frequency response objective - bode and nyquist plots for control analysis - determination of...
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Frequency ResponseFrequency Response
OBJECTIVEOBJECTIVE- Bode and Nyquist plots for control
analysis- Determination of transfer function- Gain and Phase margins- Stability in frequency response
Magnitude and Phase AngleMagnitude and Phase Angle
Transfer function is
)(
)()(
sR
sYsG R(s)
)(sGY(s)
By replacing we can find its magnitude
)(
)(
)(
)()(
jR
jY
sR
sYsG
and its phase angle
)()()( sRsYsG
)).....(()(
)).....(()(
)(
)()(
21
21
n
m
i
o
pspsps
zszszsK
s
ssG
Higher order transfer function
Can be presented in magnitude-phase form as
)()( jGjG
js
A system with transfer function of G(s) is subjected to a sinusoidal input.Determine the time response of the system.
SolutionThe input in phasor form (magnitude-phase form) can be presented as
As the transfer function is
Hence the output is by replacing
In time domain which give the time response of the system, the output is
)(si
tati sin)(
)()(
)(sG
s
s
i
o
)j(Gaa)j(G)j(o 0
)sin()()( tjGato
)(sG )(so
0)( aji
js
ExampleExample
First orderFirst order
1)(
s
KsG
Frequency response
1
1)(
jjG
Its magnitude
221
1)(
jG
and phase angle
)(tan1
tan 11
Transfer function
by replacing js
Second orderSecond order
Transfer function
22
2
2)(
nn
n
ss
KsG
Frequency response by replacing
222
2
21
21)(
nn
nn jjG
Its magnitude
222 21
1)(
nn
jG
and phase angle
21
1
2tan
n
n
js
Higher OrderHigher Order
Cascade form)().....()()( 21 sGsGsGsG n
Frequency response by replacing
)().....()()( 21 jGjGjGjG n
Or in phasor form
)....()(....)()(
)(.....)()()(
2121
2211
nn
nn
jGjGjG
jGjGjGjG
js
Example:
Find the frequency response of the following transfer function
50336
5)(
23
ssssG
)()()( 21 sGsGsG
ExampleExample
Where
15.0
1.0
2
2.0)(1
ss
sG and 254
25)(
22
sssG
Respective magnitude and phase angle
222125.01
1.0
5.01
1.0)(
jG )5.0(tan)( 1
1 jand
2222
540251
1
n).()j(G
and
2
12
51
5)4.0(2tan)(
j
)()()(.)(
)()().()()(
2121
2211
jjjGjG
jjGjjGjG
ExampleExample
222 540251
1
n).(
Bode PlotBode Plot
Consider higher order system
)()()......()().()()( 2211 jjGjjGjjGjG nn
)(log....)(log)(log)(log 21 jGjGjGjG n
Logarithmic form
In dB.
)(log20....)(log20)(log20)(log20 21 jGjGjGjGLM n
Phase angle
)().....()()( 21 jjjj n
Bode PlotBode Plot
0
0.2
0.4
0.6
0.8
1
Magnitu
de (
dB
)
100
101
102
103
0
0.2
0.4
0.6
0.8
1
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Bode Plot for constant gainBode Plot for constant gain
KsG )(Log magnitude in dB
KLM log20
Its phase angle
0
>> bode([200],[1]);grid
Let K=200
Bode Plot for constant gainBode Plot for constant gain
45
45.5
46
46.5
47
47.5
Magnitu
de (
dB
)
100
101
102
103
104
-1
-0.5
0
0.5
1
Phase (
deg)
Bode Plot G(s)=200
Frequency (rad/sec)
Bode Plot of pole on the originBode Plot of pole on the origin
ssG 1)(
)(1)( jjsG Log magnitude in dB
)log(201log20 LM
)rad.s( 1 dB)(LM
1 0
10 -20
102 -40
103 -60
)rad.s( 1 dB)(LM
01
10-1 20
10-2 40
10-3 60
1 0
2 -6
22 -12
23 -18
)rad.s( 1 dB)(LM
Slope of –20dB/decade or –6dB/octave.
