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**Schiffman Design Files - Complete**## Schiffman Phase Shifter Conference Paper Writeup

**References**

Schiffman, B. M., "A New Class of Broadband Microwave 90-Degree Phase Shifters," IRE Trans. Microwave Theory Tech., vol. MTT-6, no. 4, pp.232-237.

Jones, E. M. T. and J. T. Bolljahn, "Coupled-Strip-Transmission-Line Filters and Directional Couplers," IRE Trans. Microwave Theory Tech., vol. MTT-4, April 1956, pp.75-81.

**Nomenclature**

*Z-matrix* - an impedance matrix of the ports of a
system, which satisfies the equation V=Z*I.

*Odd mode* - When the current flow through one
conductor is contra to the flow of the current through
the other conductor. Also known as differential mode.

*Even mode* - when the current flow though one
conductor has the same polarity as the current through
the other conductor. Also known as common mode.

*Z _{oo}* - Odd mode impedance of the
system.

*Z _{oe}* - Even mode impedance of the
system.

*Image impedance, Z _{I}* - Total impedance
of the system as seen though one port; a combination of
the even and odd mode impedances.

*Image transfer constant, a+jB* - The complex
ratio of the steady-state apparent power entering and
leaving a network terminated in its image impedance.

**Theory**

The Schiffman phase shifter was introduced by B. M. Schiffman in1958 as a new class of broadband 90-degree phase shifters. In his paper written for the IRE Transactions on Microwave Theory and Techniques, he introduces several classes of phase shifters and the theory behind them. For this project, a coupled line element with connected ends and an error-correcting reference line is used. To derive the theory for this layout, Schiffman referred to a paper written by E. M. T. Jones and J. T. Bolljahn titled "Coupled-Strip-Transmission-Line Filters and Directional Couplers." Jones and Bolljahn developed the phase and impedance equations for coupled lines of several configurations, one of which is coupled lines with one end connected. The derivation of the equations is outlined below.

Starting with two coupled lines suspended midway between two ground planes, as in a stripline configuration, and driven by constant current generators, Figure 1 indicates the direction of the current flow in each line for even and odd mode excitation. As a result, voltage is induced by the current generators for the even and odd modes of operation. For the even mode, the voltage produced on the two conductors is shown by equations (1) and (2), and the odd mode voltage shown in equations (3) and (4).

At each terminal, the voltage is the sum of all the mode voltages. Using Figure 1 as a reference, the voltage at each terminal is presented by equations (5)-(8).

Substituting equations (1)-(4) into equations (5)-(8), where appropriate, the equations for the voltage at each terminal in terms of the even and odd mode impedances and the phase is obtained. The phase length of the coupled lines is represented by "theta=k*l".

At each terminal, the total current from Figure 1 at each terminal is shown in equations (13)-(16). The total current equations can be solved for the individual mode currents and substituted into equations (9)-(12).

So far, the equations presented apply for any generic set of coupled lines. Coupled lines can be modified a number of ways by modifying each terminal with a short or open, or other connection to produce the desired result. For this project, the coupled lines are connected as shown in Figure 2 below.

Using the equations above, a Z-matrix (impedance matrix) can be developed for the coupled lines. A Z-matrix relates the voltage and current of a system, as shown in equation (21).

For the case shown in Figure 2, applying the boundary conditions to the 4x4 impedance matrix will reduce it to a 2x2 matrix, relating port 1 and port 2 of the coupled lines. The boundary conditions for the coupled lines in Figure 2 are:

1. V_{3} = V_{4}

2. I_{3} = -I_{4}

Consider boundary condition 2, substituting into equation
(19) and (20). The result yields that i3=0 and i4 = I4,
and this is substituted into equation (9)-(12). This
yields equations for the voltage in terms of
I_{1}, I_{2}, and I^{4}. Next,
considering boundary condition 1, the voltage equations
V_{3} and V_{4} can be set equal to each
other to solve for the relationship between
I_{4}, I_{2} and and I_{1}. This
relationship is substituted back into the equations for
V_{1} and V_{2}, and results in equations
(22) and (23).

