eleg 205 fall 2017 lecture #14mirotzni/eleg205/lecture15.pdf · chapter 10: steady-state....
TRANSCRIPT
![Page 1: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/1.jpg)
ELEG 205Fall 2017
Lecture #15
Mark Mirotznik, Ph.D.Professor
The University of DelawareTel: (302)831-4221
Email: [email protected]
![Page 2: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/2.jpg)
Chapter 10: Steady-StateSinusoidal Analysis
)cos()( φω += tAtV
![Page 3: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/3.jpg)
Review of Complex NumbersHow can we convert a complex number from rectangular to polar notation?
22ir AAA +=
)(tan 1
r
i
AA−=φ
![Page 4: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/4.jpg)
Review of Complex NumbersHow can we convert a complex number from polar to rectangular notation?
)sin(
)cos(
φ
φ
AA
AA
i
r
=
=
![Page 5: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/5.jpg)
Review of Complex NumbersAj
ir eAjAAA φ⋅=+=Bj
ir eBjBBB φ⋅=+=Summary
( )BAjeBA
C φφ −⋅=
( )BAjeBAC φφ +⋅=( ) ( )iirr BAjBAC +++=
( ) ( )iirr BAjBAC −+−=
Addition and Subtraction Multiplication and Division
![Page 6: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/6.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
485
832π
π
j
j
je
jeC−
+
+−=
![Page 7: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/7.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
485
832π
π
j
j
je
jeC−
+
+−=
42185
83)sin(2)cos(2ππ
ππjj
ee
jjC−
+
+−+=
![Page 8: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/8.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
42185
83)sin(2)cos(2ππ
ππjj
ee
jjC−
+
+−+=
485
83)0(2)1(2πj
e
jjC+
+−⋅+−⋅=
![Page 9: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/9.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
)4
sin(8)4
cos(85
85ππ j
jC++
+−=
485
83)0(2)1(2πj
e
jjC+
+−⋅+−⋅=
![Page 10: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/10.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
657.5657.1085
228
2285
85j
j
j
jC++−
=++
+−=
)4
sin(8)4
cos(85
85ππ j
jC++
+−=
![Page 11: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/11.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
657.5657.1085j
jC++−
=
−
−
−
⋅+
⋅+=
657.10657.5tan
22
58tan
22
1
1
657.5657.10
85j
j
e
eC
![Page 12: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/12.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
−
−
−
⋅+
⋅+=
657.10657.5tan
22
58tan
22
1
1
657.5657.10
85j
j
e
eC
488.0
0122.1
065.12434.9
j
j
eeC⋅⋅
=−
![Page 13: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/13.jpg)
Review of Complex Numbers: ExampleConvert to single complex number in polar notation
5.17819.0 jeC −⋅=
488.0
0122.1
065.12434.9
j
j
eeC⋅⋅
=−
![Page 14: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/14.jpg)
Phasors
)sin()cos( xjxe jx +=Recall:( )[ ]
[ ][ ]
φ
ω
ωφ
φωφω
jm
tj
tjjm
tjmm
eVV
eV
eeVeVtVtV
⋅=
⋅=
⋅⋅=
⋅=+= +
~
~Re
ReRe)cos()(
Where is called a Phasor
![Page 15: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/15.jpg)
Phasors
φjm eVV ⋅=~
What does the Phasor tell us about the sinusoidal voltage or current
![Page 16: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/16.jpg)
Phasors
φjm eVV ⋅=~
What does the Phasor tell us about the sinusoidal voltage or current
For example if a Phasor voltage is ojeV 305~ ⋅=
and the frequency was known as kHzf 5=could you tell me what the voltage waveform was in time?
![Page 17: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/17.jpg)
PhasorsFor example if a Phasor voltage is
ojeV 305~ ⋅=and the frequency was known as kHzf 5=could you tell me what the voltage waveform was in time?