Its phase angle
90)(tan0tan 11 >>bode([1],[1 0]);grid
Bode Plot of pole on the originBode Plot of pole on the origin
Bode Plot of pole on the originBode Plot of pole on the origin
-50
-40
-30
-20
-10
0
10
Magnitu
de (
dB
)
100
101
102
103
-91
-90.5
-90
-89.5
-89
Phase (
deg)
Bode Plot G(s)=1/s
Frequency (rad/sec)
Bode Plot for Real PoleBode Plot for Real Pole
1
1)(
ssG
Frequency response
221
1
1
1
1
1)(
j
j
j
jjG
Logarithmic magnitude in dB
22
221log20
1
1log20
LM
Phase angle
)(tan1tan 11
For low frequency, 1
01log20 LM
and 0
log20LM
1For high frequency,
and90)(tan 1
1.0Can also be defined as a tenth of the corner frequency i.e.
The corner frequency is 1c
Can also be defined as a tenth of the corner frequency i.e. 10
Bode Plot for Real PoleBode Plot for Real Pole
A straight line approximation for a first order system of transfer functionA straight line approximation for a first order system of transfer function
1
1)(
ssG
Bode Plot for Real PoleBode Plot for Real Pole
The actual Bode plot can be obtained by using the exact equation for the log magnitude and phase angle. There are small differences between the actual and approximate as shown by the following table
Using MATLAB we can display the Bode plot of an open-loop transfer function of
And the MATLAB command used is given by
1
1)(
ssG
» nr=[1];» dr=[1 1];» sys=tf(nr,dr)» bode(sys)
Bode Plot for Real PoleBode Plot for Real Pole
Actual and approximate value of log magnitude for an open-loop Actual and approximate value of log magnitude for an open-loop transfer function of transfer function of
for a frequency range between 0.1-20 rad/sfor a frequency range between 0.1-20 rad/s
1
1)(
ssG
Bode Plot for Real PoleBode Plot for Real Pole
Actual and approximate value of phase angle for an open-loop transfer Actual and approximate value of phase angle for an open-loop transfer function of function of
for a frequency range between 0.1-20 rad/sfor a frequency range between 0.1-20 rad/s1
1)(
ssG
Bode Plot for Real PoleBode Plot for Real Pole
dBLM 32log2011
1log20
4511tan 1
Bode Plot for Real PoleBode Plot for Real Pole
Bode Plot for Complex polesBode Plot for Complex poles
22
2
2)(
nn
n
ss
KsG
222 21log20)(log20 nnjGLM
2
1
1
2tan
n
n
For low frequency ,
10n
1n 01log20 LM
and 0)10(tan 1
For high frequency 1n
nnLM log40log20 2
180tan2
1
and
Is the corner frequency
We define the low frequency as a tenth of the corner frequency i.e.
n
While the high frequency as ten time of the corner frequency i.e. n10
Bode Plot for Complex polesBode Plot for Complex poles
Based on straight line approximation the Bode plots for LM and phase angle Based on straight line approximation the Bode plots for LM and phase angle are shownare shown
Bode Plot for Complex polesBode Plot for Complex poles
We can observe the Bode plots in MATLAB by considering
In MATLAB
>>zeta=0.25;wn=1;num=[1];den1=[1 2*zeta*wn wn*wn];
sys1=tf(num,den1); >>zeta=0.5;wn=1;num=[1];den2=[1 2*zeta*wn wn*wn];
sys2=tf(num,den2);
>>zeta=0.75;wn=1;num=[1];den3=[1 2*zeta*wn*wn*wn];sys3=tf(num,den3); >> bode(sys1,sys2,sys3);grid
22 2
1)(
nnsssG
1n 75.0and5.0,25.0for and
Bode Plot for Complex polesBode Plot for Complex poles
Bode Plot for Complex polesBode Plot for Complex poles
Obtain the bode plot of the following block diagram if )200)(10(
)4(5000)(
sss
ssGo
)(sGo+
_
ExampleExample
5000 1/s s+4 10
1
s 200
1
s
1200
1.
110
1.1
4.
1.10
1200
1.
110
1.1
4.
1.
