the following pair of linear partial differential equations.
We solve Eq. (2.2) as an eigenvalue problem of the matrix. Denoting the eigenvalue as γ, we obtain where R 0 is called the characteristic resistance [3].
The eigenvector for γ = −jβ is expressed by The important point to note is that the term corresponding to the current should be −1 so that the term corresponding to the voltage R 0 is positive. Therefore, γ = −jβ, and the eigenvector given by Eq. (2.6) represents the backward waves. Using The first terms on the right of Eqs. (2.7a) and (2.7b) represent the forward waves, and the second terms on the right of Eqs. (2.7a) and (2.7b) represent the backward waves.
Then, we obtain a circuit element of length l from the lossless telegrapher's equations [Eqs. (2.7a) and (2.7b)] and obtain its cascade matrix.
By substituting x = 0 into Eqs. (2.7a) and (2.7b), we obtain Therefore, the constants of integration N 1 and N 2 are expressed by By substituting x = −l into Eqs. (2.7a) and (2.7b), we obtain  Equation (2.12) represents the cascade matrix of a circuit element of length l obtained from the lossless telegrapher's equations, and we call this element the unit element.

Transient Response in Circuit Constructed Using Unit Elements
Only commensurate transmission line circuits constructed using unit elements can be expressed by the z-transform. Hence, transient responses can be obtained only for such circuits. In this section, we examine the phenomena that occur in the transient responses of the commensurate transmission line circuit using a unit element as shown in Fig. 2  The four waves expressed by Eqs. (2.7a) and (2.7b) are based on current. In this section, however, we use voltagebased equations following a general textbook [4]. Using two constants of integration, K a (s) and K b (s), instead of Eqs. (2.7a) and (2.7b), the voltage and current are given by The voltage and current along the unit element are given by Eqs. (2.15a) and (2.15b), respectively, regardless of the termination conditions. The values of K a (s) and K b (s) are determined depending on the termination conditions. The termination conditions on the power side (x = 0) and at the load side (x = l) are respectively given by ) 0 Here, the voltage and current at x = 0 are expressed by and the voltage and current at x = l are expressed by By substituting Eqs. (2.17a)-(2.17d) into Eqs. (2.16a) and (2.16b), the forward voltage wave K a (s) and backward voltage wave K b (s) are given by Here, E(s)/2 is half the voltage from the voltage source and is also the incident voltage. The term t G is the instantaneous voltage transmission coefficient for the voltage wave traveling from the voltage source to the unit element and is expressed by The terms r R and r G are the instantaneous reflection coefficients for the voltage waves traveling from the unit element to the load and to the voltage source, respectively, and are expressed by appear to be expressed by the z-transform. Next, we examine whether V(x) given by Eq. (2.15a) can be expressed by the z-transform.
First, we denote the incident voltage from the voltage source to the unit element as E t (s). Using Eqs. (2.18a) and From Eqs. (2.18a) and (2.18b), the forward wave K a (s) (traveling to the right) and the backward wave K b (s) (traveling to the left) at an arbitrary point x on the unit element are expressed by Here, u indicates the propagation velocity of the voltage wave and u −1 2l is the time required for the wave to travel a distance 2l. Therefore, we assume Similarly, u −1 x is the time required for the wave to travel a distance x. Therefore, we assume gives the product of the reflection coefficients of the voltage waves at the load and power sides, which is equivalent to the rate of decrease in the voltage when the voltage wave makes a round trip along the unit element. Equation (2.23b) indicates that the voltage wave lags by a time T when it makes a round trip along the unit element.
Using the above definitions, Eqs. (2.21a) and (2.21b) can be expressed by the z-transform as follows.
To obtain Eq. (2.24a) in the form of a series, the equation is expressed as a polynominal of z −1 . The equation obtained by applying the inverse Laplace transform to the polynominal represents waveforms that appear at intervals of T, exhibiting an effect similar to sampling. Carefully examining this phenomenon, we expect that an electromagnetic wave is decomposed into several electromagnetic waves, and if it is periodic, the decomposed electromagnetic waves are superimposed to reconstruct an electromagnetic wave. Numerical examples are introduced in the next session.
The backward voltage wave K b (s) can be expressed by the z-transform, similarly to the forward voltage wave K a (s). Moreover, the forward and backward current waves can also be expressed by the z-transform. The process for obtaining these waves is not explained here. Thus, a total of four waves, i.e., the forward and backward voltage and current waves, can be expressed by the z-transform. Thus, all the transient responses can be obtained.

