1、原理电路图的设计 2、实验数据表格 3、教师签字有效，和《实验报告》 一起交。 实验《预习报告》要求： 一班赶紧去约实验一（第6周做）

Methods of Analysis #37. No.6 NO.5 Chapter 3 Methods of Analysis #37. No.6 So far, we have analyzed relatively simple circuits by applying Kirchhoff’s laws in combination with Ohm’s law. We can use this approach for all circuits, but as they become structurally more complicated and involve more and more elements, this direct method soon becomes cumbersome. In this chapter we introduce several powerful techniques of circuit analysis, such as: Nodal Analysis and Mesh Analysis（结点电压法和网孔电流法）. These techniques give us two systematic methods of describing circuits with the minimum number of simultaneous equations(联立方程).

Find I1 , I2 , I3 ,I4 , I5 , I6 For Example 如何确定KCL和KVL的独立方程数? When setting mesh currents,it’s conventional to assume that each mesh current flows clockwise. 如何确定KCL和KVL的独立方程数?

2. Branch-Current Analysis 支路电流法 3. Mesh Analysis 网孔电流法  Main Points： Graph theory 图论 2. Branch-Current Analysis 支路电流法 3. Mesh Analysis 网孔电流法 4. Loop Analysis 回路电流法 5. Nodal Analysis 结点电压法

§3-1 Graph theory 图论

电路图 circuit diagram 电路的图 graph 拓扑名词解释 Topological terms 1、电路的图 graph 电路图 circuit diagram 电路的图 graph 引入电路的图首先是为了 研究如何列出独立的KCL、KVL方程，及方程个数。

2、结点 (node)、支路 (edge) 3、有向图： 标有支路电流参考方向的图。(电压一般取关联参考方向)

4、连通图 connected graph： 图中任意两点间至少存在一条路径的图，否则是非连接通图。 5、子图 ：如果有一个图G，从图G中去掉 某些支路和某些节点所形成的图H，称为 图G的子图。

若各条支路除所联接的结点外不再交叉，则为平面图。否则为非平面图。 6、平面图 planar graph： 若各条支路除所联接的结点外不再交叉，则为平面图。否则为非平面图。 非平面图 平面图

7.树 spanning trees：包含连通图G中的所有节点，但不包含回路的连通子图，称为G的树。    n= 5   

9. 连支 link：包含于图，但又不属于树T的支路。 8. 树支 branch：树的支路。 9. 连支 link：包含于图，但又不属于树T的支路。 Example: 树支 : 1, 3, 5, 6 连支 : 2, 4, 7, 8

树支数= n - 1，连支数 l = b - (n-1) = b - n + 1 ………. 图有许多不同的树（nn-2），但无论哪一个树，树支数总是(n-1) 树支数= n - 1，连支数 l = b - (n-1) = b - n + 1 （b----支路数 n-----节点数） 对5个结点的电路，能列出4个独立的KCL方程，因为每个方程中均包含着其余方程没有涉及的支路电流，所以任一方程不可能由另外3个方程导出。 可以证明： 对于n个结点的电路，可以列出(n-1)个独立的KCL方程。

? KVL独立方程数 = b - n + 1 10. 基本回路(单连支回路) fundamental loop: (1) 对任一个树，每加一个连支，便形成一个 只包含一个连支的回路 ——基本回路。 ? (2) 基本回路数 = 连支数 =独立回路数 abca cedc acda abedca 四个基本回路 KVL独立方程数 = b - n + 1

例 说明：平面图的全部网孔就是一组独立回路， 数目恰好是该图的独立回路数。 共有8条支路，u、i共16个未知数，需要16个独立方程 节点数n=5 支路数b=8 网孔数为4 例 共有8条支路，u、i共16个未知数，需要16个独立方程 KCL:4个独立方程（n-1） KVL:4个独立方程（b-n+1) （即：网孔数） VAR:8个独立方程

§3-2 Branch-Current Analysis 支路电流法 以支路电流 ik （b条）为求解变量列方程。求出各支路电流 后，再（用VCR）求各支路电压。 VCR： KCL： KVL： uk = f ( ik ) 依据：

In a circuit with b edges and n nodes, there are 2b variables should be valued. Then: KCL for n nodes: only n-1 equations are independent. KVL for loops: only b-n+1 equations are independent. (only KVL for meshes) That is called 2b analysis. VAR for edges: b equations.  So, (n-1)+(b-n+1)+b=2b, 2b equations to value 2b variables.

