MICROVE DEVICES: Microwave Waveguides

4. Microwave Waveguides

Microwave Waveguides

A waveguide consists of a hollow metallic tube of a rectangular or circular shape used to guide an electromagnetic wave. Waveguides are used principally at frequencies in the mi crowave range; inconveniently large guides would be required to transmit radio-frequency power at longer wavelengths.

In waveguides the electric and magnetic fields are confined to the space within the guides. Thus, no power is lost through radiation, and even the dielectric loss is negligible, since the guides are normally air-filled. However, there is some power loss as heat in the walls of the guides, but the loss is very small.

It is possible to propagate several modes of electromagnetic waves within a waveguide. These modes correspond to solutions of Maxwell’s equations for particular waveguides. A given waveguide has a definite cutoff frequency for each allowed mode.

If the frequency of the impressed signal is above the cutoff frequency for a given mode, the electromagnetic energy can be transmitted through the guide for that particular mode without attenuation. Otherwise the electromagnetic energy with a frequency below the cutoff frequency for that particular mode will be attenuated to a negligible value in a relat ively short distance.

The dominant mode in a particular guide is the mode having the lowest cutoff frequency. It is advisable to choose the dimensions of a guide in such a way that, for a given input signal, only the energy of the dominant mode can be transmitted through the guide.

The process of solving the waveguide problems may involve three steps: 27

i. The desired wave equations are written in the form of either rectangular or cyl indrical coordinate systems suitable to the problem at hand.

ii. The boundary conditions are then applied to the wave equations.

iii. The resultant equations usually are in the form of partial differential equations in either time or frequency domain. They can be solved by using the proper method.

4.1 Rectangular Waveguides

A rectangular waveguide is a hollow metallic tube with a rectangular cross section. The conducting walls of the guide confine the electromagnetic fields and thereby guide the electromagnetic wave. A number of distinct field configurations or modes can exist in waveguides.

When the waves travel longitudinally down the guide, the plane waves are reflected from wall to wall. This process results in a component of either electric or magnetic field in the direction of propagation of the resultant wave; therefore the wave is no longer a transverse electromagnetic (TEM) wave.

Figure 4.1.1 shows that any uniform plane wave in a lossless guide may be resolved into Transverse electromagnetic (TE) and Transverse Magnetic (TM) waves When the

Figure 4.1.1: Plane wave reflected in a waveguide

wavelength λ is in the direction of propagation of the incident wave, there will be one 28

component λn in the direction normal to the reflecting plane and another λp parallel to the plane given as

λn =λ

cosθ(4.1.1)

and

λp =λ

sinθ(4.1.2)

where θ is the angle of incidence and λ is the wavelength of the signal in unbounded me dium.

A plane wave in a waveguide resolves into two components: one standing wave in the direction normal to the reflecting walls of the guide and one traveling wave in the direc tion parallel to the reflecting walls. In lossless waveguides the modes may be classified as either transverse electric (TE) mode or transverse magnetic (TM) mode.

In rectangular guides the modes are designated TEmn or TMmn· The integer m denotes the number of half waves of electric or magnetic intensity in the x direction, and n is the number of half waves in the y direction if the propagation of the wave is assumed in the positive z direction.

4.1.1 Solutions of Wave Equations in Rectangular Coordinates A rectangular coordinate system is shown in figure 4.1.2

Figure 4.1.2: Rectangular waveguide

The electric and magnetic wave equations in frequency domain are given by ∇2~E = γ2~E (4.1.3)

∇2H~ = γ2H~ (4.1.4)

where γ =pjω(σ + jωε) = α + jβ. These are called the vector wave equations.The rectangular components of E or H satisfy the complex scalar wave equation or Helmholtz equation.

∇2ψ = γ2ψ (4.1.5)

∂ x2 +∂2

where ∇2 =∂2

∂ y2 +∂2 ∂ z2.