Resonance of Circuit Constructed Using Unit Element
In the previous section, we proved that the reflection and transmission of the voltage and current waves take place in the transient responses of the circuit constructed using the unit element shown in Fig. 2.1. In this section, we focus on the steady-state, in particular, resonance.
Steady-state responses can be obtained using cascade matrices. In an electrical circuit, two cascade matrices can be connected by a cascade connection, and coils and capacitances can also be connected. Have you ever considered the reason why such a connection is possible?
In general, connecting two objects and unifying their physical phenomena into one is not easy. Connecting two scattering matrices in circuit theory is also not simple.
Electrical circuits are expressed by linear equations of voltage and current. Therefore, cascade connection is possible in electrical circuits because both the voltage and current satisfy continuity. Connecting circuit elements by determining their termination conditions is equivalent to connecting two cascade matrices, which requires the use of the continuities of voltage and current. However, the continuities of voltage and current waves differ in terms of their incidence, reflection, and transmission, as shown in Ref. 3 (Chapter 1). Voltage and current satisfy continuity alone, whereas active power satisfies continuity and the conservation law, as shown in Ref. 3 (Chapter 1).
Because cascade matrices have the properties described above, electrical circuits should be analyzed using cascade matrices. We now obtain a steady-state response using the cascade matrix in Eq. (2.12). First, the forward input impedance at the power side can be simply obtained: the input impedance Z in is V G /I G and the load resistance is R R . They are expressed using Eq. (2.12) as The internal resistance of the voltage source is R G . If Z in given by Eq. (2.27b) is equal to R G , impedance matching in the steady-state is achieved, resulting in no reflection: resonance occurs. Here, we determine the resonance conditions. The term Z in becomes a pure resistance expressed by a real number when (i) sinβl = 0, (ii) cosβl = 0, or (iii) the argument in the denominator is equal to that in the numerator. The values of the resistance in these cases are determined as follows. (i) When sinβl = 0, i.e., βl = nπ, When the argument of the denominator is equal to that of the numerator, It is shown by Eq. (2.28c) that impedance matching is achieved and there is no reflection when the impedances of resistors used in the circuit are all equal. Note that the phase of unit elements used for transmission line circuits is essential for resonance.
The phase is obtained by converting the time t into the angular frequency ω by the Laplace transform. Therefore, the time term is not required for obtaining resonance, that is, resonance is not expressed by Heaviside's operational calculus that includes timedependent terms [5].

Active and Reactive Powers in Resonance
It was demonstrated that resonance can be obtained for the circuit constructed using a unit element shown in Fig.  2.1. Now, we examine how the voltage and current along the unit element behave at resonance. In other words, we know that the active power is constant at an arbitrary point on the unit element and we now attempt to examine the nature of the reactive power. The power of a point theorem used in geometry is effective for this purpose and is discussed as a first step below.

Power of a point theorem in geometry
First, we consider the power of a point theorem using a theorem related to circles in geometry.

Voltage and current along unit element
The transient responses of a circuit using a unit element were discussed in Section 3. In this section, we focus on the steady-state of such a circuit. The voltage V(x) and current I(x) along the unit element of the circuit shown in Here, Equations (2.30a) and (2.30b) express the voltage and current, respectively, as complex functions of a point x on the unit element. Therefore, the complex and related powers at a point x are defined as follows.
Complex power: The normalized voltage V N (x) and current I N (x) are expressed by 36b) Thus, the normalized voltage V N (x) is expressed as the vector from point P to point A or C on the circumference. Therefore, the vector indicates that V N (x) is a standing wave and that the magnitude of the voltage depends on the position of the point x. Power is considered to be stored in the unit element because the normalized voltage V N (x) and current I N (x) are vectors PA and PB, respectively. If PA and PB are not equal to PA 0 and PB 0 , respectively, the phases of the two vectors are different and their product is the apparent power. The apparent power is divided into the active and reactive powers and is obtained by dividing the current into the inphase and orthogonal components of the voltage. Namely, the complex power is divided into the active and reactive powers by dividing vector PB, representing the normalized current I N (x), into two vectors, PC and CB. The active and reactive powers are obtained as a product of the lengths of two vectors as follows.
Active power:  This indicates that the active power on the unit element is constant.

Summary
It is conventional knowledge that there are two types of circuit response, i.e., transient and steady-state. Moreover, the transient response can be expressed by the Laplace transform, which is used to determine the time constant. The transient response also attenuates exponentially and transits to a steady-state. In circuits, both voltage and current waves are divided by reflection and transmission at junctions connected to a component with a difference impedance. When lossless transmission lines and circuits are used, a maximum available power, which is defined as the maximum active power supplied from the power source, can be supplied to the load, and resonance is achieved.
When comprehensively examining the above findings, we wondered what phenomena occur in the transient response and how resonance is obtained. To answer these questions, we examined the resonance conditions for a transmission line circuit constructed using a unit element. As a result, we found resonance conditions (i)-(iii), as described in Section 4. Condition (iii) is reasonable but self-evident. Condition (i) indicates that resonance occurs because the voltage and current at the input are equal to those at the load, which is also considered to be reasonable. However, condition (ii) is considered to be unique to circuits and indicates that resonance occurs even though the voltage and current at the input are different from those at the load.
A unit element at resonance under condition (ii) is commonly used in microwave circuits and serves as an ideal transformer, called a quarter-wave impedance transformer. Readers who have studied electrical circuits are familiar with the action of ideal transformers and may be interested in why it is obtained by achieving resonance. In contrast, those who have not studied electrical circuits may find the action of ideal transformers more mysterious than the resonance. Note that the ultimate operation ∞ is required for ideal transformers [4].
As explained, circuit theory involves characteristic operations such as the ultimate operation ∞, a zero-energy phenomenon known as reactive power, and the connection of cascade matrices to integrate a circuit. We suggest that readers pay attention to these operations. Resonance is a phenomenon in which electromagnetic waves are collected in terms of their phase. It is different from what happens with lenses, which concentrate light waves into a focal point, or the phenomenon of gathering electromagnetic waves scattered by reflection and transmission using a "broom". Rather, at resonance, electromagnetic waves in resonance are sampled as if they were collected at regular time intervals. If the waves are in-phase, resonance occurs. If the waves are out-of-phase, the number of reflected waves increases and the active power supplied by the power source decreases.
Therefore, the reactive power, which is unnecessary, is also collected even under resonance conditions (i). Therefore, we introduce such numerical examples in the next session.
[Acknowledgments] We are deeply grateful to Professor Hiroshi Tanimoto at Kitami Institute of Technology for his continued cooperation in our discussion via e-mail. We also thank Dr. Jie Ren for her continued support in the development of the figures and calculations.