Expert answer:AC Steady State (Phasor) Analysis

Expert answer:Laboratory Experiment #6 AC Steady State (Phasor) Analysis . I have already upload two files if you need any information send me , thanks the every thing explain for the lab report is on the PDF, and the second file is how will you do the lab report and some information about the steps. Also on table 3 page 51 there are blank you have to make calculation for it, and there is a table 2 on page 49 you have to flit outI have finished the procedure I need someone to help me with calculation and how the results came you can also check AC SS Review for more information. thank you
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Laboratory Report Format
The laboratory report will be broken into eight sections as follows:
1. Cover Page. This page precedes the body of the report and details who
authored the report, the title of the experiment, the date that it was
performed and the names of the laboratory group members. Each student
will turn-in their own laboratory report.
2. Abstract. This summarizes, in general terms, what was accomplished in
the laboratory experiment. The abstract will be no more than a half page
long.
3. Objective: This portion of the report indicates, in a brief and succinct
manner, what the goal of the experiment is. The objective should be no
more than one or two sentences in length.
4. Theory: This section provides the mathematical formalism pertaining to the
stated problem that will be used to analyze the experimental observations.
Completing this section is not just a matter of transcribing formulas, but
in addition provides an explanation of why these particular formulas were
used. Appropriate mathematical derivations are typically moved to the
Appendix section.
5. Procedure: This section carefully describes in the experimenter’s own
words, the manner in which the laboratory exercise was performed from
start to finish. Include all relevant circuit diagrams. (There should be at
least one schematic accompanying each laboratory report.)
6. Results and Analysis: This is the heart of the report. Information on what
was measured and how the measurements were taken belong in this
section. Tables, charts and graphs must be properly labeled with units
specified. Sample calculations and mathematical derivations will be
included in this section as well.
7. Conclusions: This summarizes the key points of the laboratory exercise by
enumerating what statements can be made that are supported by the
“Results and Analysis” section. Sweeping generalities such as “I learned a
lot from this experiment.”, “This laboratory was helpful in helping me
understand the theory.” or “Human error was responsible for the
inaccuracies encountered in the data,” are meaningless statements.
8. Appendix: This section contains reference material that would interrupt
the flow of the report if contained in the main text.
Equipment: Name, model and manufacturer of the equipment, which was
used in your experiment.
Derivations: Lengthy derivations of equations used in the analysis.
Pre-laboratory Exercises: If a pre-laboratory section has been included in the
laboratory experiment and/or additional questions are given in the laboratory
exercise, answers to these questions are to be appended to the report.
Principles of Proper Report Writing for ENGR 223
1. Figures must be completely labeled. Each figure must have a caption
below it (with figure number) describing what the figure represents.
Graphs should contain a title, have properly labeled x and y axes
(including units), and also require proper captions below. Data points
must be identified with a symbol (diamond, circle or dot) and be
connected with a smooth line. Excel software, which was introduced in
ENGR 111, may be used when graphing.
2. Circuit diagrams are to be drawn using standard schematic symbols.
Use ruler or graphics software when drawing these diagrams. Be sure to
label all components and their values. Put a caption that describes
what the figure represents, with a figure number, under each diagram .
3. Tables must also be labeled, but the caption goes above the table.
4. DO NOT double-space lines. Use normal spacing, such as used in this
laboratory manual. Use 12 point New Times Roman or Bookman Old
Style type.
5. DO NOT use binders or folders. A single staple in the upper left hand
corner will do. Use standard, white, 8.5 inch by 11 inch paper stock
with straight edges. The writing/figure frame is 6” x 9” centered
vertically 1.5 inches from the left (binding) margin.
6. DO NOT use personal pronouns, unfamiliar acronyms, or informal
language in an engineering report.
7. Spelling, grammar and sentence structure are always important
indicators of how much effort was given to the report. Be sure to proof
read your report before turning it in.
8. Neatness, readability and understandability are also the hallmark of a
good report. Ask yourself the following questions: “Does this report
make sense?” “Is this report written in a clear, concise and logical
manner?” “Are the equations and calculations neatly written?”
9. Follow the prescribed laboratory report format.
10.
Keep a copy of your report in the event that the original is
lost or misplaced.
11.
If you are having difficulty writing the report due to problems with
grammar, sentence structure and the like, make an appointment to see
the writing specialists at the Writing Center.
ENGR 223B, Fall 2017
Outline
 Basic characteristics of sinusoidal functions
 Complex forcing function
 Phasors
Sinusoids
 Sine wave function x(t ) = X M sin ωt
T = 2π
+
=