)3050002cos(5)( ottv +⋅= π
ojeV 305~ ⋅= kHzf 5=
![Page 18: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/18.jpg)
Phasor Transform of Sinusoidal Sources
)cos()( φω += tVtV m
Time-Domain Phasor-Domain
φjm eVV ⋅=~
![Page 19: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/19.jpg)
Phasor Transform of Sinusoidal Sources
)cos()( φω += tIti m
Time-Domain Phasor-Domain
φjm eII ⋅=~
![Page 20: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/20.jpg)
Phasor Transform of Sinusoidal Sources: Examples
)3
1000cos(10)( π+= ttV
Time-Domain Phasor-Domain
310~ πjeV ⋅=
![Page 21: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/21.jpg)
Phasor Transform of Sinusoidal Sources: Examples
Time-Domain Phasor-Domain
)40000,10cos(5.1)( otti +=ojeI 405.1~ ⋅=
![Page 22: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/22.jpg)
Phasor Transform of Sinusoidal Sources: Examples
)4
1000sin(7)( π−= ttV
Time-Domain Phasor-Domain
?~ =V
![Page 23: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/23.jpg)
Inverse Phasor Transform
?)( =tV
Time-Domain Phasor-Domain
58~ πjeV ⋅=
kHzf 5=
![Page 24: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/24.jpg)
Inverse Phasor Transform
)5
50002cos(8)( ππ +⋅⋅= ttV
Time-Domain Phasor-Domain
58~ πjeV ⋅=
kHzf 5=
![Page 25: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/25.jpg)
Inverse Phasor Transform
?)( =tV
Time-Domain Phasor-Domain
jV 15~ =MHzf 1=
![Page 26: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/26.jpg)
Inverse Phasor Transform
)2
102cos(15)( 6 ππ +⋅=tV
Time-Domain Phasor-Domain
215
15~
πje
jV
⋅=
=
MHzf 1=
![Page 27: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/27.jpg)
Phasors with Circuit Components
=== tjtjtjjm e
RVeIeeIti I ωωωφ~
Re~ReRe)(
R)(ti tj
tjjm
vm
eV
eeV
tVtvv
ω
ωφ
φω
~Re
Re
)cos()(
=
=
+=
= tjtj eRVeI ωω~
Re~Re
![Page 28: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/28.jpg)
Phasors with Circuit Components
R)(ti tj
tjjm
eV
eeVtv v
ω
ωφ
~Re
Re)(
=
=
= tjtj eRVeI ωω~
Re~Re
= tjtj eRVeI ωω~
Re~Re
![Page 29: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/29.jpg)
Phasors with Circuit Components
R)(ti tj
tjjm
eV
eeVtv v
ω
ωφ
~Re
Re)(
=
=
= tjtj eRVeI ωω~
Re~Re
RVI~~ =
IVR ~~
=
![Page 30: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/30.jpg)
Phasors with Circuit Components
R tjeVtv ω~Re)( =
IVR ~~
=
The ratio of the phasor voltage to phasor current is called the impedance (symbol Z)
tjeIti ω~Re)( =
![Page 31: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/31.jpg)
Phasors with Circuit Components
IVRZ ~~
==
The ratio of the phasor voltage to phasor current is called the impedance (Z)
IVZ ~~
=
In general For a resistor
![Page 32: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/32.jpg)
C
[ ] tjtjtj eCjVeVdtdCeI
dttdvCti
ωωω ω ⋅⋅==
=
~Re~Re~Re
)()(
tjeVtv ω~Re)( = tjeIti ω~Re)( =
Phasors with Circuit Components
![Page 33: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/33.jpg)
C
tjtj
tjtj
eCjVeI
eCjVeIωω
ωω
ω
ω
⋅⋅=
⋅⋅=~Re~Re
~Re~Re
tjeVtv ω~Re)( = tjeIti ω~Re)( =
Phasors with Circuit Components
![Page 34: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/34.jpg)
C
CjVI ω⋅= ~~
tjeVtv ω~Re)( = tjeIti ω~Re)( =
CjIVZ
ω1
~~==
Phasors with Circuit Components
![Page 35: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/35.jpg)
C tjeVtv ω~Re)( =
tjeIti ω~Re)( =
CjIVZ
ω1
~~==I
VZ ~~
=
In general For a capacitor
Phasors with Circuit Components
![Page 36: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/36.jpg)
L tjeVtv ω~Re)( =
tjeIti ω~Re)( =
[ ]
⋅==
=
∫
∫tjtjtj eV
LjdteV
LeI
dttvL
ti
ωωω
ω~1Re~Re1~Re
)(1)(
Phasors with Circuit Components
![Page 37: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/37.jpg)
L tjeVtv ω~Re)( =
tjeIti ω~Re)( =
⋅=
⋅=
tjtj
tjtj
eVLj
eI
eVLj
eI
ωω
ωω
ω
ω
~1Re~Re
~1Re~Re
Phasors with Circuit Components
![Page 38: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/38.jpg)
L tjeVtv ω~Re)( =
tjeIti ω~Re)( =
⋅= tjtj eVLj
eI ωω
ω~1Re~Re
VLj
I ~1~ω
= LjIVZ ω== ~~
Phasors with Circuit Components
![Page 39: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/39.jpg)
L tjeVtv ω~Re)( =
tjeIti ω~Re)( =
LjIVZ ω== ~~
IVZ ~~
=
In general For an inductor
Phasors with Circuit Components