Equation 4.1.5 is solved by the method of separation of variables, the solution is assumed in the form of

ψ = X(x)YZ(z) (4.1.6)

where the variables are functions of x, y and z only respectively.

substitution of equation 4.1.6 to equation 4.1.5 and dividing it by equation 4.1.6 we get; d2X

dx2+1Yd2Y

dy2+1Zd2Z

dz2= γ2(4.1.7)

Since the sum of the three terms on the left-hand side is a constant and each term is in dependently variable, it follows that each term must be equal to a constant. Let the three terms be k2x;, k2y;, and k2z;, respectively; then the separation equation is given by

−k2x −k2x −k2x = γ2

The general solution of each differential equation in equation 4.1.7

d2X

dx2= k2xX (4.1.8)

d2Y

dy2= k2yY (4.1.9)

d2Z

dz2= k2z Z (4.1.10)

will be of the form

X = Asin(kxx) +Bcos(kxx) (4.1.11)

Y = Csin(kyy) +Dcos(kyy) (4.1.12)

X = E sin(kzz) +F cos(kzz) (4.1.13)

The total solution of the Helmholtz equation in rectangular coordinates is ψ = {Asin(kxx) +Bcos(kxx)} Csin(kyy) +Dcos(kyy) (4.1.14) {E sin(kzz) +F cos(kzz)}

The propagation of the wave in the guide is conventionally assumed in the positive z dir ection. It should be noted that the propagation constant γg in the guide differs from the intrinsic propagation constant γ of the dielectric. Let

γ2g = γ2 +k2x +k2y = γ2 +k2c(4.1.15)

where kc =

k2x +k2yis the cut-off wave number.

For a lossless dielectric, γ2 = −ω2µε. Thus, q

γg = ±

ω2µε −k2c(4.1.16)

There are three cases for the propagation constant γg in the waveguide.

i. Case I: There will be no wave propagation (evanescence) in the guide if ω2µε = k2cand γg = 0. This is the critical condition for cutoff propagation. The cutoff frequency is expressed as

fc =1 2π√µε

k2x +k2y

ii. Case II: The wave will be propagating in the guide if ω2µε > k2cand

γg = ±jβg = ±jω√µε

1−

fc f

This means that the operating frequency must be above the cutoff frequency in order for a wave to propagate in the guide.

iii. Case III: The wave will be attenuated if ω2µε < k2cand

γg = ±αg = ±ω√µε31

s fc

f−1

This means that if the operating frequency is below the cutoff frequency, the wave will decay exponentially with respect to a factor of −αgz and there will be no wave propagation because the propagation constant is a real quantity.

Therefore the solution to the Helmholtz equation in rectangular coordinates is given by ψ = {Asin(kxx) +Bcos(kxx)} Csin(kyy) +Dcos(kyy) e−jβgz(4.1.17)

4.2 TE Modes in Rectangular Waveguides

Figure 4.2.1 shows a rectangular waveguide of cross-section a × b along the z-axis The

Figure 4.2.1: Coordinates of rectangular waveguide

TEmn modes in a rectangular guide are characterized by Ez = 0. i.e., the z component of the magnetic field, Hz, must exist in order to have energy transmission in the guide. Consequently, from a given Helmholtz equation,

∇2H~ z = γ2H~ z (4.2.1)

a solution of the form

Hz =

Am sin

mπx a

+Bm cos

mπx a

Cn sin

nπy b

+Dn cos

nπy b

e−jβgz

(4.2.2)

will be determined in accordance with the given boundary conditions, where kx =mπaand ky =nπbare replaced.

For a lossless dielectric, Maxwell’s curl equations in frequency domain are ∇×~E = −jωµH~ (4.2.3)

∇×H~ = jωε~E (4.2.4)

Equations 4.2.3 and 4.2.4 have their components as,

∂ y−∂Ey

∂Ez ∂Ex

∂ z= −jωµHx

∂ z−∂Ez

∂ x= −jωµHy

∂Ey

∂ x−∂Ex

∂ y= −jωµHz

∂ y−∂Hy

∂Hz ∂Hx

∂ z= jωεEx

∂ z−∂Hz

∂ x= jωεEy

∂Hz

∂ x−∂Hx

∂ y= jωεEz

substituting ∂∂ z = −jβg and Ez = 0 the above equations are simplified βgEy = −ωµHx (4.2.5)

