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= Rectangular Waveguides =



Relevant geometry: We consider here a waveguide of width $W$ in the $x$ -direction, height $H$  in the $y-$ direction, and infinite length in the $z-$ direction.

The goal of this section is to determine the plane-wave solutions for the electric and magnetic fields in a rectangular waveguide. These have the form following the derivation developed by Griffiths:

and must satisfy both Maxwell’s Equations

as well as the boundary conditions $E_{||}=0, B_\perp=0$ at walls of the waveguide.

The outline of this derivation is as such:


 * 1) Substitute Eqs.($$),($$) into Eqs.($$),($$) to get 6 equations.
 * 2) Rearrange those equations to express $E_{0,x},E_{0,y},B_{0,x},B_{0,y}$  all as only functions of $E_{0,z},B_{0,z}$.
 * 3) Use the Helmholtz equations

(obtained by taking the divergence of Eqs.($$),($$), applying Eqs.($$),($$), and inserting Eqs.($$),($$)) to solve for either $E_{0,z}$ or $B_{0,z}$ , depending on which mode was selected (the other will be zero). Finally, insert both $E_{0,z},B_{0,z}$ into the expressions for $E_{0,x},E_{0,y},B_{0,x},B_{0,y}$.

Part 1
Substituting Eqs. ($$),($$)) into Eq.($$) gives the three equations

Substituting Eqs.($$),($$)) into Eq.($$) gives the other three equations

Part 2
Solve for $E_{0,x}$ via Eqs.($$),($$)) by eliminating $B_{0,y}$ . Similarly, solve for $E_{0,y}$  with Eqs.($$),($$)), $B_{0,x}$  with Eqs.($$),($$)), and $B_{0,y}$  with Eqs.($$),($$)). Together these give

Part 3
Now that we have $E_{0,x},E_{0,y},B_{0,x},B_{0,y}$ all as only functions of $E_{0,z},B_{0,z}$, we just need to find $E_{0,z},B_{0,z}$  to get everything. We do this by employing the source-free Helmholtz equations Eqs.($$),($$)).

The elements of interest from Eqs.($$),($$)) are

Inserting Eqs.($$),($$)) into Eqs.($$),($$)) then gives

At this point, we decide whether to solve for TE waves ($E_z=0$ ) or TM waves ($B_z=0$ )

TE Waves
For TE waves, $E_z=0$, and so Eq.($$) is trivially satisfied. To solve Eq.($$), we assume separation of variables of the form

Inserting Eq.($$) into Eq.($$) gives

Now, because Eq. ($$) must hold in general, the first two terms - being functions of different variables - must actually be constant. So we define

The solutions to Eqs.($$),($$) are

Now the boundary condition $B_\perp=0$ means that

Looking at Eq.($$), we see that Eq.($$) is only satisfied if $\partial_x B_{0,z} (x=0)=\partial_x B_{0,z} (x=a)=0$. This means that in Eq.($$), the $X$ term must be a $\cos$  in order for its derivative to be zero at the boundaries. The same argument applies for the $Y$ term, and thus in Eqs. ($$),($$) we have $c_1=c_3=0$. Inserting Eqs.($$),($$) into Eq.($$), we then get

Finally, the boundary conditions at $x=W,y=H$ place the following constraints on $k_x,k_y$ :

Now that we have $B_{0,z}$ and $E_{0,z}$, all of the other components of $E,B$  can be found. Of particular note, however, is that there exists a minimum frequency that generates propagating in this waveguide. Inserting Eqs. ($$),($$) into Eq.($$), we get

Inserting Eqs.($$),($$) into Eq.($$) then gives

If $n_x=n_y=0$, then $B_{0,z}$ is constant, and thus Eqs.($$) - ($$) imply that that all components of the electric and magnetic fields - with the exception of $B_z$ , which is constant - are equal to zero. Therefore at least one of $n_x,n_y$ need to be nonzero to exclude this trivial solution. The lowest frequency for a TE mode is then determined by which dimension of the waveguide is smaller. Say $W<H$. Then, $\omega$ achieves is smallest value for $n_x=0, n_y=1$, also referred to as $TE_{10}$. Substituting these values into Eq.($$), we get

The smallest value that $\omega$ can then take to keep $k_z$  real is then

yielding a “cutoff frequency” of

Note that the TM modes are found with a similar treatment.

= Rectangular Waveguides with Dielectrics =

Inserting a homogeneous dielectric of permittivity and permeability $\epsilon,\mu$ in a rectangular waveguide merely changes the speed of light to the more general

which causes the cutoff frequency to become

Therefore, in general, inserting a dielectric into a waveguide lowers its cutoff frequency.

