Ch-4: Microwave resonators & Waveguide components

Page Number: 193 Example 4.1

In [30]:
from __future__ import division
from math import sqrt 
#Given
a=5 #cm
a1=a/100 #m
b=2 #cm
b1=b/100 #m
c=15 #cm
c1=c/100 #m

#(i) Air filled cavity
m=1 
n=0 
p=1 
c=3e8  #for air
fr=(1/2)*c*sqrt((m/a1)**2+(n/b1)**2+(p/c1)**2)  #hz
print 'Resonant frequency for an air filled cavity: %0.3f'%(fr/10**9),'Ghz'

#(ii) Dielctric filled cavity
er=2.56 
fr1=(1/2)*(c/sqrt(er))*sqrt((m/a1)**2+(n/b1)**2+(p/c1)**2) #hz
print 'Resonant frequency for dielectric cavity: %0.3f'%(fr1/10**9),'Ghz'

Resonant frequency for an air filled cavity: 3.162 Ghz
Resonant frequency for dielectric cavity: 1.976 Ghz

Page Number: 193 Example 4.2

In [31]:
 
#Given
a=0.38 #cm
a1=a/100 #m
b=0.76 #cm
b1=b/100 #m
f=50e9 
c=3e8 

#Length for TE102
m=1 
n=0 
p=2 
l=1/sqrt((f/c)**2-(1/(4*b1**2))) #m
print 'Length c: %0.3f'%(l*100),'cm'

Length c: 0.653 cm

Page Number: 194 Example 4.3

In [32]:
from math import pi 
#Given
c=3e8  #m/s
a=2.286 #cm
a1=a/100 #m
b=1.024 #cm
b1=b/100 #m
f=10e9 #hz
sig=6e7 
u=4e-7*pi 
w=2*pi*f 
eet=377 

#Shortest cavity length
lamc=2*a1 #m
fc=c/lamc #hz
lam=c/f #m
lamg=lam/sqrt(1-(fc/f)**2) #m
sc=lamg/2 #m
print 'Shortest cavity length: %0.3f'%(sc*100), 'cm'

#Qw of the resonator operating in TE101 mode
rs=sqrt((w*u)/(2*sig)) #ohm
lamr=c/f 
x=(((a1*b1)/(sc**2))+((sc**2+a1**2)/(2*sc*a1))+(b1*sc/a1**2)) 
qw=(2*pi*eet*a1*b1*sc)/(rs*(lamr**3)*x) 
print 'Qw of the resonator operating in TE101 mode %0.3f'%qw

Shortest cavity length: 1.988 cm
Qw of the resonator operating in TE101 mode 7990.324

Page Number: 195 Example 4.4

In [33]:
 
#Given
c=3e8  #m/s
a=4.8 #cm
a1=a/100 #m
b=2.2 #cm
b1=b/100 #m
f=5e9 #hz
er=2.25 
tandel=4e-4 
sig=5.813e7 
oneby=3e8 
u=4e-7*pi 
w=2*pi*f 
eet=377 

#Length at p=1
m=1 
n=0 
p=1 
z=(f*2*sqrt(er))/c 
cp1=p/sqrt((z**2)-((m/a1)**2)-((n/b1)**2)) 
print 'Length of resonator at p=1: %0.3f'%(cp1*100), 'cm'

#At p=2
cp2=cp1*2 
print 'Length of resonator at p=2: %0.3f'%(cp2*100), 'cm'

#Qw
rs=sqrt((w*u)/(2*sig)) #ohm
lamr=c/(f*sqrt(er)) 
x=(((a1*b1)/(cp1**2))+((cp1**2+a1**2)/(2*cp1*a1))+(b1*cp1/a1**2)) 
qw=(2*pi*(eet/sqrt(er))*a1*b1*cp1)/(rs*(lamr**3)*x) 
qd=1/tandel 
q=(qw*qd)/(qw+qd) 
print 'Q for TE101 mode: %0.3f'%q

Length of resonator at p=1: 2.200 cm
Length of resonator at p=2: 4.400 cm
Q for TE101 mode: 1925.612

Page Number: 196 Example 4.5

In [34]:
 
