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Title: Circuit Analysis Chapters 1-8
Description: From the book, Basic Engineering Circuit Analysis 11th Edition, good notes on the first eight chapters. Good for your basic or intro Circuit Analysis class.
Description: From the book, Basic Engineering Circuit Analysis 11th Edition, good notes on the first eight chapters. Good for your basic or intro Circuit Analysis class.
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1
...
One candela is the luminous intensity, in a given direction,
of a source that emits monochromatic radiation of frequency 540 × 1012 Hz
Standard SI Prefixes:
From
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Chapter 1 Page 1
1
...
○ In B, at any point in the wire, 3 amperes of charge pass from right to left each second
...
Chapter 1 Page 2
Types of Current:
• Alternating Current (AC)
○ Common current
○ Fridge, stove, washer, dryer, etc
...
Voltage:
• Difference in energy level of a unit charge located at each point
• Between two points
• AKA Electromotive force, or potential
• Example:
○ Dropping a bowling ball into water
...
○ Figure 2
Bulb uses energy = charges leaving the bulb have less energy than those
entering
Charges expend energy as they move through the bulb
Charges gain energy as they pass through the battery
Current enters positive end of bulb = absorbing energy
Current leaving positive end of battery = supplying energy
Chapter 1 Page 4
○ Figure A
Energy being supplied to the box
2 A, or 2 C of charge, are moving from A to B each second
With 3 V, each coulomb loses 3 J as it passes form A to B
Box is absorbing 6 J of energy per second
Positive current enters positive terminal on box
○ Figure B
Energy being supplied by the box
2 A, or 2 C of charge, are moving from A to B each second
With 3 V, each coulomb gains 3 J as it passes form A to B
Box is supplying 6 J of energy per second
Positive current enters negative terminal on box
○ Negative current in one direction = positive current in opposite direction
○ Negative voltage in one direction = positive voltage in opposite direction
• Voltage (joules per coulomb) is the energy required to move a positive charge of 1 C
through an element (box)
...
4
Chapter 1 Page 6
1
...
□ In the Independent Voltage Source, we assume the voltage source delivers
v volts regardless of what is connected to the terminals
...
□ Physically impossible
□ Can also be done to the Independent Current Source, but with infinite
voltage
○ Independent Voltage Source
Two-terminal element that maintains a specified voltage between its
terminals regardless of the current through it shown by the graph
Terminal A is v(t) volts positive with respect to B
○ Independent Current Source
Two-terminal element that maintains a specified current regardless of the
voltage across its terminals shown by the graph
i(t) is the specified current and the arrow indicates the positive direction of
current flow
• Dependent Sources:
○ Generate a voltage or current that is determined by a voltage or current at a specified
location in the circuit
○ Important to calculate/describe behavior of many electronic circuit elements
○ Metal-oxide-semiconductor field-effect transistors (MOSFETs) and bipolar
transistors
Used in lots of electronic equipment
Modeled with dependent sources
Analysis of electronic circuits involve the use of these controlled elements
○ The symbol for a dependent (or controlled) source is a diamond
Four types of dependent sources
In A and D, we are transforming voltage to voltage, and current to current,
respectively
...
Therefore, μ and β are dimensionless constants
...
Chapter 1 Page 8
2
...
1
• Example 2
...
2 Kirchhoff's Laws
Sunday, January 31, 2016
1:54 PM
Kirchhoff's Law
• Lumped-Parameter Circuit
○ Wires in circuits are assumed perfect conductor
○ Interconnections in circuits have zero resistance
○ Wire doesn't consume energy
Energy in circuits is lumped in each circuit element
• Node
○ A point of connection of two or more circuit elements
• Loop
○ Any closed path through the circuit in which no node is encountered more than once
• Branch
○ Portion of a circuit containing only a single element and the nodes at each end of the
element
...
5
Kirchhoff's Voltage Law (KVL)
• KVL
○ Algebraic sum of the voltages around any loop is zero
□ If sign '-' first, then voltage is negative
□ If sign '+' first, then voltage is positive
Example 2
...
11
Chapter 2 Page 10
2
...
17
Multiple Current Sources in a Single-Node Pair
Chapter 2 Page 12
•
Multiple Resistors in a Single-Node Pair Circuits
•
Chapter 2 Page 13
2
...
19
Simplifying Resistor Combinations
• To determine equivalent resistance at a pair of terminals of a network
○ Begin at the end of the circuit/opposite the terminal/right side
○ Combine resistors in series and/or parallel as needed to reduce the network to a single
resistor at the pair of terminals
• Example 2
...
