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Journal of Signal Processing

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

Nobuo Nagai, Takashi Yahagi

Published Jan 1, 2016• 0 citations

Abstract

In the steady-state of lossless circuits, resonance can be obtained using the Laplace transform. We focus on commensurate transmission line circuits constructed using eight unit elements as an example and determine the resonance, transient response, and nonstationary response of the circuits using a z-transform. In old quantum theory, light quanta are assumed to satisfy the heat conduction equation. When considering the response of light quanta as a physical phenomenon from the viewpoint of circuit theory, we can interpret that the heat conduction equation describes the propagation of waves on a lossy distributed RC transmission line, in which light quanta repeatedly collide with each other. We demonstrate that light quanta are different from photons, which are interpreted as satisfying the Maxwell's equations and propagating on a lossless transmission line.

DOI: 10.2299/jsp.20.55

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In the steady-state of lossless circuits, resonance can be obtained using the Laplace transform. We focus on commensurate transmission line circuits constructed using eight unit elements as an example and determine the resonance, transient response, and nonstationary response of the circuits using a z-transform. In old quantum theory, light quanta are assumed to satisfy the heat conduction equation. When considering the response of light quanta as a physical phenomenon from the viewpoint of circuit theory, we can interpret that the heat conduction equation describes the propagation of waves on a lossy distributed RC transmission line, in which light quanta repeatedly collide with each other. We demonstrate that light quanta are different from photons, which are interpreted as satisfying the Maxwell's equations and propagating on a lossless transmission line.

A lossless transmission line of length l is called a unit element. By showing numerical examples, we demonstrate that resonance and eigen-oscillation can be obtained using a unit element as a circuit element. Because resonance and eigen-oscillation are responses in the steady state, the transient response that occurs until the circuit reaches the steady state should be considered. Although the transient response has been conventionally considered a phenomenon of exponential attenuation that depends on a time constant, we numerically demonstrate that the transient response of a unit element is a phenomenon of discrete attenuation that can be expressed using a z-transform. In addition, resonance and eigen-oscillation involve reactive power, which is not related to energy, because the transient response of the circuit using a unit element is closely related to the phases expressed by the z-transform. We also numerically show that the reactive power is confined in the circuit.

In this lecture series, we attempt to construct a theoretical system of quantum theory that emphasizes the wave nature (in other words, that is based on circuit theory) and assumes quanta to be circuit elements. Circuit equations given by voltage and current have steady-state and transient responses. The steady-state response is expressed by complex numbers. For lossless circuits, the active and reactive powers can be distinguished, and the active power is expressed by a conserved quantity. In physics, physical phenomena are considered to occur as a result of some motion, suggesting the importance of motion. In circuit theory, a transient response is generated when a steady state transits to another steady state. In the transient response, the active and reactive powers cannot be distinguished. Therefore, the transient response is considered to be a physical phenomenon without a conserved quantity and to require special techniques involving difficult problems, such as renormalization.

Physical phenomena showing both particle and wave natures at the same time exhibit resonance. In addition, resonance is a physical phenomenon occurring in the steady state. Transient and steady-state responses can be discriminated when transmission lines are used. We demonstrate that (1) extended telegrapher's equations can be obtained using the Riccati differential equation, (2) lossless transmission lines called Pasteur, Tellegen, and bi-isotropic (BI) media can be characterized using the extended telegrapher's equations, and (3) the resonance conditions can be determined for each of these media.

To apply circuit theory to quantum theory, we review the problems of circuit theory described in the past sessions. In concrete, the physical properties of circuits, such as losslessness, steady-state response, and reactive power, are reexamined. It is shown that the Laplace transform is more advantageous than Heaviside's operational calculus and that active power can be conserved when a lossless circuit is used. It is also demonstrated that a forward wave exp(-jβx) travels clockwise because exp(jωt) in the Laplace transform is required for separating the variables of partial differential equations related to waves.

In quantum theory, a wave function expresses the state of matter waves and is not considered to physically exist in general. Equations that are satisfied by a wave function are called wave equations. In classical physics, wave equations are well known and are derived from the Maxwell equations. However, the Maxwell equations do not define important physical properties, such as the lossless property of circuits. The lossless property is represented by the cascade matrix defined in analog circuit theory. The definition of the lossless property in wave equations became possible owing to the telegrapher's equations introduced by Heaviside. The cascade matrix can be determined from uniform transmission lines that can be obtained from the telegrapher's equations and those extended using the Riccati differential equation.