= Coplanar Waveguides =

Coplanar waveguides (CPW), defined by a center conductor with two neighboring ground planes deposited on a dielectric substrate, act as transmission lines at radio frequencies (RF), e.g. they support transverse electromagnetic (TEM) modes (Wen 1969). We will review the conventional approach to deriving approximate expressions for the characteristic impedance $Z_0$ CPWs, given a design frequency and a substrate with an isotropic relative dielectric constant $\epsilon_r$.

Conformal Mapping – Semi-infinite Substrate
Following Wen (1969), we use a lowest order quasi-static approximation, assuming TEM mode wave propagation. Further, we take the conductors to be of zero thickness, substrate to be semi-infinite in the extent, and the ground planes to be semi-infinite in width. This allows us to employ conformal mapping methods outlined in (Smythe 1950) to solve for a capacitance to compute the characteristic impedance $(Z_0)$ for a lossless CPW.

We consider two complex planes $Z, Z_1$, and define $Z$ as a function of $Z_1$  such that at least one point in the $Z$ -plane corresponds to one point in the $Z_1$  plane. The function $Z = f(Z_1)$ is an analytic function of $Z_1$, where the modulus of quotient of the differentials $dZ, dZ_1$  is given by (Smythe 1950)

$$   \left|\frac{dZ}{dZ_1}\right| = \frac{\left|dZ\right|}{\left|dZ_1\right|}  = \frac{dS}{dS_1} = h$$

where$ S, S_1$ refer to differential lengths in the $Z, Z_1$ -planes. Eq. implies that Z and $Z_1$ are related by a constant scaling factor $h$  given by the map f(Z_1). If we consider a triangle with infinitesimal lengths$ dS_1^1, dS_1^2, dS_1^3 $ in the Z_1-plane and $ dS_2^1, dS_2^2, dS_2^3 $ in the $ Z_2 $ -plane, as in Fig.~\ref{fig:conformal_mapping_triangle}, we see that the mapping between $ Z_1 $  and $ Z_2 $  is conformal, preserving the angles  $\left(\phi_1^{mn}, \phi_2^{mn}\right) $, between curves in the two planes. This statement is analogous to Eq. , where the differential lengths are preserved under a conformal mapping.





We will apply such a mapping to compute the characteristic impedance of a CPW by mapping the upper half plane bounded below by the center conductor and ground planes, with a semi-infinite section of dielectric as in Fig.~\ref{fig:conformal_mapping}. $$\begin{aligned} \frac{dZ}{dZ_1} = \frac{A}{\left[\left(Z_1^2 - b_1^2\right)\left(Z_1^2 - a_1^2\right)\right]^{1/2}} \end{aligned}$$

where $ A $ is constant chosen to be 1 in this case. We multiply and integrate both sides of Eq. by $ dZ_1 $ to find an explicit solution to $ Z = a+ib $.

$$\begin{aligned} Z = a + ib = \int_0^{b_1}\frac{dZ_1}{\left[\left(Z_1^2 - b_1^2\right)\left(Z_1^2 - a_1^2\right)\right]^{1/2}} = \int_0^{b_1}\frac{dZ_1}{Z_1^2}\frac{1}{\left[\left(1 - b_1^2/Z_1^2\right)\left(1 - (a_1^2/b_1^2)b_1^2/Z_1^2\right)\right]^{1/2}} \end{aligned}$$

Changing integration variables from $ Z_1 $ to $ t=b_1/Z_1 $  in Eq.~\ref{eq:intZ1} yields the complete elliptic integral of the first kind $ K(k) $.

$$\begin{aligned} a + ib = \int_{0}^{1}\frac{dt}{\left[\left(1 - t^2\right)\left(1-k^2t^2\right)\right]^{1/2}} = K(k), \ \ \ \ k = a_1 / b_1 \end{aligned}$$

From Eq., and its complement, $ K'(k) = K(k') $, where $ k'=\sqrt{1 - k^2} $ we obtain the ratio $ a / b $  in terms of elliptic integrals of $ a_1 $  and $ b_1 $. $$	\frac{a}{b} = \frac{K(k)}{K'(k)} $$ In the $ Z $ -plane, under the quasi-static approximation, we have an electric field $ E $ across the top and bottom conductors with a capacitance per unit length $ C $  given in terms of the ratio in Eq. ~\cite{Wen1969} $$   C = 2\frac{K(k)}{K(k')}\epsilon_0\left(\epsilon_r + 1\right) = 4\epsilon_0\epsilon_{\mathrm{eff}}\frac{K(k)}{K'(k)}, $$

where $\epsilon_0 $  is the electric permittivity of free space and  $\epsilon_{\mathrm{eff}} $  is the effective dielectric constant defined as

$$   \epsilon_{\mathrm{eff}}= \frac{\epsilon_r + 1}{2} $$

The phase velocity, derived under the same approximation, is given in terms of the speed of light in vacuum, $ c $

$$   v_{ph} = c / \sqrt{\epsilon_{\mathrm{eff}}} $$

The characteristic impedance of the CPW, $ Z_0 $, then follows from the capacitance per unit length and the phase velocity.