#Given
c=3e8  #m/s
a=2 #cm
a1=a/100 #m
b=2.5 #cm
b1=b/100 #m

print 'TE modes' 
h01=3.832 
fr=(c/(2*pi))*sqrt((h01/a1)**2+(pi/b1)**2) #hz
print 'Resonant frequency for mode TE010: %0.3f'%(fr/10**9),'Ghz'

h11=1.841 
fr1=(c/(2*pi))*sqrt((h11/a1)**2+(pi/b1)**2) #hz
print 'Resonant frequency for mode TE111: %0.3f'%(fr1/10**9),'Ghz'

h21=3.054 
fr2=(c/(2*pi))*sqrt((h21/a1)**2+(pi/b1)**2) #hz
print 'Resonant frequency for mode TE211: %0.3f'%(fr2/10**9),'Ghz'

print 'TM modes:' 
l1=0 
h011=2.405 
fr3=(c/(2*pi))*sqrt((h011/a1)**2+(pi*l1/b1)**2) #hz
print 'Resonant frequency for mode TM010 %0.3f'%(fr3/10**9),'Ghz'

l2=1 
fr4=(c/(2*pi))*sqrt((h011/a1)**2+(pi*l2/b1)**2) #hz
print 'resonant frequency for mode TM011: %0.3f'%(fr4/10**9),'Ghz'

l3=1 
h111=3.832 
fr5=(c/(2*pi))*sqrt((h111/a1)**2+(pi*l3/b1)**2) #hz
print 'Resonant frequency for mode TM111: %0.3f'%(fr5/10**9),'Ghz'

TE modes
Resonant frequency for mode TE010: 10.940 Ghz
Resonant frequency for mode TE111: 7.438 Ghz
Resonant frequency for mode TE211: 9.442 Ghz
TM modes:
Resonant frequency for mode TM010 5.742 Ghz
resonant frequency for mode TM011: 8.305 Ghz
Resonant frequency for mode TM111: 10.940 Ghz

Page Number: 196 Example 4.6

In [35]:
 
#Given
QTM010=1.202 
QTE101=1.11 

r=QTM010/QTE101 
print 'Ratio of Qs of cylindrical and rectangular resonators: %0.3f'%r

Ratio of Qs of cylindrical and rectangular resonators: 1.083

Page Number: 197 Example 4.7

In [36]:
#Given
f=7.07e9 #hz
a=3 #cm
a1=a/100 #m
sig=5.8e7 
er=2.25 
tandel=4e-4 
ur=1 
n=377 
w=2*pi*f 
u=4e-7*pi 

#Q of resonantor
rs=sqrt(w*u/(2*sig)) #ohm
qw=(0.7419*n)/(rs*sqrt(2.25)) 
qd=1/tandel 
q=(qw*qd)/(qw+qd) 
print 'Q of resonator: %0.3f'%q

Q of resonator: 1931.819

Page Number: 198 Example 4.8

In [40]:
 
#Given
a=5 #cm
a1=a/100 #m
b=4 #cm
b1=b/100 #m
c=10 #cm
c1=c/100 #m
sig=5.8e7 
u0=4e-7*pi 
er=3 
eet=377 

ur=1 
spl=3e8 
tandel=2.5e-4 

#TE101 mode
m=1 
n=0 
p=1 
fr=(spl/(2*sqrt(er*ur)))*sqrt((m/a1)**2+(n/b1)**2+(p/c1)**2) #hz
print 'Resonant frequency:  %0.3f'%(fr/10**9), 'Ghz'

w=2*pi*fr 
rs=sqrt((w*u0)/(2*sig)) #ohm
lamr=spl/(fr*sqrt(er)) 
x=(((a1*b1)/(c1**2))+((c1**2+a1**2)/(2*c1*a1))+((b1*c1)/a1**2)) 
qw=(2*pi*(eet/sqrt(er))*a1*b1*c1)/(rs*(lamr**3)*x) 
print 'Q for TE101 mode: %0.3f'%qw

qd=1/tandel 
q=(qw*qd)/(qw+qd) 
print 'Q for lossy dielectric:  %0.3f'%q

#Value of qw is calculated wrong in book as lamr comes to be 0.08 not 0.89 m

Resonant frequency:  1.936 Ghz
Q for TE101 mode: 10916.466
Q for lossy dielectric:  2927.360

Page Number: 198 Example 4.9

In [41]:
 
#Given
c=3e8  #m/s
a=2.286 #cm
a1=a/100 #m
b=1.106 #cm
b1=b/100 #m

#For fr1=9.3e9 
fr1=9.3e9 #hz
lamr1=c/fr1 #m
c1=(2*a1)/sqrt((((2*a1)/lamr1)**2)-1) 

#For fr2=10.2e9 
fr2=10.2e9 #hz
lamr2=c/fr2 #m
c2=(2*a1)/sqrt((((2*a1)/lamr2)**2)-1) 

r=c1-c2 
print 'Range of piston movement: %0.3f'%(r*100), 'cm'

Range of piston movement: 0.710 cm

Page Number: 199 Example 4.10

In [44]:
 
#Given
a=3 #cm
a1=a/100 #m
d=10 #cm
d1=d/100 #m
df=2.5e6 
er=2.25 
p11=1.841 
c=3e8  #m/s