There is no need to solve for V_{3} and
V_{4} in terms of I_{1} and
I_{2}. Since port 3 and port 4 are physically
connected, they are no longer thought of as ports and are
not considered in the resulting Z-matrix. Therefore, the
Z-matrix of the coupled line configuration shown in
Figure 2 is extracted from equations (22) and (23), and
is shown in equation (24).

Jones and Bolljahn also provide equations relating the
image impedance, Z_{I}, and the image transfer
constant, a+jB, to the resulting Z'-matrix. This is also
used by Schiffman in developing the Schiffman phase
shifters. These relationships are shown in equations (25)
and (26).

Because of the symmetry of the Z'-matrix, it can be noted
that Z^{'}_{11} =
Z^{'}_{22}. Taking that into
consideration, equation (25) can be reduced. Substituting
the appropriate elements of the Z'-matrix into the
reduced equation (25) and solving for Z_{I}, the
result is shown in equation (27). This matches the
results obtained by Jones and Bolljahn, as well as
Schiffman.

Solving for the image transfer constant, it is important to note the trigonometric relationship between the hyperbolic cosine and cosine, as well as the symmetry of the impedance matrix. As a result, equation (26) can be re-written. Using other trigonometric identities to solve for the image impedance, the result is shown in equation (28). In this case, the variable B is the only part of the phase constant that needs to be considered.

Equations (27) and (28) are the two equations that result
in Schiffman's theory for a broadband phase shifter.
Specifying the ratio Z_{oe}/Z_{oo} and
the image impedance, each of which can be defined
independently, a network with the desired phase response
can be designed, along with a suitable length of uniform
transmission line as a reference. The uniform
transmission line is used to adjust the overall phase
difference. As mentioned, the Schiffman phase shifter is
a 90deg phase shifter. In order to obtain a different
phase shift, the reference line length is adjusted. For
this project, the Schiffman phase shifter is used as a
difference between the phase of the coupled lines and the
uniform transmission line. When connected in a system,
the result is that the phase of the branch of the Butler
matrix going through the coupled lines will be phase
advanced compared to the branch with the uniform
transmission line.

**Design Methodology**

To design the Schiffman phase shifters, the coupled lines need to be designed first. The width, length, and spacing of the lines will reflect the even and odd mode impedances of the lines, and the product of those impedances must equal the square of the system impedance, as in equation (27). The system impedance for this project is 50ohm. Using Ansoft Designer, the ideal model for coupled lines can be used, with the even and odd mode impedance variables set up to ensure that equation (27) is satisfied. As a result, the coupling factor for the coupled lines will be determined by the even and odd mode impedances. The goal is to optimize for flatness, maintain the relationship in equation (27), and obtain the desired phase difference of 45deg at 11GHz.

It is important to note that there is a distinct difference between the Schiffman phase shifter designed and manufactured, and the re-designed Schiffman phase shifter. The major difference is the approach to the design. The second approach will be discussed thoroughly, since it is the most comprehensive approach to designing a Schiffman Phase Shifter.

Once the ideal even and odd mode impedances are determined, the next step is creating them in Ansoft HFSS. Coupled line impedances translate to a physical width, length, and spacing of the lines. To ensure that the design meets the even and odd mode impedance requirements, the two modes are simulated in HFSS.

Line is suspended in Rogers RO3003 between two ground planes (not shown) in stripline configuration. Since even and odd mode analysis is performed by splitting a circuit down its line of symmetry, only one coupled line is analyzed, making sure that the distance between the line and the boundary is half of the intended spacing. Figure 3 shows the two mode setup, with the transmission line suspended in the dielectric between two ground planes, similar to the diagram shown in Figure 1. The ports of the line are defined, and through simulation, the dimensions of the line are adjusted in order to obtain the theoretical even and odd mode impedances determined earlier, as well as making sure that the phase of each line is approximately 90deg. Most importantly, if a dimension for one mode is changed, the same change must be done for the other mode, and the results analyzed together.

