 # Chapter 2 Resistive Circuits (電阻性電路)

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Chapter 2 Resistive Circuits (電阻性電路)
Introduction to Circuits Theory and Digital Electronics Chapter 2 Resistive Circuits (電阻性電路)

2.1 電阻串聯與並聯 以等效電路(equivalent resistances) 取代串聯或並聯電阻。 利用等效電路來分析電路。

Series Resistances (串聯電阻器)
Ohm’s law KVL open/closed

Series Resistances (串聯電阻器)

Parallel Resistances (並聯電阻器)
Ohm’s law KCL

Example 2.1 Find a single equivalent resistance

Series VS. Parallel Circuits

Exercise 2.1 (b)

2.2 Circuit Analysis using Series/Parallel Equivalents

Example 2.2 Circuit Analysis

Solve the remaining circuits
Check by KCL &KVL Power?

Exercise 2.2 (a)

2.3 Voltage/Current-Divider Circuits
Voltage-Division Principle (分壓定律)：電壓分配至串聯電阻之比例為其電阻值與總電阻值之比。

Example 2.3 Application of the Voltage-Division Principle

Current-Division Principle (分流定律)：電流分配至兩並聯電阻比例為另一電阻與總電阻值之比。

Example 2.5 Application of the Current-Division Principle

Exercise 2.4 (b)

Figure The voltage-division principle forms the basis for some position sensors. This figure shows a transducer that produces an output voltage vo proportional to the rudder angle θ.

2.4 Node-Voltage Analysis
Although they are very important concepts, series/parallel equivalents and the current/voltage division principles are not sufficient to solve all circuits.

Node Voltage Analysis 選定參考節點(reference node)並標示其他結點的電壓符號。

KCL: Node 2 (流出 node 2 的淨電流為0) KCL: Node 3

Example 2.6 KCL: Node 1 KCL: Node 2 KCL: Node 3

Example 2.7 KCL: Node 1 KCL: Node 2 KCL: Node 3

Example 2.7 三元一次方程組 Matrix Form

G V I Solve inverse matrix Matlab Example 2.7 clear
V=G\I

Example 2.9 KCL: Node 1 KCL: Node 2 解聯立方程式求 voltages

Circuits with Voltage Sources

Circuits with Voltage Sources

Circuits with Voltage Sources

Circuits with Voltage Sources

Exercise 2.13 Write a set of independent equations for the node voltage in Fig. 2.27.

KVL: KCL: Supernode enclosing 10-V source KCL for node 3 KCL at reference node

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.

Node-Voltage Analysis with a Dependent Source

Node-Voltage Analysis with a Dependent Source

Example 2.10 KCL Node 1: KCL Node 2: KCL Node 3:

Example 2.11 Node-Voltage Analysis with a Dependent Source

Example 2.10 Node-Voltage Analysis with a Dependent Source
KVL: KCL: Supernode enclosing controlled source KCL for node 3 KCL at reference node KCLs are dependent KVL: integrated with any two of the three KCL equations for independence.

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.

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.

2.5 Mesh Current Analysis (網目電流分析法)

Mesh Current Analysis

Mesh Current Analysis 利用 KVL 環繞網目(mesh)，方程式中的未知數是電流，不論網目中電流方向如何預設，只要 KVL 及歐姆定律正確使用即可。 一般進行網目分析時網目電流均取順時針方向。 多個網路電流( mesh currents) 都流過一個電路元件時，我們設通過此元件的電流為這些網路電流的和(注意方向與正負)。

Mesh Current Analysis KVL for mesh 1 KVL for mesh 2

Choosing the Mesh Currents

Example 2.12 KVL for mesh 1 KVL for mesh 2 KVL for mesh 3

Exercise 2.18 KVL for mesh 1 KVL for mesh 2 KVL for mesh 3 KVL for mesh 4

Example 2.13 For mesh 1

Mesh Currents in Circuits Containing Current Sources

Mesh Currents in Circuits Containing Current Sources

Supermesh KVL for mesh 1? KVL for mesh 2?

Circuits with Controlled Sources
KVL for supermesh Source current Controlling voltage

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.

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.

2.6 Thévenin Equivalent Circuits (戴維寧等效電路)

Thévenin Equivalent Circuits (戴維寧等效電路)

Thévenin Equivalent Circuits

Example 2.16

Finding the Thévenin Resistance Directly
When zeroing a voltage source, it becomes a short circuit. When zeroing a current source, it becomes an open circuit.

Finding the Thévenin Resistance Directly
We can find the Thévenin resistance by zeroing the sources in the original network and then computing the resistance between the terminals.

Example 2.17

Exercise 2.28 Find by zeroing the sources

Note: we can not find the Thévenin resistance by zeroing the dependent source.
a. Determine the open-circuit voltage Vt = voc. b. Determine the short-circuit current isc. c. Thévenin resistance Rt

Norton Equivalent Circuits (諾頓等效電路)

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 ).

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 .

Example 2.19 Norton Equivalent Circuit
KCL 分壓定律 (voltage-divider principle)

Exercise 2.29 Norton Equivalent Circuit

Source Transformations
Source Transformation 為外部等效(external equivalence)非內部等效。

Source Transformations
If nodes a & b are open External equivalence Source Transformation 要注意電流源、電壓源方向(以維持等效)。

Source Transformations
If nodes a & b are open Not internal equivalence

Source Transformations

Example 2.20 Source Transformations

Maximum Power Transfer

Maximum Power

Example 2.21 Find the load resistance for maximum power transformation

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

2.7 SUPERPOSITION PRINCIPLE (疊加原理)

Suppose the response is the voltage across

Suppose the response is the voltage across

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).

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.

Example 2.22 Find

2.8 Wheatstone Bridge 惠斯登電橋測直流電阻

WHEATSTONE BRIDGE KCL node a KCL node b KVL upper loop KVL lower loop

WHEATSTONE BRIDGE

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.

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?