![Page 40: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/40.jpg)
Phasors with Circuit ComponentsSummary
Impedance, Z, Ω
Resistor
Inductor
Capacitor
LjZ ω=
CjZ
ω1
=
RZ =
![Page 41: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/41.jpg)
Phasors with Circuit ComponentsSummary
Impedance,Z, Ω
Admittance,Y, sieman
Resistor
Inductor
Capacitor
LjIVZ ω== ~~
CjIVZ
ω1
~~==
RIVZ == ~~
RVIY 1~~==
LjVIY
ω1
~~==
CjVIY ω== ~~
![Page 42: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/42.jpg)
Combining Impedances
1 mH
0.5 mH
1 µF
10 ΩeqZ
Convert all of the components to their impedance values and determine the equivalent impedance of the entire network
kHzf 10=
![Page 43: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/43.jpg)
1 mH
0.5 mH
1 µF
Z=10 ΩeqZ
Convert all of the components to their impedance values and determine the equivalent impedance of the entire network
kHzf 10=
jjCj
Z 9.15101000,102
116 −=
⋅⋅⋅== −πω
jjLjZ 8.62101000,102 3 =⋅⋅⋅== −πω
jj
LjZ
4.31105.0000,102 3
=⋅⋅⋅=
=−π
ω
Combining Impedances
![Page 44: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/44.jpg)
62.8j Ω
31.4j Ω
-15.9j Ω
10 ΩeqZ
Convert all of the components to their impedance values and determine the equivalent impedance of the entire network
kHzf 10=
jjjjjZeq 8.4908.99.158.62
4.31104.3110
+=−++⋅
=
Combining Impedances
![Page 45: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/45.jpg)
62.8j Ω
31.4j Ω
-15.9j Ω
10 ΩeqZ
Convert all of the components to their impedance values and determine the equivalent impedance of the entire network
kHzf 10=
Ω==+= ⋅⋅ ojjeq eejZ 7.7939.1 6.506.508.4908.9
Combining Impedances
![Page 46: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/46.jpg)
Solving Sinusoidal Steady State Circuits
STEP #1: Transform all sources to the phasor domain using the phasor transform
![Page 47: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/47.jpg)
Solving Sinusoidal Steady State CircuitsSTEP #2: Transform all components (i.e. resistors, inductors and capacitors) into their complex impedance values.
![Page 48: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/48.jpg)
Solving Sinusoidal Steady State Circuits
STEP #3: Using Kirchhoff's laws and Ohm’s law for impedances solve for the unknown phasor quantities.
I~
![Page 49: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/49.jpg)
Solving Sinusoidal Steady State Circuits
STEP #4: Transform phasor results back into the time domain.
tjeIti ω~Re)( =
![Page 50: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/50.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
Find vc(t)
)(tvc
![Page 51: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/51.jpg)
Solving Sinusoidal Steady State CircuitsExamples
STEP #1 and #2: Transform all sources to the phasor domain using the phasor transform and transform all components to their impedances
)cos()( tAtv ω= CR
)(tvc
Time-Domain Phasor-Domain
0~ jAeV = Cjω1
R
cV~
![Page 52: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/52.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
)(tvc
Time-Domain Phasor-Domain
0~ jAeV = Cjω1
R
cV~
STEP #3: Using Kirchhoff's laws and Ohm’s law for impedances solve for the unknown phasor quantities.
Solve this circuit like you would if it was only just filled with resistors and DC sources.
![Page 53: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/53.jpg)
Solving Sinusoidal Steady State CircuitsExamples
Phasor-Domain
0~ jAeV = Cjω1
R
cV~
STEP #3: Using Kirchhoff's laws and Ohm’s law for impedances solve for the unknown phasor quantities.
CjR
CjVVc
ω
ω1
1~~
+=Voltage divider
![Page 54: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/54.jpg)
Solving Sinusoidal Steady State CircuitsExamples
Phasor-Domain
0~ jAeV = Cjω1
R
cV~
STEP #3: Using Kirchhoff's laws and Ohm’s law for impedances solve for the unknown phasor quantities.
11
1
1~~
+⋅=⋅
+=
CRjA
CjCj
CjR
CjVVc ωωω
ω
ωVoltage divider
![Page 55: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/55.jpg)
Solving Sinusoidal Steady State CircuitsExamples
Phasor-Domain
0~ jAeV = Cjω1
R
cV~
STEP #3: Using Kirchhoff's laws and Ohm’s law for impedances solve for the unknown phasor quantities.