Once the even and odd mode analysis is complete for the single coupled line, both coupled lines are built as mirrors of each other, verifying correct dimensions and spacing. As mentioned, the coupled lines of a Schiffman phase shifter are connected at one end, so that connection and an extension of the ports are created.

The complete Schiffman phase shifter is shown in Figure 4. The top connection can be optimized for width to improve the reflections and transmissions of the phase shifter.

The next step is creating the reference line to go with the phase shifter. Since each coupled line has a phase of 90deg, the total phase of the coupled lines is 180deg. The desired phase shift of the Schiffman relative to the reference line is 45deg. Therefore, the total length of the reference line is 225deg. However, looking at Figure 4, with the addition of the port extensions and the top connection, the length of the reference line needs to be adjusted in order to compensate for these additions. This can also be done through simulation using Ansoft Designer and HFSS. Ansoft Designer can be used to estimate the length of the line to include the extra phase added by the port lines and connection, and then re-simulated in HFSS to verify the dimensions.

Once both are created, they can be simulated together to verify correct operation. The two pieces remain separate since when placed in the system, they are positioned far apart due to the topology of the 4x4 Butler Matrix System A.

**Results**

**Initial Design for Manufacture**

As mentioned earlier, the first design of the Schiffman
phase shifter was approached in a different matter
compared to the re-designed phase shifter. The first
design is not necessarily incorrect since the results
obtained are valid and meet the criteria of the design,
but the approach differed from the conventional approach.
To obtain the even and odd mode impedances that will
yield the desired phase outcome were obtained by starting
with a base model using ideal components, where
Z_{oe}=Z_{oo}=50ohm, with an electrical
length of 90deg for each coupled section. Adjusting the
even and odd mode impedances until a flat phase
difference between 10 and 12GHz is obtained, keeping in
mind that Zoe>Zoo, results in Zoe=66ohm and Zoo=42ohm.
The Schiffman phase shifter is a differential phase
shifter, in that the resulting phase is compared to a
reference line. For a 45deg phase shift, the reference
line is 180deg(total electrical length of the phase
shifter) plus a 45deg extra reference length.

The ideal results are: Over 10-12GHz, the phase difference between the reference line and the Schiffman phase shifter is 45deg flat, with a return loss of -34.88dB at worst over the bandwidth.

The even and odd mode impedance values obtained from the ideal model are applied to a model with includes the 121.5 mil substrate. The phase difference between the Schiffman phase shifter and the reference line are compared between 10GHz and 12GHz. There is a slight change in the return loss, with -31.07dB at worst, and no significant change in the phase difference (flat at 45deg). The Designer model yields widths and lengths for the copper traces that will be applied to the HFSS model.

The Schiffman phase shifter is designed primarily in HFSS. The Designer model gives the length and width of the coupled lines. Port lines and the strip connecting the coupled lines is manually added and designed in HFSS. Chamfering was utilized to reduce reflection. The result was a phase shift centered at 45deg and with a return loss of -15.69dB at worst over the bandwidth. The phase difference changed from the Designer model slightly, and the greatest change was seen in the reflection due to the additional copper.

**Tolerance Analysis**

Though the designed Schiffman phase shifter shows desirable results, there are several factors to consider that may affect its performance during manufacture. Manufacturing of any microwave circuit is subject to machining and materials tolerances. This includes the variations in the machining location of the mode suppressing vias, variation in the thickness of the substrate, and the variation of the dielectric constant of the material. All these things need to be taken into consideration and analyzed, and attempt to predict what may happen once the circuit is manufactured to account for differences. The tolerance of the factors mentioned above was obtained from the manufacturer and suppliers of the material. Each of these cases was examined, and the results summarized and compared in Table 1 below.

Table 1: Tolerance analysis performed to check the effects of manufacturing variations, and compare to the designed results.

