Keyi Yuan, Electric Circuit (2018 Spring)

Electric Circuits Discussion 1

Keyi Yuan Teaching assistant Mar.13 2018

1 Keyi Yuan, Electric Circuit (2018 Spring) Contents

• Introduction

• Homework1

• Discussion Review Part 1: Basic Concepts

• Discussion Review Part 2: The way to analysis the circuits (KCL, KVL)

• Discussion Review Part 3: Circuit Theorems

• Extension Part 1: Wye - Delta transformations

2 1. Introduction

3 Introduction

• Professor: Yu Liu • Teaching assistant: Keyi Yuan, Kaihao Guo • Course Page: http://pspal.shanghaitech.edu.cn/courses_Spring2018_EE111.html • Discussion Class Time: 16:00-17:30, Tuesday; 19:30-21:00, Friday • Discussion Class Classroom: TBD • Office Hour Time: After Discussion Class or TBD • Office Hour Address: TBD • Email: [email protected], [email protected] • QQ: 1326697979

Welcome all students no matter which class are you in to attend this discussion class!

4 Introduction

• The time homework hand-in: Monday in class (Before 10:15) • The time homework hand-out: Monday in discussion class • No late or cheating homework will be accepted. • The deadline you can appeal your homework score: 22:00 Sunday of the week you get the homework. (Only for this 2 TAs) • You should save your homework and hand-in it again after Final-Exam. (5 credits)

5 Introduction

• For Discussion Class: Quiz? Not compulsive. Review what you have learnt. Some extended knowledge. And some surprises ! ! ! • You need to have: Differential Equations, Calculator (991ES-Plus)

Please feel free to let us know how to improve this course by any way!

6 Introduction

7 Introduction

• For Homework: Hand in it before class! A4; Bind all your paper. Transform the fraction to the float. Write all units! Write your process clearly.

8 2. Homework 1

9 Problem 1

10 Problem 2

11 Problem 3

12 Problem 4

13 Problem 5

14 Problem 7

15 3. Basic Concepts

16 Basic Concepts • What is Charge?

How can you get the relationship between charge, current and time?

What is DC and AC?

17 Example 1

• Determine the total charge entering a terminal between and if the

current passing the terminal is . = 1 = 2 ( = 3 − t A

18 Basic Concepts

• How can you get the relationship between charge, current and time? • is defined between node and node.

• What is the formula about power and energy?

• (Important !) Sign Convention

19 Basic Concepts

• (Important !) Circuit Elements 1. A voltage-controlled (VCVS). 2. A current-controlled voltage source (CCVS). 3. A voltage-controlled (VCCS). 4. A current-controlled current source (CCCS).

20 Example 2

• Calculate the power supplied or absorbed by each element in the figure.

21 4. The way to analysis the circuits

22 Circuit Analysis

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

• In other words, a branch represents any two-terminal element.

• The circuit figure shows 5 branches in the circuit. There is two sources one 20V voltage source, one 4A current source, three resistors. Circuit Analysis

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

• When a short circuit has two nodes it actually becomes one node. Circuit Analysis

• Loop • A loop is any closed path in a circuit.

• Loop counts starting at a node passing through a set of nodes and returning to the starting node without passing through any node more than once.

• A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop (Mesh). Circuit Analysis

• Relation • A network with b branches, n nodes and l independent loops will satisfy the fundamental theorem of network topology,

• b = l + n – 1

Branch: 5 Node: 3 Independent Loop: 3 Circuit Analysis

KCL KVL Review: Nodal Analysis – Three Steps

• Given a circuit with n nodes, the nodal analysis is accomplished via three steps: 1. Select a node as the reference (i.e., ground) node. Define the node (except reference node and the ones set by the voltage sources). Voltages are relative to the reference node.

2. Apply KCL at nodes with unknown voltage, expressing current in terms of the node voltages (using the I-V relationships of branch elements). Special cases: floating voltage sources.

3. Solve the resulting simultaneous equations to obtain the unknown node voltages. Review: Steps

• Mesh analysis follows these steps:

1. Assign mesh currents i1,i2,…in to the n meshes 2. Apply KVL to each of the n mesh currents. 3. Solve the resulting n simultaneous equations to get the mesh currents. Example 3 • Find vo in the circuit, which has § 4 nodes § 3 meshes § 2 dependent sources

• Nodal § 3 node-voltage equations § 2 constraint equations

• Mesh § 1 supermesh § 4 constraint equations

3 0 Example 3 Example 3 Nodal vs. Mesh

• In principle both the nodal analysis and mesh analysis are useful for any given circuit. • What then determines if one is going to be more efficient for solving a circuit problem?

