Dielectric Phase Shifters
Dielectric Phase Shifters: Construction and Working Principle
A dielectric phase shifter can be realized by placing a lossless dielectric slab inside a waveguide parallel to the direction of propagation and at the position of maximum electric field. The dielectric slab changes the velocity of the microwave signal compared with its velocity through an empty waveguide. This difference in wave velocity produces a differential phase change, which is the basic operating principle of the dielectric phase shifter.
Construction of a Dielectric Phase Shifter
A dielectric phase shifter consists of a waveguide containing a lossless dielectric slab. The slab is placed inside the waveguide parallel to the direction of propagation of the electromagnetic wave. Its position is selected according to the electric field distribution of the dominant mode so that the dielectric interacts effectively with the microwave signal.
The dielectric slab occupies a certain length of the waveguide and changes the propagation characteristics of the electromagnetic wave within that region. The wave therefore travels through two different propagation environments: the dielectric loaded section and the corresponding empty waveguide section. The difference between the phase accumulated in these two sections produces the required differential phase shift.
Placement of the Dielectric Slab
The dielectric slab is placed at the position of maximum electric field inside the waveguide. This arrangement provides effective interaction between the dielectric material and the electromagnetic field. Since the dielectric primarily affects the electric field distribution, placing the slab at the maximum electric field region produces a significant change in the propagation characteristics of the wave.
For the dominant TE10 mode of a rectangular waveguide, the electric field has a definite spatial distribution across the guide. The dielectric slab is positioned according to this field distribution so that the phase shifting effect is obtained efficiently.
Effect of the Dielectric Slab on Wave Velocity
When the dielectric slab is introduced into the waveguide, the velocity of the microwave signal through the dielectric loaded section becomes different from the velocity through an empty waveguide. The dielectric changes the electromagnetic properties of the medium and consequently changes the propagation characteristics of the wave.
The differential phase change is produced because the same physical length of waveguide now produces a different phase accumulation. The wave travelling through the dielectric slab experiences a different phase progression compared with a wave travelling through an empty waveguide. This difference in phase progression is used to obtain the required phase shift.
Comparison with an Empty Waveguide
Consider an empty waveguide and a waveguide section containing a dielectric slab of the same physical length. In the empty waveguide, the propagation characteristics are determined by the waveguide dimensions and the operating wavelength. When the dielectric slab is introduced, the propagation velocity and guide wavelength are changed.
Therefore, the phase accumulated by the wave in the dielectric loaded section is different from the phase accumulated in the corresponding empty guide. The difference between these two phase changes is known as the differential phase shift.
Tapered Ends of the Dielectric Slab
The dielectric slab is tapered at both ends to reduce reflections of the microwave wave from the dielectric slab. If the dielectric material were introduced abruptly into the waveguide, the sudden change in propagation conditions could produce significant reflections. These reflections would affect the matching and transmission characteristics of the phase shifter.
The tapered ends provide a gradual transition between the empty waveguide and the dielectric loaded section. This gradual transition reduces the discontinuity experienced by the electromagnetic wave and consequently reduces the reflected wave.
Matching of the Two Ports
The two ports of the dielectric phase shifter are designed to be matched. The reduction of reflections from the dielectric slab helps maintain good matching at the input and output ports. When the ports are properly matched, most of the incident microwave power is transmitted through the phase shifter while the primary effect of the dielectric slab is to introduce the required phase change.
Thus, the dielectric phase shifter can provide phase control without introducing significant reflections. The tapered dielectric structure is an important part of achieving this matched operation.
Basic Operating Principle
The operation of the dielectric phase shifter is based on the difference between the propagation velocity through the dielectric loaded section and the propagation velocity through an empty waveguide. A microwave signal entering the phase shifter propagates through the waveguide in the dominant TE10 mode. When the signal reaches the dielectric loaded region, the dielectric changes the propagation characteristics of the wave.
As a result, the phase accumulated over the dielectric loaded length is different from the phase that would have been accumulated over the same length of an empty waveguide. The difference between these two phase changes produces the differential phase shift. By adjusting the length of the dielectric section, different values of phase shift can be obtained.
Dominant TE10 Mode
The dielectric phase shifter considered here operates with the dominant TE10 mode of the rectangular waveguide. The TE10 mode determines the electric field distribution inside the waveguide and therefore determines the appropriate position for the dielectric slab.