20010
45000)(
jj
j
j
jj
j
jjGo
By replacing with s=j and making the open-loop transform function into corner frequency form
ExampleExample
dBLM 401.0
110log20)1.0(
90)1.0(
The initial LM and phase at start frequency is given by
ExampleExample
For the magnitude plot of the Bode plot, we build a table to show the For the magnitude plot of the Bode plot, we build a table to show the contribution of gradient/slope by the zero and poles at respective corner contribution of gradient/slope by the zero and poles at respective corner frequencies. The pole at origin will provide a slope of -20db/decade for all frequencies. The pole at origin will provide a slope of -20db/decade for all range of frequencyrange of frequency
ExampleExample
- For the phase plot, we need to know the change of gradient of the phase at low and - For the phase plot, we need to know the change of gradient of the phase at low and high frequencies of respective corner frequencyhigh frequencies of respective corner frequency- The pole at the origin does not contribute to gradient of the phase angle as its phase - The pole at the origin does not contribute to gradient of the phase angle as its phase angle is a constant 90angle is a constant 90oo
ExampleExample
The LM vs. freq. plot can be determined the LM value at the corner frequency, this can be obtained by simple trigonometry
2log1log
LM (dB/dec)1LM
2LM
LM
LMLM
12
12
loglog
1212 loglog LMLMLM
Using the trigonometry’s formula, we tabulate the values at the initial, corner frequencies and final value.
ExampleExample
dB.loglogLM 810420402 Eg:
As for the LM slope, we apply a trigonometry's formula to obtain the
phase angle slope at the low and high frequency for each cornerfrequency as in table below
2log1log
1
2
(deg/decade)
12
12
loglog 1212 loglog
ExampleExample
Eg: 9010400902 .log.log
Bode plot for the magnitude and phase using straight line approximationsBode plot for the magnitude and phase using straight line approximations
ExampleExample
We can obtain the actual Bode plot using MATLAB as We can obtain the actual Bode plot using MATLAB as >>bode([5000 20000],[1 210 2000 0],{10^-1,10^4});>>bode([5000 20000],[1 210 2000 0],{10^-1,10^4});
ExampleExample
-100
-50
0
50
Magnitude (
dB
)
10-1
100
101
102
103
104
-180
-135
-90
-45
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Determination of transfer functionDetermination of transfer function
Constant gain
Example:
If 36yK dB, determine the transfer function.
dB
yK
(rad.s-1)
yKK log20
36yK dB
1.6320
36antilog
K
Pole/zero at originPole/zero at origin
Example:101 10yK 40 LMIf rad.s-1, dB and slope of dB/decade.
dB
yK
1
(rad.s-1)
40 LM dB/decade
2s
K
yKK
log
21
20
101 10yKWe know rad.s-1, dB
6.3120
10antilog102
K
Real pole/zeroReal pole/zero
Example:
dB
45 201 LM dB/decade
402 LM dB/decade
203 LM dB/decade
(rad.s-1) 250 40 0.01
140
1250
jj
jK
4501.0
log20)01.0(
KLM
78.120
45antilog10 2
K
)s(s
)s(.s
s
s.
40
250280
140
1250
781
ExampleExample
Pair of complex polesPair of complex poles
Example:
Assume damping ratio of 0.5.
dB
60
0.1 50 400
602 LM dB/decade
(rad.s-1)
401 LM dB/decade
5025011400
2 j
j
K
2" nKK
60)"log(20)1.0( KLM
100020
60antilog"
K
4.050
10002K
250050)400(
160
50)5.0)(50(2)400(
4004.0)(
2
22
sss
ssssG
ExampleExample
Nyquist PlotNyquist Plot
)j(G )( j
0and phase angleNyquist plot is a plot of magnitude, for frequency
on s-plane.
