基本概念

基本定义

An electric circuit is an interconnection of electrical elements.

Charge is an electrical property of the atomic particles of which matter consists, measured in coulombs (C).

$$e=-1.602\times {10}^{-19}\text{ C}$$

Electric current is the time rate of change of charge, measured in amperes (A).

$$\begin{array}{rl}i& \triangleq \frac{\mathrm{d}q}{\mathrm{d}t}\\ Q& \triangleq {\int }_{t_0}^{t}i\mathrm{d}t\end{array}$$

A direct current (dc) flows only in one direction and can be constant or time varying.

An alternating current (ac) is a current that changes direction with respect to time.

Voltage (or potential difference) is the energy required to move a unit charge from a reference point (-) to another point (+), measured in volts (V).

$$v_{ab}\triangleq \frac{\mathrm{d}w}{\mathrm{d}q}$$

Power is the time rate of expending or absorbing energy, measured in watts (W).

$$p\triangleq \frac{\mathrm{d}w}{\mathrm{d}t}$$

or

$$p=\frac{\mathrm{d}w}{\mathrm{d}t}=\frac{\mathrm{d}w}{\mathrm{d}q}\cdot \frac{\mathrm{d}q}{\mathrm{d}t}=v\cdot i$$

Passive sign convention is satisfied when the current enters through the positive terminal of an element and $p=+vi$. If the current enters through the negative terminal, $p=-vi$.

$$\mathrm{\Sigma }p=0$$

Energy is the capacity to do work, measured in joules (J).

$$1\text{ Wh}=3600\text{ J}$$

An ideal independent source is an active element that provides a specified voltage or current that is completely independent of other circuit elements.

An ideal dependent (or controlled) source is an active element in which the source quantity is controlled by another voltage or current.

1. A voltage-controlled voltage source (VCVS).
2. A current-controlled voltage source (CCVS).
3. A voltage-controlled current source (VCCS).
4. A current-controlled current source (CCCS).

基本定律

$$R=\rho \frac{l}{A}$$

Ohm's law states that the voltage v across a resistor is directly proportional to the current i flowing through the resistor.

$$v=iR$$

The resistance R of an element denotes its ability to resist the flow of electric current; it is measured in ohms (Ω).

$$R=\frac{v}{i}$$

A short circuit is a circuit element with resistance approaching zero.

$$v=iR=0$$

An open circuit is a circuit element with resistance approaching infinity.

$$i=\underset{R\to \mathrm{\infty }}{lim}\frac{v}{R}=0$$

Conductance is the ability of an element to conduct electric current; it is measured in mhos (℧) or siemens (S).

$$G=\frac{1}{R}=\frac{i}{v}$$

A branch represents a single element such as a voltage source or a resistor.

A node is the point of connection of two or more branches.

A loop is any closed path in a circuit.

$$b=l+n-1$$

Two or more elements are in series if they exclusively share a single node and consequently carry the same current. Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them.

Kirchhoff's current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.

$${\mathrm{\Sigma }}_{n=1}^N{i}_n=0$$

Kirchhoff's voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero.

$${\mathrm{\Sigma }}_{m=1}^M{v}_m=0$$

The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

$$R_{eq}={\mathrm{\Sigma }}_{n=1}^N{R}_n$$

The equivalent conductance of any number of resistors connected in parallel is the sum of the individual conductances.

$$G_{eq}={\mathrm{\Sigma }}_{n=1}^N{G}_n$$

Wye-Delta Transformations

Each resistor in the Y network is the product of the resistors in the two adjacent Δ branches, divided by the sum of the three Δ resistors.

$$\begin{array}{r}{R}_1=\frac{R_b{R}_c}{R_a+R_b+R_c}\\ R_2=\frac{R_a{R}_c}{R_a+R_b+R_c}\\ R_3=\frac{R_a{R}_b}{R_a+R_b+R_c}\end{array}$$

Each resistor in the Δ network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.

$$\begin{array}{r}{R}_a=\frac{R_1{R}_2+R_2{R}_3+R_3{R}_1}{R_1}\\ R_b=\frac{R_1{R}_2+R_2{R}_3+R_3{R}_1}{R_2}\\ R_c=\frac{R_1{R}_2+R_2{R}_3+R_3{R}_1}{R_3}\end{array}$$

The Y and Δ networks are said to be balanced when

$$R_1=R_2=R_3=R_Y,R_a=R_b=R_c=R_{\mathrm{\Delta }}$$

Under these conditions, conversion formulas become

$$R_Y=\frac{R_{\mathrm{\Delta }}}{3}\text{or}{R}_{\mathrm{\Delta }}=3R_Y$$

基本分析方法

节点法

Nodal Analysis

Steps to Determine Node Voltages:

1. Select a node as the reference node. Assign voltages v₁, v₂, ..., v_n-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.
2. Apply KCL to each of the n-1 nonreference nodes. Use Ohm's law to express the branch currents in terms of node voltages.
3. Solve the resulting simultaneous equations to obtain the unknown node voltages.

网孔法

Mesh Analysis

Steps to Determine Mesh Currents:

1. Assign mesh currents i₁, i₂, ..., i_n to the n meshes.
2. Apply KVL to each of the n meshes. Use Ohm's law to express the voltages in terms of the mesh currents.
3. Solve the resulting n simultaneous equations to get the mesh currents.

解题步骤

1. Carefully define the problem.

2. Present everything you know about the problem.

3. Establish a set of alternative solutions and determine the one that promises the greatest likelihood of success.

4. Attempt a problem solution.

5. Evaluate the solution and check for accuracy.

6. Has the problem been solved satisfactorily? If so, present the solution; if not, then return to step 3 and continue through the process again.