• There are two factors that dictate the best choice: § The nature of the particular network § The information required Nodal Analysis if…

• If the network contains: § Many parallel connected elements § Current sources § Supernodes § Circuits with fewer nodes than meshes

• If node voltages are what are being solved for • Non-planar circuits can only be solved using nodal analysis

• This format is easier to solve by computer § Easy to program Mesh Analysis when…

• If the network contains: § Many series connected elements § Voltage sources § Supermeshes § A circuit with fewer meshes than nodes • If branch or mesh currents are what is being solved for. • NOT SUITABLE FOR NONPLANAR CIRCUITS • Mesh analysis is the only suitable analysis for transistor circuits. Mesh Analysis and Dependent Sources

• A dependent source is a current source or voltage source that depends on the voltage or current of another element in the circuit. When a dependent source is contained within an essential mesh, the dependent source should be treated like an independent source. After the mesh equation is formed, a dependent source equation is needed. This equation is generally called a constraint equation. This is an equation that relates the dependent source’s variable to the voltage or current that the source depends on in the circuit. The following is a simple example of a dependent source.

(https://en.wikipedia.org/wiki/Mesh_analysis)

Discussion 2 36 Supernode & Supermesh

- +

VLL Va Vb

I1 R2 R4 I2

Q:When Supernode & Supermesh? Supernode: Only voltage source can be contained

S15 upermesh: Only current source can be conLteactuiren2ed. 5. Circuit Theorems Linearity Property

Response Excitation System x f(x)

• Linearity is a combination of § homogeneity (scaling) property Excitation Response α · x System α · f(x) α is constant § additivity property

Excitation Response System x l + x2 f x l + f x2 Linear Circuit

• In a circuit, § Excitation: Sources § Response: Voltage or current

• A linear circuit consists of only linear elements (resistorscapacitors and inductors), linear dependent sources, and independent sources. § Linear means I-V characteristic of elements/sources are straight lines when plotted. Superposition

• The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone. Source Transformation

• A source transformation is the process of replacing a voltage

source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa. • These transformations work because the two sources have equivalent behavior at their terminals: § If the sources are turned off, resistance at the terminals are both R § If the terminals are short circuited, the currents need to be the same. Dependent Sources

• Source transformation also applies to dependent sources • The same relationship between the voltage and current holds here: Example 4 Find the current in the 10kΩ resistor in the circuit in the figure below by making a succession of appropriate source transformations. Example 6 Thevenin’s Theorem

• Thevenin’s theorem states that a linear two terminal circuit may be replaced with a voltage source in series with a resistor: § The voltage source’s value is equal to the open circuit voltage at the terminals. § The resistance is equal to the resistance measured at the terminals when the independent sources are turned off. Norton’s Theorem Three ways to use Thevenin’s Theorem Something puzzles you

• It is possible for the result of this analysis to end up with a negative resistance; § This implies the circuit is supplying power § This is reasonable with dependent sources [Only With dependent source] Example 5 Example 6 Obtain the Norton equivalent circuits of the circuit in the figure to terminals a and b. [With dependent and independent source] Maximum Power Transfer

• In many situations we want to transfer maximum power to the load.

52 For realistic circuits ,we often find their Thévenin equivalent circuit to define which RL makes maximum load power

RTh

+ iL 2

+ 2 æ VTh ö

v RL p = i R =ç ÷ R – L L L L VTh ç R ÷ è Th + RL ø –

dp é(R + R )2 - R ´2(R + R )ù =V 2 ê Th L L Th L ú = 0 dR Th 4 L ë (RTh + RL ) û

Þ RTh = RL 53 Find RL that makes the load power maximum

54 Example 7 Determine the maximum power that can be delivered to the variable resistor R in the circuit. 6. Wye-Delta Transformations

56 Extension: Wye-Delta Transformations

• There are cases where resistors are neither parallel nor series. • Consider the bridge circuit shown here. • This circuit can be simplified to a three-terminal equivalent. Prove: Wye-Delta Transformations Prove: Wye-Delta Transformations Prove: Wye-Delta Transformations Prove: Wye-Delta Transformations Delta to Wye

• The conversion formula for a delta to wye transformation are: Wye to Delta

• The conversion formula for a wye to delta transformation are: Example 8

• Find the equivalent resistance Example 8

• Find the equivalent resistance Quiz

• Find Vth and Rth. Quiz-Solution

• Find Vth and Rth.