(rad.s-1) 0
)( jG
)j(
)0( jG
)0( j
1)( 1jG
)( 1j
)( jG
)( j
However we can obtain the sketch of the plot by obtaining the following vectors:
(i) at
0
(ii) at
(iii) 180or0 , crossing on the real axisat, crossing on the imaginary axis
(iv) 90at
First orderFirst order
1)(
s
KsG
Frequency response
221
1
1)(
j
Kj
KjG
Magnitude
221)(
KjG and
Phase angle
)(tan 1
(i) At, 0 KK
jG
01
)( ,, and 0)0(tan 1
0)( jG(ii) At , dan
11 tan1
tan 90tan 1 . i.e. or 270
90(iii) No crossing in the real axis as ,
90(iv) No crossing in the imaginary axis as ,
KjG 222)(
is a circle.
First orderFirst order
Example:
Nyquist plot of
125.0
5)(
ssG
4
20)(
ssG
Frequency response
20625.01
5)(
jG and )25.0(tan 1
At 0 501
5)0(
jG 0)0(tan 1 and
At
,
01
5)(
jG
11 tan1
tan 90and
,
.
>> nyquist([5],[.25 1])
First orderFirst order
Plot Nyquist G(s)=5/(0.25s+1)
Paksi Hakiki
Paks
i Khayali
-1 0 1 2 3 4 5 6
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
System: sys Real: 0.917 Imag: 1.93 Freq (rad/sec): -8.46
System: sys Real: 5 Imag: 0
Freq (rad/sec): 0
System: sys Real: 0.000335 Imag: -0.0409 Freq (rad/sec): 489
System: sys Real: 1.24 Imag: -2.16 Freq (rad/sec): 6.98
System: sys Real: 4.52
Imag: -1.47 Freq (rad/sec): 1.3
System: sys Real: 3.64 Imag: 2.22 Freq (rad/sec): -2.45
First orderFirst order
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Nyquist Diagram
Real Axis
Imag
inar
y A
xis
)()(1 jHjG
Second OrderSecond Order
Second OrderSecond Order
22
2
2)(
nn
n
ssKsG
Frequency response
nn jjG
21
1)(
2
Rearrange
222
2
21
21
nn
nn
j
)j(G
Magnitude
222
21
1)(
nn
jG
Phase angle
2
1
1
2
tan
n
n
and (i) at 0 1)( jG, 0
0)( jG 1801
tan2
1
(ii) At , and .
(iii) No crossing on the real axis.
01
2
n
n (iv) Crossing of the imaginary axis when , .
2
1)( njG 9002tan 1 and
Second OrderSecond Order
Example:
)1)(12(
1)(
2
ssssG
Frequency response
2322
32
322
)23()31(
)23()31(
)23()31(
1
)1)(12(
1)(
j
jjjjG
Magnitude
2322 )23()31(
1)(
jG
Phase angle
2
31
31
23tan
Second OrderSecond Order
(i) At , 0and1)0( jG
0)( jG 2701
tan2
31
(ii) At , and
0
(iii) Real axis crossing at 023 3 ,, 0 2.123 2.1and rad.s-1 or
Magnitude
3.00)2.131(
1)2.1(
22
jG
(iv) Imaginary crossing at 031 2 31 =0.6 rad.s-1 ,
Magnitude 7.0)3323(0
1)6.0(
2
jG
»dr1=[2 1]; » dr2=[1 1 1];
» dr=conv(dr1,dr2);» nr=1;
» nyquist(nr,dr)
Second OrderSecond Order
,
Plot Nyquist G(s)=1/(2s+1)(s2+s+1)
Paksi Hakiki
Paksi K
hayali
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
System: sys Real: -0.00357 Imag: -0.74 Freq (rad/sec): 0.579
System: sys Real: -0.297 Imag: 0.0104 Freq (rad/sec): -1.21
System: sys Real: 0.999
Imag: 0.00365 Freq (rad/sec): -0.00126
Second OrderSecond Order
Nyquist PathNyquist Path
For Nyquist path
j
r
s-plane
Contour of )()(1 sHsG which will give the closed loop poles for till by replacing
js to become )()(1 jHjG
j
)j(H)j(G 1
F(j)-plane
However
)(1)()(11)()( jFjHjGjHjG
Hence on F(j)-plane, we represent GH(j)-plane to plot )()(1 jHjG
)()( jHjG
j
-1+j0
)()(1 jHjG
.
