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Problem 6.23PP
Nyquist plots and the classical plane cun/es; Determine the Nyquist plot, using Matlab, for the 
systems given below, with K = and verity that the beginning point and end point for the jw > 0 
portion have the correct magnitude and phase;
(a) The classical cunre called Cayley’s Sextic, discovered by Maclaurin in 1718;
1KG(s) = K-
( » + ! ) ’ ■
(b) The classical cunre called the Cissoid, meaning Ivy-shaped; 
1
JTG(a) = K
i ( i + 1)
(c) The classical cunre called the Folium of Kepler, studied by Kepler in 1609; 
1
CG(a) = K
( a - l ) ( s - H ) 2 '
(d) The classical cunre called the Folium (not Kepler's); 
1KG(s) = K
( j - l ) ( j - F 2 )
(e) The classical cunre called the Nephroid, meaning kidney-shaped;
KG{s) = K -
(1 -1 )3
(f) The classical cunre called Nephroid of Freeth, named after the English mathematician T J. 
Freeth;
KG(s) = K ( i-F l) ( j3 -F 3 )
4 ( r - l ) 3
(g) A shitted Nephroid of Freeth; 
_____
Step-by-step solution
step 1 of 14
(a)
Consider the open loop transfer function of the classical cunre called Cayley’s Sextic.
1KG{s) = K -
Substitute j a for s.
KG (ja>) = K — L ^
[ j a + l )
Determinethemagnitudeat ^ s Q a n d Jtf = l 
1KG (jta)=-
■ ( i / T i v ) ’
X G (0 ) = I
Determine the phase at ^ s Q a n d Jtf = l
za :g (o) = o°
Enter the following code in MATtAB to plot the Nyquist plot for = i 
» numG=1;
» denG=conv(conv([1 1],[1 1]),[1 1]);
» sysG=tf(numG,denG);
» nyquist(sysG)
Step 2 of 14
The Nyquist plot for the cunre is shown in Figure 1.
The magnitude of the beginning point is 1 and phase is Q* and hence has the correct magnitude 
and phase.
Step 3 of 14
(b)
Consider the open loop transfer function of the classical cunre called Cissoid, meaning ivy­
shaped.
Substitute jo> for s.
= AT
Jta(ja>+\) 
1
- a + j a
step 4 of 14
Determine themagnitudeat ^ s Q a n d Jtf = l 
1
KG{Ja>)
■
Determine the phase at ^ s Q a n d Jtf = l
ZKGO v») = - t a i T ' ^ - ^ j
- ( i )
ZATG(0) = 0°
Step 5 of 14
Enter the following code in MATtAB to plot the Nyquist plot for = i 
» numG=1;
» denG=[1 1 0];
» sysG=tf(numG,denG);
» nyquist(sysG)
The Nyquist plot for the cunre is shown in Figure 2.
The magnitude of the beginning point is oo and phase is 0® and hence has the correct 
magnitude and phase.
Step 6 of 14
(c)
Consider the open loop transfer function of the classical cunre called Folium of Kepler;
1KG(s) = K -
( 4 - l ) ( 4 + l ) '
Substitute j a for s.
K G ( ja ) - K —
(yV »-l)(y«> + l)’
^ ( l+ a i ‘ ) . ^ ( l - a ‘ )^ + 4 a ‘
step 7 of 14
Determine themagnitudeat ^ s Q ^ n d Jtf = l
1K G {ja) = -
^ { l+ a ’ ) ^ j { l - a ? f + 4 a ‘
K G {0 )-\
Determine the phase at ^ s Q ^ n d Jtf = l 
ZK G (ja )= -lS iy ‘
Enter the following code in MATtAB to plot the Nyquist plot for = i 
» numG=1;
» denG=conv([1 -1],[1 2 1]);
» sysG=tf(numG,denG);
» nyquist(sysG)
Step 8 of 14
The Nyquist plot for the cunre is shown in Figure 3
Hence the magnitude and phase of the beginning point are verified.
Step 9 of 14
(b)
Consider the open loop transfer function of the classical cunre called Folium;
1KG(s) = K j
(» -1)(4+2)
Substitute J a for s.
1K G (ja) = K -
= K -
{ J a - l ) ( J a * 2 )
1
Determine themagnitudeat ^ s Q ^ n d Jtf = l 
K G ( ja ) = - '
'4 + a
JCG(0) = i = 0.S
Determine the phase at ^ s Q ^ n d Jtf = l
■ - tan" '(r» ) - t a n " ' j
Enter the following code in MATtAB to plot the Nyquist plot for Jtf = i 
» numG=1;
» denG=conv([1 -1],[1 2]);
» sysG=tf(numG,denG);
» nyquist(sysG)
Step 10 of 14
The Nyquist plot for the cunre is shown in Figure 4.
Hence the magnitude and phase of the beginning point are verified.
Step 11 of 14
(e)
Consider the open loop transfer function of the classical cunre called Nephroid, meaning kidney 
shaped;
KG(s) = K
2 (j + 1)(4’ - 4 j + i )
( 4 -1 ) ’
Substitute J a for s.
( y o - l )
KG(0) = 2
Determine the phase at at = 0bnd Jtf = l
^ G [ ja ) ■ tan"' (nr)+tan"' ̂ |-̂ ĵ-3tan"' {-a )
= 0»
Enter the following code in MATtAB to plot the Nyquist plot for s | 
» numG=2*conv([1 1],[1 -4 1]);
» denG=conv(conv([1 -1],[1 -1]),[1 -1]);
» sysG=tf(numG,denG);
» nyquist(sysG)
Step 12 of 14
The Nyquist plot for the cunre is shown in Figure 5.
Nyaebl Dtoaraw
Hence the magnitude and phase of the beginning point are verified.
(f)
Consider the open loop transfer function of the classical cunre called Nephroid of Freeth.
4 ( j - l )
Substitute J a for s.
4(y = 0bnd Jtf = l 
ZK G (ja ) ■ tan"' (ar)-3tan"' (-®)
= 0
Enter the following code in MATtAB to plot the Nyquist plot for Jtf = i 
> numG=2*conv([1 1],[1 0 3]);
» denG=4*conv{conv([1 -1],[1 -1]),[1 -1]);
» sysG=tf(numG,denG);
» nyquist(sysG)
Step 13 of 14
The Nyquist plot for the cunre is shown in Figure 6.
Hence the magnitude and phase of the beginning point are verified.
Step 14 of 14
(g)
Consider the open loop transfer function of the classical cunre called shifted Nephroid of Freeth.
r V (^’ + 0KG(s) = K-^------ V
 ̂ ’ 4 (4-1)’
Substitute J a for s.
(y»-l)
■ ( v / n v ) ’
Determine themagnitudeat a> = 0bnd Jtf = l
(V T iV )
a:g (o) = i
Determine the phase at a> = 0bnd Jtf = l 
ZKG(ja) ■ -3tan"' (-