Circuit Analysis Fundamentals

Author: Chief Editor: Li Huazhong
Publisher:
Publishing Date: 1999-09-01
Features: This book is written based on the fundamental requirements of the electrical course in higher industrial schools and also takes into account the needs of cultivating applied talents. The book is divided into eleven chapters, including Basic Concepts and Laws of Circuits, Equivalent Transformation of Circuits, General Analysis Methods and Theorems for Linear Circuits, Analysis of Sinusoidal Steady-State Circuits, Coupled Inductors and Ideal Transformers, Three-Phase Circuits, Periodic Non-Sinusoidal Circuits and Signal Spectra, Frequency Characteristics and Resonance of Circuits, Transient Analysis of Linear Circuits, Complex Frequency Domain Analysis Methods for Linear Circuits, and Two-Port Networks. The book aims to highlight key points, provide detailed explanations, and stay close to practical applications. For the convenience of self-study, exercises and thought questions are generally provided after each chapter, along with problems. This book can serve as a textbook or reference for relevant majors in higher industrial schools, including undergraduate and associate degree programs in electrical engineering, mechatronics, and other fields. It can also be used as a reference for engineering and technical personnel, as well as for night schools and correspondence courses in corresponding fields.
Excerpt: A two-terminal element is called a two-terminal element, while an element with three or more terminals is called a multi-terminal element. The three basic circuit elements are passive two-terminal elements. Ideal sources are active two-terminal elements. Coupled inductors, ideal transformers, and controlled sources are multi-terminal elements. A circuit composed of ideal circuit elements and ideal conductors is called a circuit model. A circuit model is an abstraction of an actual circuit. In this book, unless otherwise specified, the circuits discussed are circuit models, often simply referred to as circuits. Figure 1-1b is the circuit model of a flashlight circuit. The introduction of ideal circuit elements ensures that all electromagnetic phenomena discussed in the circuits occur only within the elements. That is, energy consumption occurs within resistive elements, while electric and magnetic fields store energy within capacitive and inductive elements. Therefore, ideal circuit elements are also called lumped-parameter elements. Circuits constructed from lumped-parameter elements are called lumped-parameter circuits. Since in lumped-parameter circuits, electromagnetic phenomena are concentrated within the elements, there is a separation of electric and magnetic field interactions. According to electromagnetic field theory, the interaction between electric and magnetic fields generates electromagnetic waves. When the geometric dimensions of a circuit are comparable to the wavelength corresponding to its operating frequency, the radiation of electromagnetic waves will be significantly enhanced. Some energy in the circuit will be radiated into space as electromagnetic waves. This contradicts the assumption that all energy consumption occurs within resistive elements. Therefore, for an actual circuit to be treated as a lumped-parameter circuit, the energy loss due to electromagnetic wave radiation must be ignored. That is, as a lumped-parameter circuit, the geometric dimension \( l \) of the circuit must be much smaller than the wavelength \( \lambda \) corresponding to its operating frequency, i.e., \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light and \( f \) is the operating frequency of the circuit. In lumped-parameter circuits, the current flowing out of one terminal of a two-terminal element at any given moment is equal to the current flowing into the other terminal at that moment, meaning that current flow through the circuit does not require time. When the geometric dimensions of a circuit can be compared to the wavelength corresponding to its operating frequency, the circuit cannot be treated as a lumped-parameter circuit and must be analyzed using distributed-parameter circuits or electromagnetic field theory. This book only discusses lumped-parameter circuits. In circuit analysis, the sources or signals acting on the circuit are called excitations (or inputs), while the voltages or currents that appear in the circuit due to the excitation are called responses (or outputs). Circuit analysis refers to the discussion and search for the relationship between excitations and responses under known circuit structure and component parameters. If the excitation and response are known, and the circuit structure and component parameters are sought, this falls under "circuit synthesis" in circuit theory. This book only discusses circuit analysis topics.
1.2 Basic Variables in Circuit Analysis
One of the main problems in circuit analysis is to understand and determine the operating state of the circuit, i.e., to solve for the currents and voltages across each element. Current and voltage are the two basic variables in circuit analysis. A basic variable is one that can conveniently represent other physical quantities in the circuit. For example, electric power is a basic physical quantity derived from the combination of current and voltage. In addition to current and voltage, charge and flux linkage are also basic variables. Current and voltage, as well as the power derived from them, are the most commonly used variables in circuit analysis.
1.2.1 Current and Its Reference Direction
Under the action of an electric field, the orderly directional movement of charge forms current. The rate of charge movement is defined as the magnitude of current as follows: The algebraic sum of charge passing through a cross-sectional area of a conductor per unit time is called current. It is represented by the symbol \( i \), i.e., \( i \) is used to describe the physical phenomenon of charge moving in a directional manner as well as to represent its magnitude. For time-varying currents (also called alternating currents), lowercase \( i \) is used; for constant currents (also called direct currents), uppercase \( I \) is used. In the International System of Units, the basic units for current, charge, and time are named ampere (abbreviated as A), coulomb (abbreviated as C), and second (s), respectively. In practical applications, current is sometimes also expressed in auxiliary units such as kiloamps (kA), milliamps (mA), and microamps (μA), with the following conversion relationships: Due to historical reasons, the actual direction of current is defined as the direction of positive charge movement. In simple DC circuits (e.g., the circuit shown in Figure 1-1), the actual direction of current is easily identifiable. However, when circuits become more complex, it becomes difficult to determine the actual direction of current. For example, in the bridge circuit shown in Figure 1-3, the current in resistor \( R_5 \) is flowing from \( a \) to \( b \), from \( b \) to \( a \), or is zero, and this must be determined through calculation. Calculation, however, requires the assignment of current directions to each element, which creates difficulties. Additionally, in AC circuits, the direction of current varies with time, making it impossible to indicate the actual direction. To facilitate calculations, the concept of "reference direction" is introduced. For a physical quantity like current, which has only two possible directions on a two-terminal element, any direction can be arbitrarily chosen as the reference direction for current. The reference direction for current is represented by an arrow. If the calculated or given current value is positive, it indicates that the reference direction is the same as the actual direction; if it is negative, it indicates that the reference direction is opposite to the actual direction. In Figure 1-4a, the current is positive, indicating that the reference direction is the same as the actual direction, i.e., the current is indeed flowing from \( a \) to \( b \). In Figure 1-4b, the current is negative, indicating that the actual direction is from \( b \) to \( a \). The reference direction for current can also be represented using double subscripts, such as "iab," which indicates the assumed reference direction from \( a \) to \( b \). Clearly, \( i_{ba} = -i_{ab} \). With the introduction of reference directions, current becomes an algebraic quantity.