Voltage divider)
1(tan22
0
1
)(11
~RCj
j
c
eRC
eACRjAV ω
ωω −
⋅+
⋅=
+=
![Page 56: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/56.jpg)
Solving Sinusoidal Steady State CircuitsExamples
Phasor-Domain
0~ jAeV = Cjω1
R
cV~
STEP #3: Using Kirchhoff's laws and Ohm’s law for impedances solve for the unknown phasor quantities.
Voltage divider)
1(tan
2
1
)(1~ RCj
c eRC
AVω
ω
−−⋅
+=
![Page 57: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/57.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
)(tvc
Time-Domain Phasor-Domain
0~ jAeV = Cjω1
R
cV~
STEP #4: Transform phasor results back into the time domain.
)1
(tan
2
1
)(1~ RCj
c eRC
AVω
ω
−−⋅
+=( )
))(tancos(1
)( 1
2RCt
RC
Atvc ωωω
−−+
=
![Page 58: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/58.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
)(tvc
( )))(tancos(
1)( 1
2RCt
RC
Atvc ωωω
−−+
=
Lets look at this solution at different frequencies.
![Page 59: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/59.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
)(tvc
( )))(tancos(
1)( 1
2RCt
RC
Atvc ωωω
−−+
=
Lets look at this solution at different frequencies.
1. At DC (ω=0) what does the capacitor voltage look like?
![Page 60: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/60.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
)(tvc
( )))(tancos(
1)( 1
2RCt
RC
Atvc ωωω
−−+
=
Lets look at this solution at different frequencies.
1. At DC (ω=0) what does the capacitor voltage look like?
( )ARCt
RC
Atvc =⋅−⋅⋅+
= − ))0(tan0cos(01
)( 1
2 Does this make sense?
![Page 61: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/61.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
)(tvc
( )))(tancos(
1)( 1
2RCt
RC
Atvc ωωω
−−+
=
Lets look at this solution at different frequencies.1. At very very high frequencies (ω=infinity) what does the capacitor voltage look like?
![Page 62: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/62.jpg)
Solving Sinusoidal Steady State CircuitsExamples
)cos()( tAtv ω= CR
)(tvc
( )))(tancos(
1)( 1
2RCt
RC
Atvc ωωω
−−+
=
Lets look at this solution at different frequencies.1. At very very high frequencies (ω=infinity) what does the capacitor voltage look like?
( )0))(tancos(
1)( 1
2=⋅∞−⋅∞
⋅∞+= − RCt
RC
Atvc Does this make sense?
![Page 63: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/63.jpg)
Solving Sinusoidal Steady State CircuitsExamples
Lets now plot the magnitude of the capacitor sinusoid as a function of frequency
ω
( )21 RC
A
ω+
A
0 0 RC20
As the frequency increases the amplitude of the output voltage (vc(t)) gets smaller.
RC10
![Page 64: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/64.jpg)
Solving Sinusoidal Steady State CircuitsExamples
Lets now plot the magnitude of the capacitor sinusoid as a function of frequency
ω
( )21 RC
A
ω+
A
0 0 RC20
As the frequency increases the amplitude of the output voltage (vc(t)) gets smaller.
RC10
THIS PLOT IS CALLED A MAGNITUDE FREQUENCY RESPONSE PLOT
![Page 65: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/65.jpg)
Solving Sinusoidal Steady State CircuitsExamples
As the frequency increases the amplitude of the output voltage (vc(t)) gets smaller.
( )))(tancos(
1)( 1
2RCt
RC
Atvc ωωω
−−+
=
![Page 66: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/66.jpg)
Solving Sinusoidal Steady State CircuitsExamples
Lets now plot the phase angle of the capacitor sinusoid as a function of frequency ω
)(tan 1 RCω−−
o90−0 RC20
As the frequency increases the phase of the output voltage (vc(t)) goes from 0 degrees to -90 degrees.
RC10
THIS PLOT IS CALLED THE PHASE FREQUENCY RESPONSE PLOT
o0
![Page 67: ELEG 205 Fall 2017 Lecture #14mirotzni/ELEG205/Lecture15.pdf · Chapter 10: Steady-State. Sinusoidal Analysis. V(t) =A. ... Solving Sinusoidal Steady State Circuits. STEP #3: Using](https://reader030.vdocuments.mx/reader030/viewer/2022040120/5e93ff7d7cedb55bbc3d9481/html5/thumbnails/67.jpg)