The dielectric slab is placed parallel to the direction of propagation and at the position of maximum electric field. This allows the slab to produce the required change in propagation velocity and consequently generate the differential phase shift.
Propagation Through the Dielectric Loaded Section
Let the propagation constant through a length l of the dielectric slab be represented by β1. For the dielectric loaded waveguide operating in the dominant TE10 mode, the propagation constant depends on the dielectric constant, the guide dimensions, and the operating wavelength.
The corresponding propagation constant for the same length of an empty waveguide is represented by β0. Since the dielectric changes the propagation characteristics of the waveguide, in general β1 is different from β0.
Differential Phase Shift
The phase accumulated through the dielectric loaded section is different from that accumulated through the corresponding empty guide. Therefore, the differential phase shift produced by the dielectric phase shifter is given by
\[ \Delta\phi=(\beta_1-\beta_0)l \]
where Δφ is the differential phase shift, β1 is the propagation constant through the dielectric loaded section, β0 is the propagation constant through the empty waveguide, and l is the length of the dielectric section.
This equation shows that the phase shift depends directly on the difference between the two propagation constants and on the length of the dielectric section. Therefore, by adjusting the length l, different phase shifts can be produced.
Propagation Constants and Differential Phase Shift
The phase shift produced by a dielectric phase shifter can be analyzed by comparing the propagation of the electromagnetic wave through a dielectric loaded waveguide with its propagation through an equivalent empty waveguide. Since the dielectric changes the velocity and wavelength of the wave, the propagation constant in the dielectric loaded section is different from that of the empty guide. The difference between the phases accumulated in the two sections produces the required differential phase shift.
Propagation Constant Through the Dielectric Loaded Section
Consider a dielectric slab of length l placed inside a rectangular waveguide and operating in the dominant TE10 mode. Let the propagation constant through the dielectric loaded section be represented by β1. The propagation constant determines the rate at which the phase of the electromagnetic wave changes as it travels through the dielectric loaded waveguide.
For a dielectric material having relative dielectric constant εr, the propagation characteristics of the waveguide are modified because the dielectric changes the effective wavelength and phase velocity. For the dominant TE10 mode, the propagation constant through the dielectric loaded section can be expressed in terms of the wavelength and the dielectric properties of the medium.
Definition of β1
The propagation constant through the dielectric loaded section is given by
\[ \beta_1=\frac{2\pi}{\lambda_{g1}} \]
where λg1 is the guide wavelength in the dielectric loaded section. The subscript 1 indicates that the wave is propagating through the section containing the dielectric slab.
For the dominant TE10 mode, the guide wavelength is determined by both the dielectric constant and the dimensions of the waveguide. Therefore, inserting the dielectric slab changes λg1 and consequently changes β1.
Guide Wavelength in the Dielectric Loaded Section
For a rectangular waveguide containing a non magnetic dielectric material, the guide wavelength in the dielectric loaded section can be written as
\[ \lambda_{g1}=\frac{\lambda_0}{\sqrt{\varepsilon_r-\left(\frac{\lambda_0}{2a}\right)^2}} \]
where λ0 is the free space wavelength, εr is the relative dielectric constant of the dielectric material, and a is the broad dimension of the rectangular waveguide.
The presence of εr in this expression shows that the dielectric material directly affects the guide wavelength. Since the propagation constant is related to the guide wavelength by β1 = 2π/λg1, a change in the dielectric constant produces a corresponding change in the propagation constant.
Propagation Constant Through the Empty Waveguide
To determine the phase shift produced by the dielectric phase shifter, the propagation through the dielectric loaded section must be compared with propagation through an equivalent length of an empty waveguide. Let the propagation constant of the empty waveguide be represented by β0.
The empty waveguide does not contain the dielectric material, so its propagation characteristics are determined only by the waveguide dimensions and the free space wavelength. For the dominant TE10 mode, its propagation constant is related to the guide wavelength of the empty waveguide.
Definition of β0
The propagation constant through the empty waveguide is given by
\[ \beta_0=\frac{2\pi}{\lambda_{g0}} \]
where λg0 is the guide wavelength of the empty waveguide. The subscript 0 indicates the unloaded or empty waveguide condition.