Example :

Steps to Branch-Current Analysis: （1）确定各支路电流ik 参考方向； （2）列独立的结点KCL方程(n-1个)； （3） 列独立的回路KVL方程(b-n+1个)， 溶入元件VCR。 （4）求解方程，求出支路电流； （5）依据支路约束关系，求解支路电压；

Example: Find the edge currents. I2 I1 I3 Uab I4 I5 I6 Apply KVL to the three meshes in turn. I6

讨论 I1 I2 I3 I4 I5 I6 Uab 1、对含无伴电流源 (unaccompanied current sources)的电路， 列支路电流方程时，可增 加一个变量：该电流源上 的电压。 （因其电压无法用I 表示） 故应补充一个方程 ij=is （已知） 。 也可不列网孔bacb的KVL方程。 这样就不会出现变量uab，仍可保证变量数与方程数一致。

繁！ I1 I2 2、验算方法：不能用ΣI=0，要用一个没用过的回路计算ΣU是否为0。 I3 Uab I4 I5 I6 3、对任何电路均可用支路电流法求解。 4、支路电流法的缺点： 繁！

§3-3 Mesh Analysis 网孔电流法 (The Mesh-Current Method) Mesh analysis provides a general procedure for analyzing circuits using Mesh currents as the circuit variables. This method is convenient and reduces the number of equations.

Example: A circuit with three meshes Im1 Im2 Im3 When setting mesh currents,it’s conventional to assume that each mesh current flows clockwise.

Steps to Determine Mesh Currents: Assign mesh currents Im1, Im2,…Imn to the n meshes. Im1 Im2 Im3

2. Apply KVL to each of 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. Im1 Im2 Im3

Im1 Im2 Im3 由KVL：（并将支路电流用网孔电流表示） Apply KVL to each mesh.

3.Solve the resulting n simultaneous equations to get the mesh currents.

Im1 Im2 Im3 归纳规律性 ∵

Im1 Im2 Im3 rearrange mutual resistances self resistances

Rkk: self resistance (positive) ，k =1 , 2 ,  , m General: where Rkk: self resistance (positive) ，k =1 , 2 ,  , m Rjk: mutual resistance (negative) , (by choosing all the meshes in the same clockwise or counterclockwise direction) 其中：ij为网孔电流（网孔电流采用同一方向）； Rjj为网孔j的自电阻(取正) Rij为网孔 i, j的互电阻(取负) usjj为网孔j的全部电压源的代数和(升为正) uSlk : the summation of the voltage rises in loop k in the direction of the loop currents.

Steps to Mesh Analysis : (1)选网孔电流为变量，并标出各变量； (2)按照规律列网孔方程； (3)解网孔电流； (4)解其他变量；

Find I1 , I2 , I3 (using Mesh Analysis) Example Find I1 , I2 , I3 (using Mesh Analysis) Im2 Im1

Mesh Analysis with Current Sources Case 1 When a current source exists only in one mesh: Example Solution:

When a current source exists between two meshes: Case 2 When a current source exists between two meshes: R1 R2 R3 + U - US1 US2 Is solutions: Set u as the voltage across the current source, then add a constraint equation. R4 R5 Example (unaccompanied current sources)

Solution : + U - R2 US2 US1 R3 R4 R5 R1 Is

Example: A circuit with three meshes Im1 Im2 Im3

Example Use Mesh Analysis to obtain I1, I2, I3 注意：增加控制量与网孔电流的关系方程

3. Mesh Analysis is only applicable to planar. Note ： For mesh analysis, we write b-(n-1) mesh equations, while for Branch-Current Analysis we write b equations. 2. Pay attention to the added constraint equation. 如：对受控源：需补充方程时，受控源的控制量要用网孔电流表示。 3. Mesh Analysis is only applicable to planar.

§3-4 Loop Analysis 回路电流法 (The Loop-Current Method)

Loop Analysis 回路电流法: 以基本回路为独立回路， 以回路电流（连支电流）为变量列方程。 是网孔电流法的推广，适用于平面或非平面电路。

Steps to Loop Analysis : 1. Given a circuit, form the corresponding graph of it; Select any spanning tree; 3. Determine all of the fundamental loops; 4. Establish link currents in the given circuit , then proceed as for mesh analysis.

ú û ù ê ë é = u i R M K In matrix form: NN u i R M K 2 1 22 21 12 11 In matrix form: (In general) Where: Rkk: sum of the resistances in fundamental loop k;(+) Rjk=Rkj: j≠k ;（+ or -） uk: sum of all voltage sources in loop k , with voltage rise treated as positive; ik: unknown loop current of loop k .

Write the Loop-Current equations . Example1 Write the Loop-Current equations . Ii3 Ii2 Ii1 Solution： (1) To draw the graph. Select a spanning tree. Determine all of the fundamental loops;

(2) To write the Loop-Current equation

Note ： Loop Versus Mesh Analysis 网孔法常用于平面电路； 回路法多用于非平面电路及 大规模网 络分析中。 问题：除网孔电流、基本回路电流外，还存在完备且独立的一组变量吗？

Example A I4 + – + – + – E1 E2 E3 U + – R4 I1 I2 R1 R2 I3 R3

§3-5 Nodal Analysis 结点电压法 (The Node-Voltage Method ) Nodal analysis provides another general procedure for analyzing circuits using node voltages as the circuit variables. This method is convenient and reduces the number of equations.