1 on pg 54
• Tolerance
○ Typically, 5% and 10%, which specifies possible min and max resistance values
• Power Rating
○ Specifies the maximum power that can be dissipated by the resistor
Typically, 1/4 W, 1/2 W, 1 W, 2 W
○
• Example 2
...
24
Wye-Delta Transformation
•
• For two networks to be equivalent at each corresponding pair of terminals, it is necessary
that the resistance at the corresponding terminals be equal
•
Circuits with Dependent Sources
• Controlled sources are used to model many important physical devices
• Problem solving strategy
○ When writing KVL and/or KCL equations for the network, treat the dependent
sources as though it were an independent source
○ Write the equation that specifies the relationship of the dependent source to the
controlling parameter
○ Solve the equations for the unknowns
...
• Example 2
...
1 Nodal Analysis
Wednesday, February 03, 2016
8:31 AM
Nodal Analysis
• Systematic method to calculate all currents and voltages in circuits that contain multiple
nodes and loops
• In nodal analysis, the variables in the circuit are selected to be the node voltages
○ All other unknown variables are expressed in terms of node voltages
• One node is selected as reference node and all other node voltages are defined with respect
to the reference node
• The reference node is called ground, symbol: (pic)
○
○
• In an N-node circuit, exactly N-1 linearly independent KCL equations are needed to
determine the N-1 unknown node voltages
• Only reference node is selected, our task is to identify remaining N-1 nodes and write one
KCL equation for each of the nodes
Circuits Containing only Independent Current Sources
• Apply KCL and Ohm's Law
• Write N-1 linearly independent KCL equations
• Three techniques to solve simultaneous equations
○ Gaussian elimination
○ Matrix analysis
○ Matlab software or calculator
• Example 3
...
• Example 3
...
5
Circuits with an Independent Voltage Source Connected Between Two Nonreference Nodes
• Example 3
...
2 Loop Analysis
Wednesday, February 03, 2016
8:32 AM
Do Chapter 3 (PP 3)
Loop Analysis
• Uses KVL to determine a set of loop currents
• In general, there are (B-M+1) linearly independent KVL equations
○ B = Number of branches
○ M = Number of nodes
• A better way…
○ The number of "window panes" tells us how many equations
○ Picture
3 window panes
○ Picture
2 window panes
○ Example
○ Loop analysis = mesh analysis
○ Start from lower left corner, go clockwise
• In loop analysis, the unknown parameters are loop currents
• The current in a branch equals to the algebraic sum of all the loop currents passing through
the branches
○ Example 3
...
15
○ Example 3
...
1 Operational Amplifiers
Wednesday, February 17, 2016
8:40 AM
Operational Amplifier
• Op-amp
• Single most important integrated circuit for analog circuit design
• Originally designed to perform mathematical operations such as addition, differentiation, integration, etc
...
The input impedance is infinite
2
...
The open-loop gain (A) is infinite
4
...
2 Voltage Summation and Subtraction
Friday, February 19, 2016
9:06 AM
Voltage Summation (Inverting Version)
•
• An op-amp circuit that produces an output equal to the (inverted) sum of three separately
scaled input signals
Voltage Summation (Non-inverting Version)
•
•
Voltage Subtraction
Chapter 4 Page 20
•
• Using the amplifier in a differential mode to obtain an output proportional to the difference
between two scaled inputs
• Example
• Example
Summary
• Op-amps are characterized by
○ High-input resistance
○ Low-output resistance
○ Very high gain
• The ideal op-amp is modeled using
○
○
• Op-amp problems are typically analyzed by writing node equations at the op-amp input
terminals
Chapter 4 Page 21
5
...
1
Chapter 5 Page 24
5
...
3
• Find
○
• We set to zero the voltage source
○
• Now we set to zero the current source
○
Example 5
...
4
• Find
○
○
○
Applying Superposition
•
Chapter 5 Page 26
5
...
6:
Chapter 5 Page 27
•
Chapter 5 Page 28
6
...
Energy is stored in electric field
○ Inductors are capable of storing energy when a current is passing through them
...
2: Capacitors
Wednesday, March 23, 2016
8:38 AM
Capacitors
• A capacitor is a circuit element that consists of two conducting surfaces separated by nonconducting, or dielectric, material
• Capacitors are categorized by the type of dielectric material used between the conducting
plates
○ Each type is more suitable for particular applications
• Unit of capacitance is coulombs per volt, or Farad (F)
○ Typical values range from thousands of micro-farads to a few pico-farads
• Capacitance of two parallel plates of area A separated by distance d
○
: electrostatic permitivity of free space
□
: dielectric constant or relative permitivity of the insulator in between
• The charge on the capacitor is proportional to the voltage across it
○
C: capacitance of the capacitor in farads
Chapter 6 Page 30
○
• Power
○
• Energy
○
Capacitors
• Capacitors only store and release electrostatic energy; they don't create energy
• The capacitor is passive element and follows passive sign convention
Chapter 6 Page 31
Characteristics of Capacitors
• Capacitor blocks DC current or capacitor is an open circuit to DC current
○
Capacitor is often used to filter out unwanted DC voltage/current
When analyzing a circuit containing only DC voltage/current source, we can
replace capacitors with an open circuit before analysis
• Capacitor has continuity of voltage
○ Voltage across capacitor is always continuous
○
• Example 6
...
