In the last lecture, we have coonsidered the case of guided wave between a pair of infinite conducting planes. In this lecture, we consdier a rectangular wave guide which consists of a hollow pipe of infinite extent but of rectangular cross section of dimension α x b. The long direction will be taken to be the z direction.
Unlike in the previous case is not zero in this case. However, as the propagation direction is along the z direction, we have We can write the Maxwell’s curlequations as
The second set of equations are obtained from the Faraday’s law, (these can be written down from above by
As in the case of parallel plate waveguides, we can express all the field quantities in terms of the derivatives of Ez and Hz . For instance, we have
The other components can be similarly written down
As before, we will look into the TE mode in detail. Since Ez = 0, we need to solve for Hz from the Helmholtz equation,
Remember that the complete solution is obtained by multiplying with We solve equation (5) using the technique of separation of variables which we came across earlier. Let
Substituting this in (5) and dividing by XY throughout, we get
where Ky is a constant. We further define
We now have two second order equations,
The solutions of these equations are well known
The boundary conditions that must be satisfied to determine the constants is the vanishing of the tangential component of the electric field on the plates. In this case, we have two pairs of plates. The tangential direction on the plates at x= 0 and x = α is the y direction, so that the y component of the electric field
Likewise, on the plates at y = 0 and y = b,
We need first to evaluate Ex and Ey using equations (1) and (2) and then substitute the boundary conditions. Since Ez = 0, we can write eqn. (1) and (2) as
Since Ey = 0 at x = 0, we must have C2 = 0 and then we get,
Further, since Ex = 0 at y=0, we have C4 = 0. Combining these, we get, on defining a constant C = C1C3
We still have the boundary conditions, Ex= 0 at y = b and Ey = 0 at x = α to be satisfied. The former gives while the latter gives where m and n are integers. Thus we have,
However so that,
For propagation to take place, γ must be imaginary, so that the cutoff frequency below which propagation does not take place is given by
The minimum cutoff is for TE1,0 (or TE0,1 ) mode which are known as dominant mode. For these modes Ex (or Ey ) is zero.
We will not be deriving the equations for the TM modes for which Hz = 0 . In this case, the solution for Ez , becomes,
As the solution is in terms of product of sine functions, neither m nor n can be zero in this case. This is why the lowest TE mode is the dominant mode
For propagating solutions, we have,
We have, Differentiating both sides, we have,
The group velocity of the wave is given by
which is less than the speed of light. The phase velocity, however, is given by
It may be noted that which in vacuum equal the square velocity of light.
For propagating TE mode, we have, from (1) and (4) using Ez = 0
where ηTE is the characteristic impedance for the TE mode. It is seen that the characteristic impedance is resistive. Likewise
We have seen that in the propagating mode, the intrinsic impedance is resistive, indicating that there will be average flow of power. The Poynting vector for TE mode is given by
Substituting the expressions for the field components, the average power flow through the x y plane is given by
Impossibility of TEM mode in Rectangular waveguides
We have seen that in a parallel plate waveguide, a TEM mode for which both the electric and magnetic fields are perpendicular to the direction of propagation, exists. This, however is not true of rectangular wave guide, or for that matter for any hollow conductor wave guide without an inner conductor.
We know that lines of H are closed loops. Since there is no z component of the magnetic field, such loops must lie in the x-y plane. However, a loop in the x-y plane, according to Ampere’s law, implies an axial current. If there is no inner conductor, there cannot be a real current. The only other possibility then is a displacement current. However, an axial displacement current requires an axial component of the electric field, which is zero for the TEM mode. Thus TEM mode cannot exist in a hollow conductor. (for the parallel plate waveguides, this restriction does not apply as the field lines close at infinity.)
In a rectangular waveguide we had a hollow tube with four sides closed and a propagation direction which was infinitely long. We wil now close the third side and consider electromagnetic wave trapped inside a rectangular parallopiped of dimension α × b × d with walls being made of perfector conductor. (The third dimension is taken as d so as not to confuse with the speed of light c).
Resonanting cavities are useful for storing electromagnetic energy just as LC circuit does but the former has an advantage in being less lossy and having a frequency range much higher, above 100 MHz.
The Helmholtz equation for any of the components Eα of the electric field can be written as
We write this in Cartesian and use the technique of separation of variables,
Introducing this into the equation and dividing by , we get,
Since each of the three terms on the right is a function of an independent variable, while the right hand side is a constant, we must have each of the three terms equaling a constant such that the three constants add up to the constant on the right. Let
As each of the equations has a solution in terms of linear combination of sine and cosine functions, we write the solution for the electric field as
Tha tangential component of above must vanish at the metal boundary. This implies,
Let us consider Ex which must be zero at . This is posible if
with Here m and n are non-zero integers.
In a similar way, we have,
with with ℓ being non-zerointeger.
We now use
This relation must be satisfied for all values of x, y, z within the cavity. This requires
Let us choose some special points and try to satisfy this equation. Let x = 0, y, z arbitrary,
This requires Bx = 0. Likewise, takingy = 0, x, z arbitrary, we require Dy = 0 and finally, z = 0,
x, yarbitrary gives Gz = 0.
With these our solutions for the components of the electric field becomes,
where we have redefined our constants Ex0, Ey0 and Ez0.
We also have,
the integers ℓ, m, n cannot be simultaneously zero for then the field will identically vanish. Note further that since must be satisfied everywhere within the cavity, we must have, calculating the divergence explicitly from the above expressions
This requires that is perpendicular to
One can see that the modes within a cavity can exist only with prescribed “resonant” frequency corresponding to the set of integers ℓ, m, n,
Though there is nothing like a propagation direction here, one can take the longest dimension (say d) to be the propagation direction.
We can have modes like corresponding to the set l, m, n for which Ez = 0. For this set
with The magnetic field components can be obtained from the Faraday’s law,
Solutions to Tutorial Assignments
1. The cutoff frequency for (m,n) mode is . Some of the lowest cutoffs are as under (in GHz) :
|m||n||Cutoff frequency (GHz)|
At 1 GHz, no propagation takes place. At 3 GHz only TE10 mode propagates (recall no TM mode is possible when either of the indices is zero.) At 8 GHz, we have TE01, TE02, TE10, TE11, TE12, TE20, TE21, TE30, TE31, TE40, TM11, TM12, TM21 and TM31, i.e. a total of 14 modes propagating.
2. The cutoff frequency is given by
The propagation of the wave is given by the group velocity which is 1.98x 108 and 2.07x108 m/s respectively for f=10 GHz and 12 GHz respectively. Thus the difference in speed is 9x 106 m/s, resulting in a time difference of approximately 10-5 s in travelling 10m.
3. The operating frequency is given by
Substitutingl = 1, m = 0, n = 3, we get d = 2.4 cm
Self Assessment Questions
Identify the propagating mode, determine the frequency of operation and find Hz and Ez .
Solutions to Self Assessment Questions
Thus the mode is TE02 mode. From the relation the operating frequency is 15.74 GHz.
Since kx = 0, Ey = 0 . Further, by definition for TE mode Ez= 0.
Here we have The ratio