Typical circuit for Nodal Analysis V1 V2 IS1 There are two essential nodes V1,V2

Steps to Determine Node Voltages: Select a node as the reference node(ground), define the node voltages V1, V2,… Vn-1 to the remaining n-1nodes . The voltages are referenced with respect to the reference node. Apply KCL to each of the n-1 independent nodes. Use Ohm’s law to express the branch currents in terms of node voltages. Solve the resulting simultaneous equations to obtain the unknown node voltages.

So at node 1 and node 2, we can get the following equations. KCL VCR

Sum of currents is entering the node G11、G22 （ self conductance ）：”+” G12、G21 （ mutual conductance ） ：”-” E ttand The right side of Eq. ： Sum of currents is entering the node

Gkk: sum of the conductance connected to node k;(+) In general： Gkk: sum of the conductance connected to node k;(+) Gjk=Gkj: negative of the sum of the conductances directly connecting nodes jand k, j≠k ; (-) iskk: sum of all independent current sources directly connected to node k , with currents entering the node treated as positive. vk: unknown voltage at node k;

Steps to Node Voltages: (1)选参考点，对结点进行编号； (2)按照规律列结点方程； (3)解结点电压； (4)解其他变量；

Example: Find the edge currents (using Nodal Analysis) V1 V2 V3

Example : Determine the voltages at the nodes. We can obtain the equations by mere inspection of the circuit Solution: 1 2 3 受控电流源

Branch-Current Analysis 支路电流法 Mesh Analysis 网孔电流法 No.7 Chapter 3 Methods of Analysis Branch-Current Analysis 支路电流法 Mesh Analysis 网孔电流法 (Loop Analysis 回路电流法) Nodal Analysis 结点电压法

Nodal analysis vs. Mesh analysis We would like to consider nodal analysis and loop analysis as mirror scenarios: In nodal analysis, the unknown parameters are nodal voltages and KCL is used to solve the problem. If there are N nodes in the network, N-1 equations are required to solve the problem. In Mesh analysis, the unknown parameters are current and KVL is employed to determine the unknowns. If there are L meshes, L independent equations are needed to describe the network.

Mesh simultaneous equations: Nodal simultaneous equations:

Nodal Analysis with Voltage Sources Case 1 If a voltage source is connected between the reference node and a nonreference node. Solution: we simply set :

The voltage source should be converted to a current source first. Case 2 If a voltage source (dependent or independent) is connected with a resistor in series. Solution: The voltage source should be converted to a current source first. A supernode may be regarded as a closed surface enclosing the voltage source and its two nodes.

Set i as the current across the voltage source ; Case 3 If the voltage source is connected between two nonreference nodes(without a series-connected resistor). 无伴电压源 Solution: Set i as the current across the voltage source ; A supernode may be regarded as a closed surface enclosing the voltage source and its two nodes. 2) Add a constraint equation ; 增加一个电压源与结点电压的关系方程

Example Write the node-voltage equations I 解：(1) 选参考点及结点电压为变量。 (2) 等效变换：电压源串联电阻电流源并联电阻

I (3)列结点方程

Note ： 1. The simultaneous equations may be solved using calculators such as Matlab. 2. Nodal Analysis is applicable to every circuits. ( planar or nonplanar )

Nodal Versus Mesh Analysis Both provide a systematic way of analyzing a complex network. When is the nodal method preferred to the mesh method? A circuit with fewer nodes than meshes is better analyzed using nodal analysis, vice versa. 2. Based on the information required. Node voltages required-----nodal analysis Branch or mesh currents required---mesh analysis Examples: E page130 Nodal Analysis and Mesh Analysis, which is more efficient? The answer depends on the circuit under consideration. The method that results in the least number of unknowns is generally efficient.

Example A I4 + – + – + – E1 E2 E3 U + – R4 I1 I2 R1 R2 I3 R3

Branch-current method i4 i2 i6 i5 Branch-current method Node-Voltage Method Example Find：i3  u2 R4 + – i3 R3 b a i1 + R1  i1 R2 R5 u2 + uS – R6 – c

Mesh-Current Method Example Find：i3 Im1 Im2 Im3 Im4  u2 R4 + – i3 R3 b i1 + R1  i1 R5 u2 + R2 uS – R6 – c

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本章小结 线性电路的一般分析方法 (1) 普遍性：对任何线性电路都适用。 (2) 系统性：计算方法有规律可循。 方法的基础 (1) 普遍性：对任何线性电路都适用。 (2) 系统性：计算方法有规律可循。 方法的基础 （1）电路的连接关系—KCL，KVL定律。 （2）元件的电压、电流关系特性 VCR 。 方法的特点 不改变电路的结构，根据KCL、KVL及元件电压和电流关系列方程、解方程。 根据列方程时所选变量的不同可分为支路电流法、回路电流法和结点电压法。

P74 3-7 《电路》: 《 Fundamentals of Electric Circuits》: P113 3.39 3.40 作业5： 《电路》: P74 3-7 《 Fundamentals of Electric Circuits》: P113 3.39 3.40

P110 3.13 P111 3.20 《电路》: P75 3-12 (mesh or loop) 作业6： 《电路》: P75 3-12 (mesh or loop) 《 Fundamentals of Electric Circuits》: P110 3.13 P111 3.20