2
Chapter 6 Page 32
• Example 6
...
3
• Example 6
...
3: Inductors
Wednesday, March 23, 2016
2:37 PM
Inductors
• An inductor is a circuit element that consists of a conducting wire usually in the form of a
coil
• Inductors are typically categorized by the type of core on which they are wound
○ Each type is more suitable for particular applications
• The unit of inductance is volt-second per ampere, or Henry (H)
• Power
○
• Energy
○
• Inductors only store and release electromagnetic energy; they don't create energy
• The inductor is passive element and follows passive sign convention
Chapter 6 Page 34
Characteristics of Inductors
• Inductor is a short circuit to DC current
○
In analyzing a circuit containing only DC voltage/current source, we can replace
inductors with a short circuit before analysis
• Inductor has continuity of current
○ Current flowing through an inductor is always continuous
•
• Example 6
...
7
• Example 6
...
4: Overview
Tuesday, April 05, 2016
7:42 PM
Capacitance and Inductance
Specifications
• Capacitors
○ Capacitance, working voltage, tolerance
○ The working voltage is specified to keep the applied voltage below the breakdown
point of the dielectric
• Inductors
○ Inductance, resistance, tolerance, current rating
○ The major difference between wire-wound resistors and inductors is the wire material
○ Low resistance materials are used in inductors
Chapter 6 Page 36
Capacitors
• Series
○
• Parallel
○
Inductors
• Series
○
• Parallel
Chapter 6 Page 37
○
Extra
• Since the same current flows in each of the series capacitors, each capacitor gains the same
charge in the same time period
○
• The voltage across each capacitor will depend on this charge and the capacitance of the
element
○
Chapter 6 Page 38
7
...
2: General Form
Wednesday, April 06, 2016
8:54 AM
•
• Any fundamental theorem of differential equations states that if
1
...
is any solution to the homogeneous equation
•
• Then
is a solution to the original equation
• The term
is called the particular integral solution, or forced response, and
called the complementary solution, or natural response
• As an example, consider the situation in which
○
(i
...
, some constant);
Then
○
1
...
□
□
□
○
is referred to as the steady-state solution
○
is the value of the decaying exponential when t = 0
○ The constant is called the time constant of the circuit
The rate at which the exponential decays
Chapter 7 Page 41
is
Analysis Techniques
• The differential equation approach
1
...
Find initial value of state variables at
(usually
)
3
...
Solve first-order differential equation
Chapter 7 Page 42
Chapter 7 Page 43
8
...
1
• Example 8
...
2: Sinusoidal and Complex Forcing Functions
Tuesday, May 03, 2016
11:50 PM
Introduction
• If the independent sources are sinusoids, then for any variable in the linear circuit the
steady state response will be sinusoidal and of the same frequency
○
• To determine the steady state solution we only need to determine the parameters B and Φ
Example 8
...
4:
• Determine the current in the RL circuit
○
○ Instead of
○ Assume
, we will apply
○ KVL:
○
=
○
○
○
○
○
Chapter 8 Page 49
○
○
Chapter 8 Page 50
8
...
4: Phasor Relationships for Circuit Elements
Thursday, May 05, 2016
12:03 AM
Resistors:
•
○
• Phasor representation for a resistor
•
○
Inductors:
•
○
• Phasor representation for an inductor
•
Chapter 8 Page 52
○
Capacitors:
•
○
• Phasor representation for a capacitor
•
○
Example 8
...
6:
Chapter 8 Page 53
•
Chapter 8 Page 54
8
...
7:
•
•
•
Chapter 8 Page 56
Example 8
...
8: Analysis Techniques
Thursday, May 05, 2016
12:47 AM
Problem Solving Strategy:
•
•
Example 8
Title: Circuit Analysis Chapters 1-8
Description: From the book, Basic Engineering Circuit Analysis 11th Edition, good notes on the first eight chapters. Good for your basic or intro Circuit Analysis class.
Description: From the book, Basic Engineering Circuit Analysis 11th Edition, good notes on the first eight chapters. Good for your basic or intro Circuit Analysis class.