27 G V I Solve inverse matrix Matlab Example 2.7 clear V=G\I
28 Example 2.9KCL: Node 1KCL: Node 2解聯立方程式求 voltages
29 Circuits with Voltage Sources 因為電壓源與 相連，所以我們無法寫出只含節點電壓的電流方程式。?
30 Circuits with Voltage Sources 當分枝處於兩非參考節點之間且包含一個電壓源時，即可使用超節點技術(supernode)。將包含電壓源的節點含括為一supernode(超節點)。
31 Circuits with Voltage Sources 流入(流出)supernode (封閉表面, closed surface)的淨電流(net current)為0.KCL: Supernode 包含10V voltage sourceNote: We obtain dependent equations (相依) if we use all of the nodes in a network to write KCL equations. (KCL S1 與 KCL S2 相依)
32 Circuits with Voltage Sources 將電壓源連結的節點寫出KVL以獲得另外的獨立方程式。KVL:
33 Exercise 2.13 Write a set of independent equations for the node voltage in Fig. 2.27.
34 KVL:KCL: Supernode enclosing 10-V sourceKCL for node 3KCL at reference node
35 KCL: Supernode enclosing 10-V source 3 variables 4 equations?KVL:(1)KCL: Supernode enclosing 10-V source(2)KCL for node 3(3)KCL at reference node(4)(2) +(3) =(4)(2) (3) and (4) are dependent(1) must be included with any two of the three KCL equations for independence.
41 Example 2.11 Node-Voltage Analysis with a Dependent Source
42 Example 2.10 Node-Voltage Analysis with a Dependent Source KVL:KCL: Supernode enclosingcontrolled sourceKCL for node 3KCL at reference nodeKCLs are dependentKVL:integrated with any two of the three KCL equations for independence.
43 Node-Voltage Analysis 1. Select a reference node and assign variables for the unknown node voltages. If the reference node is chosen at one end of an independent voltage source, one node voltage is known at the start, and fewer need to be computed.2. Write network equations. First, use KCL to write current equations for nodes and supernodes. Write as many current equations as you can without using all of the nodes. Then if you do not have enough equations because of voltage sources connected between nodes, use KVL to write additional equations.
44 3. If the circuit contains dependent sources, find expressions for the controlling variables in terms of the node voltages. Substitute into the network equations, and obtain equations having only the node voltages as unknowns.4. Put the equations into standard form and solve for the node voltages.5. Use the values found for the node voltages to calculate any other currents or voltages of interest.
45 2.5 Mesh Current Analysis (網目電流分析法) 待求如何簡少未知數，方便求解？
53 Mesh Currents in Circuits Containing Current Sources 要利用KVL 時電流源上的電壓為何？將電流源上的電壓設為0 是常犯的錯誤。
54 Mesh Currents in Circuits Containing Current Sources 電流源在旁邊KVL for mesh 2
55 Supermesh KVL for mesh 1? KVL for mesh 2? 結合 meshes 1 and 2 為一個 supermesh (超網目).KVL for supermeshKVL for mesh 3Additional Equation
56 Circuits with Controlled Sources KVL for supermeshSource currentControlling voltage
57 Mesh-Current Analysis 1. Define the mesh currents. Select a clockwise direction for each of the mesh currents.2. Write network equations.First, use KVL to write voltage equations for meshes. Express current sources in terms of the mesh currents. Finally, if a current source is common to two meshes, write a KVL equation for the supermesh.
58 3. If the circuit contains dependent sources, find expressions for the controlling variables in terms of the mesh currents.4. Put the equations into standard form. Solve for the mesh currents by use of determinants or other means.5. Use the values found for the mesh currents to calculate any other currents or voltages of interest.
72 Thévenin/Norton-Equivalent-Circuit Analysis 1. Perform two of these:a. Determine the open-circuit voltage Vt = voc.b. Determine the short-circuit current In = isc.c. Zero the sources and find the Thévenin resistance Rt looking back into the terminals(if NO dependent source is in the circuit ).
73 2. Use the equation Vt = Rt In to compute the remaining value. 3. The Thévenin equivalent consists of a voltage source Vt in series with Rt .4. The Norton equivalent consists of a current source In in parallel with Rt .
74 Example 2.19 Norton Equivalent Circuit KCL分壓定律 (voltage-divider principle)
84 Example 2.21 Find the load resistance for maximum power transformation
85 2.7 SUPERPOSITION PRINCIPLE (疊加原理) The superposition principle states that the total response is the sum of the responses to each of the independent sources acting individually.In equation form, this is
86 2.7 SUPERPOSITION PRINCIPLE (疊加原理) 要獲得某一 independent source 所造成的response，則將其他 independent source zeroing (reduce the source value to zero)。
89 Linearity Ohm’s law is a linear equation. The controlled source ics=Kix is also a linear equation.Superposition principle does not apply to circuits that have element(s) described by nonlinear equation(s).
90 Dependent source do not contribute a separate term to the total response. We must not zero dependent source in applying superposition。However, dependent source affect the contributions of the independent sources.
95 Example 2.23 If R1=1-kΩ, R3= 0~1100-Ω steps by 1-Ω R2: 1k, 10k, 100k or 1MΩ(a) What is the value of Rx such that the bridge is balanced with R3=732 Ω, R2=10k Ω?(b) What is the largest value of Rx for which bridge is balanced.
96 Example 2.22 If R1=1-kΩ, R3= 0~1100-kΩsteps by 1-Ω R2: 1k, 10k, 100k or 1MΩ(c) Suppose R2=1M Ω. What is the increment between values of Rx for which the bridge can be precisely balanced?