Guide Wavelength in the Empty Waveguide
For the dominant TE10 mode of an empty rectangular waveguide, the guide wavelength is given by
\[ \lambda_{g0}=\frac{\lambda_0}{\sqrt{1-\left(\frac{\lambda_0}{2a}\right)^2}} \]
Here, λ0 is the free space wavelength and a is the broad dimension of the rectangular waveguide. Since the empty guide has no dielectric loading, the relative dielectric constant is effectively unity.
Using this guide wavelength, the propagation constant of the empty guide becomes
\[ \beta_0=\frac{2\pi}{\lambda_{g0}} \]
Thus, β0 represents the normal phase progression of the electromagnetic wave through the empty waveguide.
TE10 Mode Propagation
The dielectric phase shifter is analyzed using the dominant TE10 mode. The propagation constant for this mode depends upon the free space wavelength, waveguide dimensions, and the dielectric medium present inside the guide.
When the waveguide is empty, the wave propagates with propagation constant β0. When the dielectric slab is introduced, the propagation constant changes to β1. Therefore, the same physical length of waveguide produces two different phase changes under the two propagation conditions.
Derivation of β1
The propagation constant is related to the guide wavelength by
\[ \beta_1=\frac{2\pi}{\lambda_{g1}} \]
Substituting the guide wavelength of the dielectric loaded waveguide gives
\[ \beta_1=\frac{2\pi}{\lambda_0}\sqrt{\varepsilon_r-\left(\frac{\lambda_0}{2a}\right)^2} \]
This expression shows that the propagation constant in the dielectric loaded section depends on the relative dielectric constant εr, the free space wavelength λ0, and the broad dimension a of the waveguide.
Derivation of β0
For the empty waveguide, the propagation constant is
\[ \beta_0=\frac{2\pi}{\lambda_{g0}} \]
Using the guide wavelength of the empty waveguide,
\[ \beta_0=\frac{2\pi}{\lambda_0}\sqrt{1-\left(\frac{\lambda_0}{2a}\right)^2} \]
This represents the phase progression of the dominant TE10 mode through the empty waveguide.
Comparison of the Two Propagation Constants
The two propagation constants are different because the dielectric slab changes the propagation characteristics of the waveguide. For the dielectric loaded section, the propagation constant is β1, while for the empty waveguide it is β0.
Therefore,
\[ \beta_1 \neq \beta_0 \]
in general. The difference between these two propagation constants determines how much additional phase is accumulated by the wave while travelling through the dielectric loaded section.
The dielectric slab therefore produces a change in the electrical length of the waveguide. Although the physical length l of the section remains unchanged, the phase progression through that section is different because the propagation constant has changed.
Phase Accumulated Through the Dielectric Section
Consider a dielectric slab having length l. When the microwave signal propagates through this section, the phase accumulated by the wave is determined by the propagation constant β1. Therefore, the phase accumulated through the dielectric loaded section is
\[ \phi_1=\beta_1 l \]
This represents the total phase progression experienced by the wave while travelling through the dielectric loaded length.
Phase Accumulated Through the Corresponding Empty Section
Now consider the same physical length l without the dielectric slab. The wave propagates through the empty waveguide with propagation constant β0. Therefore, the phase accumulated through the corresponding empty section is
\[ \phi_0=\beta_0 l \]
The difference between φ1 and φ0 represents the additional phase introduced by the dielectric loaded section.
Derivation of Differential Phase Shift
The differential phase shift is obtained by subtracting the phase accumulated through the empty waveguide from the phase accumulated through the dielectric loaded section. Therefore,
\[ \Delta\phi=\phi_1-\phi_0 \]
Substituting
\[ \phi_1=\beta_1l \]
and
\[ \phi_0=\beta_0l \]
gives
\[ \Delta\phi=\beta_1l-\beta_0l \]
Taking l as common,
\[ \boxed{\Delta\phi=(\beta_1-\beta_0)l} \]
This is the main expression for the differential phase shift produced by the dielectric phase shifter. It shows that the phase shift is determined by the difference between the propagation constants of the dielectric loaded and empty waveguides and by the length of the dielectric section.