$$   Z_0 = \frac{1}{Cv_{ph}} $$

Upon substitution of Eq. and into, we find

$$   Z_0 = \frac{1}{4\epsilon_0\epsilon_{\mathrm{eff}}}\frac{K(k')}{K(k)}\frac{\sqrt{\epsilon_{\mathrm{eff}}}}{c} = \frac{1}{\sqrt{\epsilon_{\mathrm{eff}}}}\sqrt{\frac{\mu_0}{\epsilon_0}}\frac{K(k')}{K(k)}\simeq\frac{30\pi}{\sqrt{\epsilon_{\mathrm{eff}}}}\frac{K(k')}{K(k)} $$

We have used $ c=1/\sqrt{\mu_0\epsilon_0} $ and approximated the impedance of free space  $\eta_0=\sqrt{\mu_0 / \epsilon_0}\simeq 120\pi\ \Omega\simeq 377\ \Omega $.

Conformal Mapping – Finite Substrate
For the finite thickness case, summarize results of Simons (2004) where he considers a multilayer CPW and treats the special case of a single layer CPW with a substrate of finite thickness $h_1$ and effective dielectric constant $\left(\epsilon_{r} - 1\right)$. The conformal mapping of the $Z_1$ -plane into the $Z$ -plane involves an intermediate step, where we map the domains in $Z_1$ to the $Z_2$  plane by function $t = f(Z_1)$  as



[fig:conformal_mapping_finite]

$$   t = \cosh^2\left(\frac{\pi Z_1}{2h_1}\right) $$ Substituting the points from the $ Z_1 $ -plane into Eq. , we have the following expressions for $ t_1, t_2, t_3 $.

$$   t_1 = 1, \ \ \ \ t_2 = \cosh^2\left(\frac{\pi a_1}{2h_1}\right), \ \ \ \ t_3 = \cosh^2\left(\frac{\pi b_1}{2h_1}\right) $$

These points correspond to 1,2,3 in Fig.~\ref{fig:conformal_mapping_finite}, with 4 $\to\infty $. Mapping the $ t_j $ points in Eq.\ref{eq:t_transforms} to the $ Z $ -plane follows from the Christoffel-Schwarz transformation~\cite{Gevorgian1995}

$$   Z = \frac{2A_1}{\sqrt{t_3 - t_1}} F(\varphi, k) + A_2 $$

where $ F(\phi, k) $ is the elliptic integral of the first kind, defined by

$$   F(\varphi, k) = \int_0^{\varphi}\frac{d\theta}{\sqrt{1 - k^2\sin^2\theta}} $$

Note, that Eq. reduces to the complete elliptic integral of the first kind, $ K(k) $ for  $\varphi=\pi/2 $. As with the infinite substrate thickness case, we will use $ K(k) $, but we mention $ F(\varphi,k) $ for completeness with respect to the Christoffel-Schwarz transformation defining the mapping of the upper half plane of $ Z_2 $  to $ Z $. The capacitance due to the dielectric follows from a similar argument as Eq. ~\cite{Simons2004}.

$$   C_1 = 2\epsilon_0\left(\epsilon_r - 1\right)\frac{K(k_1)}{K(k_1')} $$ $$   k_1 = \frac{\sinh\left(\pi a_1/2h_1\right)}{\sinh\left(\pi b_1/2h_1\right)}, k_1' = \sqrt{1 - k_1^2} $$

There is an additional contribution to the total capacitance, $ C_{\mathrm{CPW}} $ due to the the air-conductor interface. This capacitance $ C_{\mathrm{air}} $ is given by a similar expression as Eq.~\ref{eq:C1_finite}~\cite{Simons2004, Gevorgian1995}.

$$   C_{\mathrm{air}} = 4\epsilon_0\epsilon_r\frac{K(k_0)}{K(k_0')}, $$ $$   k_0 = a_1 / b_1, \ \ \ \ k_0'  = \sqrt{1 - k_0^2} $$