βgEx = −ωµHy (4.2.6)

∂Ey

∂ x−∂Ex

∂ y= −jωµHz (4.2.7)

∂Hz

∂ y+ jβgHy = jωεEx (4.2.8)

jβgHx −∂Hz

∂ x= jωεEy (4.2.9)

∂ x−∂Hx

∂Hz

∂ y= 0 (4.2.10)

Solving the above six equations we get the TE-mode field equations of

Ex = −jωµ k2c

∂Hz

∂ y(4.2.11)

Ey =jωµ k2c

∂Hz

∂ x(4.2.12)

Ez = 0 (4.2.13)

Hx = −jβg k2c

Hy = −jβg k2c

∂Hz

∂ x(4.2.14) ∂Hz

∂ y(4.2.15)

Hz = Equation 4.2.2 (4.2.16)

where k2c = ω2µε = β2g has been replaced.

Differentiating equation 4.2.2 w.r.t x and y and then substituting equations 4.2.11 through 4.2.16 will yield a set of field equations. The boundary conditions are applied to the newly found field equations in such a manner that either the tangent E field or the normal H field vanishes at the surface of the conductor.

Since Ex = 0, then ∂Hz

∂ y = 0 at y = 0,b, then Cn = 0. Also for Ey = 0, then ∂Hz

∂ x = 0 at

y = 0,a, then Am = 0.

It is generally concluded that the normal derivative of Hz must vanish at the conducting surface i.e.,∂Hz

∂n= 0 (4.2.17)

at the guide walls. Therefore the magnetic field in the positive z direction is given by

Hz = H0z cos

mπx a

cos

nπy b

e−jβgz(4.2.18)

where H0zis the constant amplitude. Substituting equation 4.2.18 in equations 4.2.11 to 4.2.16 yields TEmn field equations as

Ex = E0x cos

mπx a

sin

nπy b

e−jβgz(4.2.19)

Ey = E0y sinHx = H0x sin

mπx a

cos

nπy b

e−jβgz(4.2.20)

e−jβgz(4.2.21)

Hy = H0y cos

mπx a

sin

nπy b

e−jβgz(4.2.22)

where m and n are equal to 0,1,2,3..... with m = n = 0 excepted. 34

The cutoff wave number kc for the TEmn modes, is given by

kc =

r mπ a

2+ nπb 2= ωc√µε

The cutoff frequency fc for the TEmn modes, is given by

fc =1

2√µε

The phase constant βg is expressed as

r m a

2+ nb 2

βg = ω√µε

1−

fc f

The phase velocity in the positive z direction for the TEmn modes νg =ωβg=νp

1−

fcf 2

where νp = √1µε is the phase velocity in an unbounded dielectric. The characteristic wave impedance of TEmn modes in the guide is

Hy= −Ey

Hx=ωµ

Zg =Ex

βg=η

qµ

1−

fcf 2

where η =

εis the intrinsic impedance in an unbounded dielectric. The wavelengthλg

in the guide for the TEmn modes is given by λg =λ

1−

fcf 2

where λ =vpfis the wavelength in an unbounded dielectric.

The cutoff frequency is a function of the modes and guide dimensions, the physical size of the waveguide will determine the propagation of the modes. Whenever two or more modes have the same cutoff frequency, they are said to be degenerate modes. In a rectangular guide the corresponding TEmn and TMmn modes are always degenerate.

The mode with the lowest cutoff frequency in a particular guide is called the dominant 35

mode. The dominant mode in a rectangular guide with a < b is the TE10 mode. Each mode has a specific mode pattern (or field pattern).

It is normal for all modes to exist simultaneously in a given waveguide. The situation is not very serious, however. Actually, only the dominant mode propagates, and the higher modes near the sources or discontinuities decay very fast.