#Resonant frequency
fr=(c/2)*(sqrt((p11/a1)**2+(pi/d1)**2)) #hz
print 'Resonant frequency: %0.3f'%(fr/10**9),'Ghz'

#Q without dielectric
q0=fr/df 
print 'Q wirhout dielectric constant: %0.3f'%q0

# Q with dielectric
fr1=fr/sqrt(er) 
qd=1e3 
q=(q0*qd)/(q0+qd) 
print 'Q with dielectric constant: %0.3f'%q

Resonant frequency: 10.341 Ghz
Q wirhout dielectric constant: 4136.446
Q with dielectric constant: 805.313

Page Number: 200 Example 4.11

In [45]:
 
#Given
f=9.375e9 #hz
sig=5.8e7 
eet=377 
c=3e8  #m/s
w=2*pi*f 
r=1.5 
u=4e-7*pi 

#Radius
a=c/(f*2.62) #m
print 'Radius of resonantor %0.3f'%(a*100), 'cm'

#O
rs=sqrt((w*u)/(2*sig)) #ohm
x=1.202*eet 
y=rs*(1+(1/r)) 
q=x/y 
print 'Q of the resonator: %0.3f'%q

#Answer for Q is calculated as 10875 in book but it is 10763.303

Radius of resonantor 1.221 cm
Q of the resonator: 10763.303

Page Number: 215 Example 4.12

In [46]:
 
#Given
f=5e9 #hz
sig=5.813e7 
er=2.25 
tandel=4e-4 
c=3e8  #m/s
h01=3.832 
u=4e-7*pi 

#Length of resonator
lamr=c/(f*sqrt(er)) 
d=sqrt((((((2*3.832)**2)+(pi*pi))*(lamr*lamr))/(2*2*pi*pi)) )
print 'Length of resonator: %0.3f'%(d*100),'cm'

#Q of resonator
n=(120*pi)/sqrt(er) 
Rs=sqrt((f*u)/sig) 
a=d/2 
Qw1=n*(((h01/a)**2+(pi/d)**2)**(3/2))
Qw2=2*Rs*(((h01*h01)/(a*a*a))+((2*pi*pi)/(d*d*d))) 
Qw=Qw1/Qw2 
Qd=1/tandel 
Q=(Qw*Qd)/(Qw+Qd) 
print 'Q of resonator: %0.3f'%Q

#Value of Qw is calculated wrong in the book, it should be 50057.91 instead of 53473.8
#Hence the value of Q also differs

Length of resonator: 5.273 cm
Q of resonator: 2381.084

Page Number: 215 Example 4.13

In [14]:
#Given
p=100  #mW
#As 2 and 3 are matched terminals
x=1/2 
y=1/sqrt(2) 
s=[x,-x, y, -x, 0, y, y, y, 0] 

#Power delivered
#Port 1
p1=p*(1-s[1]**2) 
print 'Power at port 1:',p1, 'mW'

#Port2
p2=p*s[2]**2 
print 'Power at port 2:',p2, 'mW'

#Port 3
p3=p*s[3]**2 
print 'Power at port 3:' ,p3,'mW'

Power at port 1: 75.0 mW
Power at port 2: 50.0 mW
Power at port 3: 25.0 mW

Page Number: 216 Example 4.14

In [48]:
 
#Given
p=40  #mW
#Since port 3 is matched
x=sqrt(2) 
s=[1, 1, x, 1, 1, -x, x, -x, 0] 
r1=40  #ohm
r2=60  #ohm
w=50  #ohm

#Reflection coefficients
T1=(w-r1)/(w+r1) 
T2=(r2-w)/(r2+w) 

#As power is fed into 1 and 2 equally
pd=p/2 

#Power delivered
#Port 1
p1=pd*(1-T1**2) 
print 'Power at port 1:  %0.3f'%p1,'mW'

#Port2
p2=pd*(1-T2**2) 
print 'Power at port 2: %0.3f'%p2,'mW'

Power at port 1:  19.753 mW
Power at port 2: 19.835 mW

Page Number: 216 Example 4.15

In [53]:
from __future__ import division 
#Given
T1=1/2 
T2=3/5 
T3=0 
T4=4/5 
p=500e-3  #W
#S matrix for magic Tee
x=1/sqrt(2) 
s=[0, 0, x, x, 0, 0, x, -x, x ,x, 0, 0, x, -x, 0, 0] 
#Using the input output relation
#[b]=[s]*[a]
b=[0.6565, 0.7576, 0.5536, 0.0892] 

#(i) Power transmitted through ports
#Port 1
p1=(1/2)*b[0]**2*(1-T1**2) 
print 'Power at port 1: %0.3f'%p1,'W'

#Port2
p2=(1/2)*(b[1]**2)*(1-(T2**2)) 
print 'Power at port 2: %0.3f'%p2, 'W'