Effect of Changing the Length of the Dielectric Section
The differential phase shift is directly proportional to the length l of the dielectric section. From
\[ \Delta\phi=(\beta_1-\beta_0)l \]
it can be seen that increasing the length of the dielectric slab increases the accumulated differential phase shift, provided that the propagation constants remain unchanged. Similarly, reducing the length of the dielectric section reduces the phase shift.
Therefore, by adjusting the length l, different phase shifts can be produced. This is the basic method by which the dielectric phase shifter can be designed to provide the required phase difference.
Relationship Between Dielectric Constant and Phase Shift
The relative dielectric constant εr affects the propagation constant of the dielectric loaded section. Since β1 depends on εr, changing the dielectric constant changes the difference between β1 and β0. Consequently, the differential phase shift also changes.
A dielectric material with different electromagnetic properties therefore produces a different phase response. This relationship is important when selecting the dielectric material for a phase shifter because the dielectric constant determines how strongly the material modifies the propagation characteristics of the waveguide.
Relationship Between Wavelength and Phase Shift
The operating wavelength also affects the phase shift because both propagation constants depend on the free space wavelength. A change in frequency produces a corresponding change in λ0, which changes the guide wavelengths and propagation constants of both the dielectric loaded and empty waveguide sections.
Therefore, the differential phase shift is frequency dependent. The design of a practical dielectric phase shifter must take this relationship into account when determining the desired operating frequency range and phase shift performance.
Relationship Between Waveguide Dimensions and Phase Shift
The broad dimension a of the rectangular waveguide appears in the expressions for both propagation constants. Therefore, the waveguide dimensions influence the guide wavelength and consequently affect the phase shift produced by the dielectric section.
The phase behavior of the dielectric phase shifter is therefore determined by the combined effect of the dielectric constant, operating wavelength, waveguide dimensions, and dielectric section length. These parameters determine the difference between β1 and β0 and hence determine the final differential phase shift.
Differential Phase Shift
The dielectric phase shifter works by creating two different propagation conditions for the microwave signal. The dielectric loaded section has propagation constant β1, while the corresponding empty waveguide has propagation constant β0. Over a length l, the two sections accumulate phases β1l and β0l respectively. Their difference produces the differential phase shift.
Thus, the fundamental result is
\[ \boxed{\Delta\phi=(\beta_1-\beta_0)l} \]
This equation provides the mathematical basis for controlling the phase shift of the dielectric phase shifter. By controlling the dielectric properties, waveguide dimensions, operating wavelength, or length of the dielectric section, the required phase shift can be obtained.
Derivation of the S Matrix of an Ideal Dielectric Phase Shifter
The dielectric phase shifter can be represented as an ideal two port microwave network. Since the device is designed to introduce a controlled phase change while maintaining the magnitude of the transmitted signal, its scattering matrix can be obtained by applying the properties of matching, losslessness, and reciprocity. The differential phase shift obtained from the dielectric loaded waveguide determines the phase term appearing in the transmission coefficients.
Two Port Representation of the Ideal Phase Shifter
Consider a two port dielectric phase shifter with Port 1 as the input port and Port 2 as the output port. Let a1 and a2 represent the incident waves at Ports 1 and 2, while b1 and b2 represent the corresponding reflected and transmitted waves. The general scattering relationship for a two port network is
\[ \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \]
Therefore, the general scattering matrix of the phase shifter is
\[ [S]= \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix} \]
At this stage, the four scattering parameters are unknown. The properties of an ideal dielectric phase shifter are now applied to determine each element of the matrix.
Applying the Matched Port Condition
The dielectric phase shifter is designed with both ports matched. This means that when a wave is incident on either port while the other port is terminated in a matched load, there is no reflected wave at the excited port.
For Port 1 to be matched, the reflection coefficient must be zero. In scattering parameter form, this gives
\[ S_{11}=0 \]
Similarly, for Port 2 to be matched,
\[ S_{22}=0 \]
Therefore, the scattering matrix becomes
\[ [S]= \begin{bmatrix} 0 & S_{12}\\ S_{21} & 0 \end{bmatrix} \]
The diagonal elements disappear because there is no reflection at either port. The remaining terms, S12 and S21, describe transmission between the two ports.
Applying the Lossless Condition
An ideal dielectric phase shifter is considered lossless. This means that no microwave power is absorbed inside the device. The power entering the phase shifter is therefore completely transmitted to the other port when the corresponding port is properly terminated.