The total capacitance, $ C_{\mathrm{CPW}} $ is just the sum of $ C_{\mathrm{air}} $  and $ C_1 $

$$   C_{\mathrm{CPW}} = C_1 + C_{\mathrm{air}} =  2\epsilon_0\left(\epsilon_r - 1\right)\frac{K(k_1)}{K(k_1')} + 4\epsilon_0\epsilon_r\frac{K(k_0)}{K(k_0')} $$

The effective dielectric constant $\epsilon_{\mathrm{eff}} $  for the finite thickness substrate CPW is given by the ratio of the total capacitance relative to the partial capacitance due to the air from~\cite{Simons2004}.

$$   \epsilon_{\mathrm{eff}}= \frac{C_{\mathrm{CPW}}}{C_{\mathrm{air}}} = \epsilon_0\left(\epsilon_r -1\right)\frac{K(k_1)}{K(k_1')}\frac{K(k_0')}{K(k_0)} + 1 $$

Characterisitic impedance of the finite thickness CPW then follows from Eq. , substituting $ C_{\mathrm{CPW}} $ for $ C $  and Eq. with $\epsilon_{\mathrm{eff}} $  in Eq. .

$$   Z_0 = \frac{1}{C_{\mathrm{CPW}}v_{ph}} = \frac{1}{cC_{\mathrm{CPW}}}\sqrt{\frac{C_{\mathrm{CPW}}}{C_{\mathrm{air}}}} = \frac{C_{\mathrm{air}}}{cC_{\mathrm{CPW}}}\sqrt{\frac{C_{\mathrm{CPW}}}{C_{\mathrm{air}}}} = \frac{1}{cC_{\mathrm{air}}}\sqrt{\frac{C_{\mathrm{air}}}{C_{\mathrm{CPW}}}} = \frac{1}{cC_{\mathrm{air}}\sqrt{\epsilon_{\mathrm{eff}}}} $$ $$   =\frac{1}{4\epsilon_0 c}\frac{K(k_0')}{K(k_0)}\frac{1}{\sqrt{\epsilon_{\mathrm{eff}}}} \simeq \frac{30\pi}{\sqrt{\epsilon_{\mathrm{eff}}}}\frac{K(k_0')}{K(k_0)} $$

From Eq. we arrive at expression of the same form as Eq. under the same approximation of the characteristic impedance of free space as before with a modified effective dielectric constant as in Eq. .

TEM Waves for Coplanar Waveguides
The electric and magnetic fields in a CPW are calculated in the even and odd modes of excitation akin to the analysis applied to coupled lines in coupled line filter design (Pozar 2011). In the odd mode a magnetic wall is placed at $y=0$, at half the distance from the left edge of the center conductor in Fig. [fig:even_odd_mode_bcs], and at $y=b$, the extent of the right ground plane in Fig. [fig:cpw_even_odd_cross_sec]. By inserting two electric walls a distance $\lambda_{odd}'/2$, half the wavelength of the slot-line mode, the problem reduces to an equivalent rectangular waveguide problem for half of the device (Simons and Arora 1982). Similarly for the even mode, we have the same boundary conditions except that the magnetic wall at $y=0$ is replaced by an electric wall and the distance between the other two electric walls becomes $\lambda_{even}'/2$.







= References =

Collin, R.E. 1965. Foundations for Microwave Engineering. Electrical Engineering Series. McGraw-Hill.

Gevorgian, S., L. J. P. Linner, and E. L. Kollberg. 1995. “CAD models for shielded multilayered CPW.” IEEE Transactions on Microwave Theory and Techniques 43 (4): 772–79. doi:10.1109/22.375223.

Pozar, D.M. 2011. “Microwave Engineering.” In, 426–36.

Simons, R. N., and R. K. Arora. 1982. “Coupled Slot Line Field Components.” IEEE Transactions on Microwave Theory and Techniques 30 (7): 1094–9. doi:10.1109/TMTT.1982.1131202.

Simons, R.N. 2004. Coplanar Waveguide Circuits, Components, and Systems. Wiley Series in Microwave and Optical Engineering. Wiley. https://books.google.com/books?id=XgNMZ5YVJdgC.

Smythe, William Ralph. 1950. Static and Dynamic Electricity. 2nd ed. McGraw Hill Book Company, Inc.

Wen, C. P. 1969. “Coplanar Waveguide: A Surface Strip Transmission Line Suitable for Nonreciprocal Gyromagnetic Device Applications.” IEEE Transactions on Microwave Theory and Techniques 17 (12): 1087–90. doi:10.1109/TMTT.1969.1127105.