#Port 4
p4=(1/2)*b[3]**2*(1-T4**2) 
print 'Power at port 4: %0.3f'%p4,'W'

#(ii) Power reflected at port 3
#Port 3
p3=p*b[2]**2 
print 'Power at port 3: %0.3f'%p3,'W'

#(iii) Power absorbed
pabs=p-(p1+p2+p3+p4) 
print 'Power absorbed: %0.3f'%pabs,'W'

#Answer for power absorbed is calculated wrong in book

Power at port 1: 0.162 W
Power at port 2: 0.184 W
Power at port 4: 0.001 W
Power at port 3: 0.153 W
Power absorbed: 0.000 W

Page Number: 236 Example 4.18

In [57]:
from numpy import mat, set_printoptions
#Given
C=10  #dB
D=30  #dB

#Parameters
bet=10**(-C/20) 
x=bet/(10**(D/20)) 
a=sqrt(1-(bet*bet)) 
#Scattering matrix
#Assuming symmetery
s=mat([[0, a ,x ,(bet*1J)],[ a, 0, (bet*1J), x],[ x ,(bet*1J), 0 ,a],[ (bet*1J) ,x ,a, 0] ])
set_printoptions(precision=3)
print 'Scattering matrix:\n',s 

Scattering matrix:
[[ 0.000+0.j     0.949+0.j     0.010+0.j     0.000+0.316j]
 [ 0.949+0.j     0.000+0.j     0.000+0.316j  0.010+0.j   ]
 [ 0.010+0.j     0.000+0.316j  0.000+0.j     0.949+0.j   ]
 [ 0.000+0.316j  0.010+0.j     0.949+0.j     0.000+0.j   ]]

Page Number: 238 Example 4.20

In [59]:
 
#Given
vswr=2 
D1=8  #mW
D2=2  #mW

#Reflection coefficient at arm 4
T=(vswr-1)/(vswr+1) 
#Powwe delivered to D1
P=(D1*100)/(1-T**2) 
P1=0.99*P 
#Power reflected at D1
W1=(P/100)*T*T 
#Power reflected at load
W2=D2-W1 
Tt=sqrt((W2*100)/(P1)) 
pt=(1+Tt)/(1-Tt) 
print 'VSWR:%0.3f'%pt
Pl=P1*(1-(Tt*Tt)) 
print 'Power delivered:' ,Pl,'mW'

#Answer for P1 should be 792 but it is given as 800

VSWR:2.008
Power delivered: 791.0 mW

Page Number: 239 Example 4.21

In [60]:
from numpy import mat, set_printoptions
#Given
I=30  #dB
Il=0.4  #dB

S12=10**(I/-20) 
S21=10**(Il/-20) 
s=mat([[0, S12],[ S21, 0] ])
set_printoptions(precision=3)
print 'Scattering matrix:\n' ,s

Scattering matrix:
[[ 0.     0.032]
 [ 0.955  0.   ]]

Page Number: 240 Example 4.22

In [61]:
from numpy import mat, set_printoptions
#Given
I=30  #dB
Il=2  #dB
p=1.3 

#Elelments
T=(p-1)/(p+1) 
S11=T 
S22=T 
S33=T 
S12=10**(-Il/20) 
S13=10**(-I/20) 
S21=S13 
S32=S13 
S23=S12 
S31=S23 
s=mat([[S11, S21, S31] ,[S12, S22, S32], [S13, S23 ,S33] ])
set_printoptions(precision=3)
print 'Scattering matrix:\n' ,s

Scattering matrix:
[[ 0.13   0.032  0.794]
 [ 0.794  0.13   0.032]
 [ 0.032  0.794  0.13 ]]

Page Number: 249 Example 4.23

In [62]:
 
#Given
f=10e9  #Hz
u=4e-7*pi 
c=3e8  #m/s
a=2.29  #cm
a1=a/100 
b=1.02  #cm
b1=b/100 

#E/H
w=2*pi*f 
EbyH=(w*u)/sqrt(((w/c)**2)+((pi/a1)**2)) 
lam=c/f 
lamc=2*a1 
d=(1/4)*(lam/sqrt(1-((lam/lamc)**2))) 
print 'Position: %0.3f'%(d*100), 'cm'

#Answer for positon is calculated wrong in book

Position: 0.993 cm

Page Number: 250 Example 4.24

In [65]:
from numpy import mat, set_printoptions
#Given
#As it is perfectly matched
S12=1/sqrt(2) 
S21=S12 
s=mat([[0 ,S12] ,[S21, 0] ])
set_printoptions(precision=3)
print 'Scattering matrix:\n' , s

Scattering matrix:
[[ 0.     0.707]
 [ 0.707  0.   ]]