Elements of Electromagnetics
Elements of Electromagnetics
7th Edition
ISBN: 9780190698669
Author: Sadiku
Publisher: Oxford University Press
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Chapter 12, Problem 29P

(a)

To determine

Find the value of the attenuation constant due to the dielectric losses and due to the conduction losses of the copper waveguide for TE10 mode.

(a)

Expert Solution
Check Mark

Answer to Problem 29P

The value of the attenuation constant due to the dielectric losses (αd) and due to the conduction losses (αc) for TE10 mode is 0.0185Np/m and 0.033Np/m respectively.

Explanation of Solution

Calculation:

Given dimensions a×b for the copper waveguide is 1cm×2cm.

Write the expression to calculate the cutoff frequency for TE10 mode.

fc=u2a        (1)

Here,

u is the phase velocity of uniform plane wave in dielectric medium and

a is the inner dimension of the waveguide.

Write the expression to calculate the phase velocity of uniform plane wave in the lossless dielectric medium.

u=cεr

Here,

c is the speed of light in vacuum which is 3×108m/s and

εr is the permittivity of the medium.

Substitute cεr for u in Equation (1).

fc=(cεr)2a=c2aεr

Substitute 1cm for a, 3×108m/s for c and 2.6 for εr in above Equation.

fc=(3×108m/s)2(1cm)2.6=(3×108)m/s22.6(1×102)m {1c=102}=9.3026×109s1=9.3026GHz {1Hz=11s,1G=109}

Write the expression to calculate the intrinsic impedance of a uniform plane wave in the medium.

η=με=377εr

Substitute 2.6 for εr in above Equation.

η=3772.6Ω=233.81Ω

Write the expression to calculate the attenuation constant due to the dielectric losses.

αd=ση21(fcf)2        (2)

Substitute 233.81Ω for η, 104S/m for σ, 12GHz for f and 9.3026GHz for fc in Equation (2).

αd=(104S/m)(233.81Ω)21(9.3026GHz12GHz)2=(104)(233.81)ΩSm120.39904=0.0185ΩΩ1m1 {1S=1Ω1}=0.0185Np/m

Write the expression to calculate the attenuation constant due to conduction losses for the TE10 mode.

αc=2Rsbη1(fcf)2(0.5+ba(fcf)2)        (3)

Here,

Rs is the real part of the intrinsic impedance of the conducting wall .

Write the expression to calculate the real part of the intrinsic impedance of the conducting wall.

Rs=πfμσc

Rs=πfμoσc {μ=μo}

Substitute 4π×107H/m for μo, 12GHz for f and 5.8×107S/m for σc in above equation.

Rs=π(12GHz)(4π×107H/m)5.8×107S/m=π(12×109)(4π×107)s1Hm15.8×107S/m {1G=109,1Hz=11s}=47374.10113Ωss15.8×107Ω1 {1H=1Ω1s,1S=11Ω}=8.1679×104Ω2

Simplify the above Equation.

Rs=0.02858Ω

Substitute 0.02858Ω for Rs, 1cm for a, 2cm for b, 233.81Ω for η, 12GHz for f and 9.3026GHz for fc in Equation (3).

αc=2(0.02858Ω)(2cm)(233.81Ω)1(9.3026GHz12GHz)2(0.5+(2cm1cm)(9.3026GHz12GHz)2)=0.05716(2×102)(233.81)1(0.601)(0.5+(2)(0.601))=0.033Np/m

Conclusion:

Thus, the value of the attenuation constant due to the dielectric losses (αd) and due to the conduction losses (αc) for TE10 mode is 0.0185Np/m and 0.033Np/m respectively.

(b)

To determine

Find the value of the attenuation constant due to the dielectric losses and due to the conduction losses of the copper waveguide for TM11 mode.

(b)

Expert Solution
Check Mark

Answer to Problem 29P

The value of the attenuation constant due to the dielectric losses (αd) and due to the conduction losses (αc) for TM11 mode is 0.02344Np/m and 0.0441Np/m respectively.

Explanation of Solution

Calculation:

Write the expression to calculate the cutoff frequency for TM11 mode.

fc=u2[1a2+1b2]12

Substitute cεr for u in above equation.

fc=(cεr)2[1a2+1b2]12=c2εr[1a2+1b2]12

Substitute 1cm for a, 2cm for b, 3×108m/s for c and 2.6 for εr in above Equation.

fc=(3×108m/s22.6)[1(1cm)2+1(2cm)2]12=(3×108)m/s22.6(111.803m1)=10.4×109s1=10.4GHz {1Hz=11s,1G=109}

Substitute 233.81Ω for η, 104S/m for σ, 12GHz for f and 10.4GHz for fc in Equation (2).

αd=(104S/m)(233.81Ω)21(10.4GHz12GHz)2=(104)(233.81)ΩSm120.2489=0.02344ΩΩ1m1 {1S=1Ω1}=0.02344Np/m

Write the expression to calculate the attenuation constant due to conduction losses for the TM11 mode.

αc=2Rsbη1(fcf)2[(ba)3+1(ba)2+1]        (4)

From part (a), the real part of the intrinsic impedance of the conducting wall is,

Rs=0.02858Ω

Substitute 0.02858Ω for Rs, 1cm for a, 2cm for b, 233.81Ω for η, 12GHz for f and 10.4GHz for fc in Equation (4).

αc=2(0.02858Ω)(2cm)(233.81Ω)1(10.4GHz12GHz)2[(2cm1cm)3+1(2cm1cm)2+1]=0.05716(2×102)(233.81)0.24889[8+14+1]=0.0441Np/m

Conclusion:

Thus, the value of the attenuation constant due to the dielectric losses (αd) and due to the conduction losses (αc) for TM11 mode is 0.02344Np/m and 0.0441Np/m respectively.

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