Nyquist PathNyquist Path
GH(j)-plane
Nyquist Stability CriterionNyquist Stability Criterion
The criteria states that the number of closed poles of a system is equal to the number of open-loop, G(s)H(s), zeros plus the number of encirclements in the clockwise direction by the Nyquist plot of the open-loop, G(s)H(s). Or, it can be represented in mathematic form as PNZ
where
Z = Number of zeros )(sF on the right-half plane of j-axis, where .
N = Number of clockwise encirclement of 01 jP = Number of open-loop poles )()( sHsG on the right-half plane of j-axis.
)()(1)( sHsGsF
• Consider , applying the Nyquist stability criteria Z=N+P=1+0=1
Hence the system has 1 closed-loop pole on the right of the imaginary axis
2)1(
)2()()(
s
sKsHsG
ExampleExample
s^2 + 10 s + 24 --------------- s^2 - 8 s + 15
ExampleExample
•The first thing we need to do is find the number of positive real poles in our open-loop transfer function, P:
•>>roots([1 -8 15]) ans = 5
3 •The second thing we need to do is find the number of positive real zeros in our open-loop transfer function, Z:
•>>roots([1 10 24])ans =-6.0000-4.0000
•Z=N+P•N=Z-P=0-2=-2•The poles of the open-loop transfer function are both positive. Therefore, we need two anti-clockwise (N = -2) encirclements of the Nyquist diagram in order to have a stable closed-loop system (Z = P + N).
•If the number of encirclements is less than two or the encirclements are not anti-clockwise, our system will be unstable.
•Let's look at our Nyquist diagram for a gain of 1: •nyquist([ 1 10 24], [ 1 -8 15])
ExampleExampleNyquist Diagram
Real Axis
Imag
inar
y A
xis
-1.5 -1 -0.5 0 0.5 1 1.5 2-1.5
-1
-0.5
0
0.5
1
1.5
Stability analysis
There is no encirclement of the -1+j0 point. This implies that the system is stable if there are no poles of G(s)H(s) in the right-half s plane; otherwise, the system is unstable.
There are one or more counterclockwise encirclements of the -1+j0 point. In this case the system is stable if the number of counterclockwise encirclements is the same as the number of poles of G(s)H(s) in the right-half s plane; otherwise, the system is unstable.
There are one or more clockwise encirclements of the -1+j0 point. In this case the system is unstable.
Nyquist Stability CriterionNyquist Stability Criterion
Is the reciprocal of the magnitude at the frequency at which the phase angle is -1800.)j(G
Phase crossover frequency, pFrequency where the phase angle of )()( jHjG is –180 o
Gain crossover frequency, gFrequency where the magnitude of )()( jHjG
180)(1
jGGM
Gain margin,)(
1
)(
1
180 pjGjGGM
)(log20)(log20)(180 pjGjGdBGM
Gain margin in Nyquist Plot
1)( gjG
GH(j)-planej
)( pjG
-1+j0
1/GM p
g
Gain margin,
Phase margin in Nyquist PlotPhase margin in Nyquist Plot
1)( gjG
j
-1+j0
GH(j)-plane
Unit circle
1)( gjG will give phase angle of )( gjG hence the phase margin
Phase margin, 180
Negative gain and phase margins Negative gain and phase margins in Nyquist Plotin Nyquist Plot
1)( pjG negativejGdBGM p )(log20)(
180
-1+j0
j GH(j)-plane
As , the gain margin
As , the phase margin negative 180
)( pjG
For
Example:
8K , shows the Nyquist plot and its respective gain and phase margin
74
52 ss
K
3
1
s
+
-
Open loop transfer function
)74)(3(
5)(
2
sss
KsGOL
Frequency response
)19()721(
5
)47)(3(
5)(
222
j
K
jj
KjG
ExampleExample
Magnitude
22222 )19()721(
5)(
KjG
Phase angle
)721(
)19(tan
2
21
(i) At, 0
9.121
)8(5)0( jG and 0
21
0tan)0( 1
j
(ii) At,
0)8(5
)(
jG and 90270tan)(2
31
j
ExampleExample
Real crossing, when .(iii) 0)19( 2
then 1.19 srad 36.00)19721(
)8(5)19(
2
jG
Imaginary crossing, when (iv) 0721 2
1.3 srad
then 44.1)319(30
)8(5)3(
2
jG
» nr=40;» dr1=[ 1 3];» dr2=[1 4 7];» dr=conv(dr1,dr2);» nyquist(nr,dr)
ExampleExample
ExampleExample
Nyquist Diagram
Real Axis
Imag
inar
y Ax
is
-1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
180)(1
jGGM
36.0)19()(180
jGjG
dB87.8)36.0log(20
)(log20180
jGGM
ExampleExample
At =2.54 1)( gjG
8352.