Example:

An air-filled rectangular waveguide of inside dimensions 7 x 3.5 cm operates in the dominant TE10 mode as shown in figure 4.2.2.

(a) Find the cutoff frequency.

(b) Determine the phase velocity of the wave in the guide at a frequency of 3.5 GHz. (c) Determine the guided wavelength at the same frequency.

Figure 4.2.2: Rectangular waveguide

Solution

(a) Find the cutoff frequency.

fc =c2a=3×108 m/s

2×7×10−2 m= 2.14 GHz

(b) Determine the phase velocity of the wave in the guide at a frequency of 3.5 GHz.

νg =c

fcf 2=3×108 m/s

1−

1−2.14 3.5

2= 3.78×108 m/s

λg = rλo 1−

1−2.14 3.5

2= 10.8 cm

4.3 TM Modes in Rectangular Waveguides

The TMmn modes in a rectangular guide are characterized by Hz = 0. i.e., the z compon ent of the electric field, Ez, must exist in order to have energy transmission in the guide. Consequently, from a given Helmholtz equation as

∇2~Ez = γ2~Ez (4.3.1)

A solution of the Helmholtz equation is in the form of

Ez =

Am sin

mπx a

+Bm cos

mπx a

Cn sin

nπy b

+Dn cos

nπy b

e−jβgz(4.3.2)

which must be determined in accordance with the given boundary conditions. The proced ures for doing so are similar to those used in finding the TE-mode wave.

The boundary conditions on Ez require that the field vanishes at the waveguide walls, since the tangent component of the electric field Ez is zero on the conducting surface. This requirement is that for Ez = 0 at x = 0,a, then Dn = 0. and for Ez = 0 at y = 0,b, then ∂Hz

∂ x = 0 at y = 0,a, then Am = 0.Thus the solution in equation 4.3.2 reduces to

Ez = E0z sin

where m and n equals to 1,2,3,4,......

mπx a

sin

nπy b

e−jβgz(4.3.3)

If either m = 0 or n = 0, the field intensities all vanish. So there is no TM01 or TM10 mode in a rectangular waveguide, which means that TE10 is the dominant mode in a rectangular waveguide for a > b. For Hz = 0, the field equations, after expanding ∇×H = jωεE, are

given by

∂Ez

∂ y+ jβgEy = −jωµHx (4.3.4)

∂Ez

∂ x+ jβgEx = jωµHy (4.3.5)

∂Ey

∂ x+∂Ex

∂ y= 0 (4.3.6)

βgHy = ωεEx (4.3.7)

−βgHx = ωεEy (4.3.8)

∂Hy

∂ x+∂Hx

∂ y= jωεEz (4.3.9)

Solving equations 4.3.4 through 4.3.9 simultaneously the resultant field equations for TM

modes are

Ex = −jβg k2c

Ey = −jβg k2c

∂Ez

∂ x(4.3.10) ∂Ez

∂ y(4.3.11)

Ez = Eq.4.3.3 (4.3.12)

Hx =jωε k2c

∂Ez

∂ y(4.3.13)

Hy = −jωε k2c

∂Ez

∂ x(4.3.14)

Hz = 0 (4.3.15)

where β2g −ω2µε = −k2cis replaced.

Differentiating equation 4.3.3 w.r.t x and y and then substituting equations 4.3.10 through 4.3.15 will yield a set of field equations. The TMmn mode field equations in rectangular waveguides are

Ex = E0x cos

mπx a

sin

nπy b

e−jβgz(4.3.16)

Ey = E0y sinHx = H0x sin

mπx a

cos

nπy b

e−jβgz(4.3.17)

e−jβgz(4.3.18)

Hy = H0y cos

where m and n are equal to 0,1,2,3.....

mπx a

sin

nπy b

e−jβgz(4.3.19)

Some of the TM-mode characteristic equations are identical to those of the TE modes, but some are different i.e.

fc =1 2√µε

r m a

2+ nb 2

βg = ω√µε

1−

fc f

νg =νp

Zg =βg

1−

fcf 2 fc

ωε= η

1−

λg =λ

1−