For a lossless network, the scattering matrix is unitary. Therefore,
\[ [S]^\dagger[S]=[I] \]
Using the matrix obtained from the matched condition,
\[ [S]= \begin{bmatrix} 0 & S_{12}\\ S_{21} & 0 \end{bmatrix} \]
The first row of the unitary condition gives
\[ |S_{12}|^2=1 \]
and the second row gives
\[ |S_{21}|^2=1 \]
Therefore,
\[ |S_{12}|=|S_{21}|=1 \]
This means that the transmission coefficients have unit magnitude. Hence, an ideal lossless phase shifter does not reduce the magnitude of the transmitted wave. Its function is to change the phase of the wave.
Applying Reciprocity
A dielectric phase shifter made using a passive dielectric material is a reciprocal network. For a reciprocal microwave network, the transmission coefficients in the two directions are equal. Therefore,
\[ S_{12}=S_{21} \]
Let the common transmission coefficient be represented by S. The scattering matrix can now be written as
\[ [S]= \begin{bmatrix} 0 & S\\ S & 0 \end{bmatrix} \]
From the lossless condition, the magnitude of this transmission coefficient must be unity. Therefore, it can be represented in exponential form as
\[ S=e^{-j\theta} \]
where θ represents the phase introduced during transmission through the phase shifter.
Determining the Transmission Phase
The remaining task is to determine the phase associated with the transmission coefficient. From the previous analysis of the dielectric phase shifter, the differential phase shift produced by the dielectric loaded section is
\[ \Delta\phi=(\beta_1-\beta_0)l \]
This differential phase shift represents the additional phase change introduced by the dielectric section relative to the corresponding empty waveguide section. Therefore, the transmission coefficient must contain this phase shift.
Since the magnitude of the transmission coefficient is unity, the coefficient can be represented as a complex exponential. For a phase delay of Δφ, the transmission coefficient is written as
\[ S=e^{-j\Delta\phi} \]
The negative sign in the exponent represents the phase delay introduced by propagation through the phase shifter. Thus,
\[ S_{12}=e^{-j\Delta\phi} \]
and, because the network is reciprocal,
\[ S_{21}=e^{-j\Delta\phi} \]
Why the Transmission Coefficient is Written as e−jΔφ
The exponential form is used because a microwave signal can be represented as a complex phasor. A complex transmission coefficient having unit magnitude and phase angle −Δφ is written as
\[ e^{-j\Delta\phi} \]
Using Euler's relationship, this can also be expressed as
\[ e^{-j\Delta\phi}=\cos(\Delta\phi)-j\sin(\Delta\phi) \]
The magnitude of this quantity is always unity, while its phase angle is −Δφ. Therefore, this representation clearly shows that the ideal phase shifter changes the phase of the transmitted signal without changing its magnitude.
Complete S Matrix of the Ideal Dielectric Phase Shifter
We have now determined every element of the scattering matrix. The matched condition gives
\[ S_{11}=S_{22}=0 \]
The lossless condition gives
\[ |S_{12}|=|S_{21}|=1 \]
The reciprocal property gives
\[ S_{12}=S_{21} \]
Finally, the differential phase shift determines the phase of the transmission coefficient as
\[ S_{12}=S_{21}=e^{-j\Delta\phi} \]
Substituting all these results into the general two port scattering matrix gives
\[ \boxed{ [S]= \begin{bmatrix} 0 & e^{-j\Delta\phi}\\ e^{-j\Delta\phi} & 0 \end{bmatrix} } \]
Interpretation of the S Matrix
The final S matrix describes the ideal behavior of the dielectric phase shifter. The zero diagonal elements indicate that both ports are matched and therefore no power is reflected from either port. The off diagonal elements have unit magnitude, indicating that the device is lossless and transmits the signal without attenuation. Their common phase term, e−jΔφ, represents the differential phase shift introduced by the dielectric phase shifter.
Thus, the matrix shows that the ideal dielectric phase shifter performs one primary operation: it transfers the microwave signal from one port to the other while introducing the required phase change. The magnitude of the transmitted signal remains unchanged under the ideal lossless assumption, while its phase is shifted by the differential phase shift determined from the propagation constants of the dielectric loaded and empty waveguide sections.