Positive gain and Phase margins Positive gain and Phase margins in Bode plotin Bode plot
g
LM (dB)
(rad.s-1)g p0
GMpLM
0o
PM g
-180o
GM – gain marginPM – phase margin
p- Gain crossover frequency
- Phase crossover frequency
Negative gain and Phase margins in Bode plotNegative gain and Phase margins in Bode plot
LM (dB)
GM g (rad.s-1)
p
PM
0
-180 o
0o
Note that, negative gain or phase margin means that the system is not stable
Example:
If 1)( sGc and K=96 determine gain and phase crossover frequencies. Consequently what is the system gain and phase margin.
cG )40)(2( sss
K+
-
ExampleExample
Frequency response
140
12
)(
2.1
140
12
)(402
96)(
jjj
jjj
jG
140
12
)(
2.1log20)(
22
jLM
)025.0(tan)5.0(tan90)( 11 j
dB22)1.0(
2.1log20)1.0(
jLM and 90)1.0( j
ExampleExample
s
1
2
1
s
40
1
s
kutub orijin
kutub hakiki
kutub hakiki
(rad.s-1) 0.1 2 40
(dB/dekad)
-20 -20 -20
(dB/dekad)
0 -20 -20
(dB/dekad)
0 0 -20
Total slope dB/decade
-20 -40 -60
Table of LM slope for the 3 poles
ExampleExample
2
1
s
40
1
s
kutub hakiki
kutub hakiki
(rad.s-1) 0.1 0.2 4 20 400
kecerunan (darjah/dekad)
0 -45 -45 0 0
kecerunan (darjah/dekad)
0 0 -45 -45 0
jumlah kecerunan(darjah/dekad)
0 -45 -90 -45 0
Table for phase angle slope from the two poles, notes that the pole at origin does not contribute to the slope as the angle is constant -90o
ExampleExample
(rad.s-1) 0.1 2 40 400 1000
Total slope(dB/dec)
-20 -40 -60 -60 -60
LM (dB) 22 -4 -56 -116 -140
To obtain the LM vs freq. plot, we determine the LM value at the corner frequency, this can be obtained by simple trignometry
2log1log
LM (dB/dekad)1LM
2LM
LM
LMLM
12
12
loglog
1212 loglog LMLMLM
Using the trignometry’s formula,
ExampleExample
As for the LM slope, we apply a trignometry’s formula to obtain the
phase angle slope at the low and high frequency for each cornerfrequency as in table below
(rad.s-1) 0.1 0.2 4 20 400 1000
Total slope(deg/dec)
0 -45 -90 -45 0 0
(deg) -90 -90 -149 -212 -271 -271
2log1log
1
2
(deg/decade)
12
12
loglog 1212 loglog
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10-1
100
101
102
103
-150
-100
-50
0
50
10-1
100
101
102
103
-300
-250
-200
-150
-100
-50
PM
GM
wg
wp
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1rad.s5.8 p1rad.s3.1 g
50PM
dB30GM
>> bode([96],[1 42 80 0])
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-150
-100
-50
0
50
10-1
100
101
102
103
-270
-225
-180
-135
-90
PM
GM
Wg Wp
>>[GM,PM,Wg,Wp] = margin(sys)
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>> num=[96];den=[1 42 80 0];>> sys=tf(num,den);>> [GM,PM,Wg,Wp] = margin(sys)Gm = 35.0000Pm = 60.5601Wg = 8.9443Wp = 1.0599
>> Gm_dB = 20*log10(Gm)Gm_dB = 30.8814
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