→ x(ωt + 2π ) = x(ωt )
ω
ω
x
t
T
x
t
[
(
)]
(
)
 Period T
 Frequency f: f=1/T
 For sinusoid functions, we have ωT=2π  ω=2π/T=2πf
 Phase angle θ
x(t ) = X M sin(ωt + θ )
Trigonometric Identities
π
cos ωt = sin(ωt + )
2
π
sin ωt = cos(ωt − )
2
− cos ωt = cos(ωt ± π )
− sin ωt = sin(ωt ± π )
sin(α ± β ) = sin α cos β ± cos α sin β
cos(α ± β ) = cos α cos β  sin α sin β
Review of Complex Number Relationships
j = −1
jθ
x + jy = re , r = x + y , θ = tan
2
x = r cos θ , y = r sin θ
e
− jθ
1
= jθ
e
2
−1
y
x
Basic Complex Operations
Addition
(a + bj ) + (c + dj ) = (a + c) + (b + d ) j
Subtraction
(a + bj ) − (c + dj ) = (a − c) + (b − d ) j
Multiplication
Division
(a + bj )(c + dj ) = (ac − bd ) + (bc + ad ) j
(a + bj )
ac + bd
bc − ad
)+( 2
)j
=( 2
2
2
(c + dj )
c +d
c +d
Sinusoidal and Complex Forcing Function
 Basic assumption: circuit network is linear
 A sinusoidal forcing function results in sinusoidal
outputs (voltage, current).
 The computational analysis of linear networks is very
complicated for sinusoidal inputs.
 The introduction of complex forcing function
j ωt
 Euler’s equation e = cos ωt + j sin ωt
 Complex voltage v(t ) = VM e jωt = VM cos ωt + jVM sin ωt
 Complex current i (t ) = I M cos(ωt + φ ) + jI M sin(ωt + φ )
Phasors
 Complex representations of current and voltage
v(t ) = VM cos(ωt + θ ) = Re[VM e j (ωt +θ ) ]
i (t ) = I M cos(ωt + φ ) = Re[ I M e j (ωt +φ ) ]
 Phasor notation V = VM ∠θ , I = I M ∠ϕ
 Sinusoidal functions would be represented as phasors
with a phase angle based on a cosine function.
 Sine functions need to represented by cosine function
A cos(ωt ± θ ) → A∠ ± θ
A sin(ωt ± θ ) → A∠ ± θ − 90
Examples of Phasor Representation
Impedance
 Impedance Z (ohms): the ratio of the phasor voltage V
to the phasor current I.
V VM ∠θ v VM
Z= =
=
∠θ v − θ i = Z∠θ z
I I M ∠θ i I M
Characteristics of Impedance
 Z is complex.
 If Z1, …, Zn are connected in series, the equivalent
impedance Zs is:
n
Zs = ∑ Zi
i =1
 If Z1, …, Zn are connected in parallel, the equivalent
impedance Zp is given by:
n
1
1
=∑
Z p i =1 Z i
Example: Calculate the Equivalent Impedance
Question: Given the frequency f=60 Hz, and
v(t ) = 50 cos(ωt + 30 ) V,
calculate the equivalent impedance and the
current i(t).
First , compute the impedance of each element at 60 Hz :
Z R = 25Ω
Z L = jωL = j( 2π × 60 )( 20 ×10 −3 ) = j 7.54Ω
Z C = − j /(ωC ) = − j/[( 2π × 60 )( 50 ×10 −6 )] = − j 53.05Ω
The elements are connected in series, we have
Z = Z R + Z L + Z C = 25 − j 45.51Ω
The current is determined by
I = V/Z = 50∠30 /(25 − j 45.51) = 50∠30 / 51.93∠ − 61.22 = 0.96∠91.22 A
or in time domain : i (t ) = 0.96 cos(377t + 91.22 ) A
Voltage-Current Relationship for Resistor
 Resistor R
V = RI, where V = VM e jθ v = VM ∠θ v and I = I M e jθi = I M ∠θ i
Voltage-Current Relationship for Inductor
 Inductor L

V = jωLI  
→ V = ωLI M e
j =1e j 90
j (θ i + 90 )
Voltage-Current Relationship for Capacitor
 Capacitor C
I = jωCV = ωCVM e
j (θ v + 90 )
Phasor Diagrams
 An example of phasor diagram for the parallel circuit
 V: reference phasor with zero phase
V = VM ∠0
KCL : I S = I R + I L + I C
VM ∠0 VM ∠ − 90
=
+
+ VM ωC∠90
ωL
R
…
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