It is important to note a few observations when performing the tolerance analysis for the mode suppression vias. For the Schiffman phase shifter, not the reference line, when the via location is 5mils lower in the x-direction and 4mils to the right in the y-direction, the transmission and reflection for the phase shifter becomes erratic. Transmission and reflection, in effect, reverse in characteristics. This seems to be an anomaly in the HFSS solution because the physical location of the holes is not close enough to the copper traces where ground losses would come into play. In simulation, there seems to be a lot of parasitic loss when the vias are at the indicated locations, but it does not seem likely that this is correct. Therefore, in the table above, these outliers in the data have been removed. The tolerance analysis shows the range of data of the analyzed cases. From the data, it seems that the dielectric thickness and mode suppression via location has the greatest affect on phase for the Schiffman phase shifter. Transmission and reflection, although affected, do not show as great of a variation as the phase.

**Re-Design for Improvement**

After the initial design, it was clear that many improvements could be made to the performance of the phase shifter, to level out the phase difference over frequency and to improve the return loss. The re-design was performed from the beginning, using the method outlined in the "Design Methodology" section. The coupled lines were designed to ensure that the square root of their product equaled the system impedance of 50ohm. As a result, when the port lines and the connection between the coupled lines was added, there was improvement in the phase difference, as well as reflections. Initially, the phase difference over frequency was approximately +/- 3deg. Flatness was achieved by varying the size and shape of the connection between the coupled lines, as well as the chamfering of the port lines. Since a complete redesign of the phase shifter was performed, from the beginning, this had to be taken into account in the timeline of the project. Once the even and odd mode analyses were complete, the entire phase shifter could be constructed and adjusted. The initial putting together did not produce desirable results, so adjustments to the connecting strip and the port lines had to be performed. Unfortunately, this is an exhaustingly iterative and time consuming process. After numerous attempts in a short time, the results obtained were not idea, but proved to be an improvement from the original design.

As can be seen in Figure 9 above, the re-design did show some improvement, most of it in transmission and reflection characteristics. The first design had reflection of less than -15.69dB over the operating frequency, while the re-design has less than -29.04dB over the same range. The phase, however, did not see as a significant improvement as was hoped. From Figure 9, it is evident that the phase is still not flat across the band. At 11GHz, the phase is centered at 44.92deg, with +2.29deg at 10GHz and -1.12deg at 12GHz. It seems that this is a reversal of what was seen in the original design, where at lower frequencies the phase difference was smaller than at higher frequencies. Table 2 below shows a comparison between the two designs.

Table 2: Table comparing the results from the original design, and the re-designed Schiffman phase shifter.

As mentioned, the re-design showed the most improvement for transmission and reflection. For the phase, it was difficult to obtain a flatter response. The efforts made to improve the phase difference were successful compared to the phase results when it was first put together for the re-design. However, comparing to the original design, there is no significant improvement. Within the time allotted for the re-design, improvement in phase could not be accomplished. With additional time, it is almost certain that more improvements could be made, though simulation and though building and experimentation.

**Manufactured Phase Shifter**

The Schiffman phase shifter designed first was the model used in the systems for manufacture. A separate component of the phase shifter and the reference line was manufactured as well in order to compare the design to the manufactured outcome. The manufactured phase shifter also incorporated the SMA launch that was designed for this project, as well as for future uses. A modification had to be made post-manufacturing to solder one of the legs of the connector to the metallic surface it was sitting on. This decision was made based on initial data that showed significant loss in the component in transmission and reflection, and most importantly in phase. Two measurements were taken of the component: gated and un-gated. Gating allows the network analyzer to bypass the connector launches and take the data of just the component, much like the simulation. In essence, the start and stop points of the measurement are being set in time to a point beyond the launch. The launch was designed well, but proved to inject a lot of noise into the circuit, as indicated by Figures 10 and 11 below.

Figure 10: Un-gated phase (left) and reflection and transmission (right) of the Schiffman phase shifter and reference line.

Figure 11: Gated phase (left) and reflection and transmission (right) of the Schiffman phase shifter and reference line.

The gated results are the best ones to analyze since they do not include the lossyness and noise introduced by the connectors. Table 3 below shows a summary of the results.

Table 3: Summary and comparison of the simulated and manufactured results for the Schiffman phase shifter, and the re-design.

As can be seen from Table 3 above, the reflection and transmission of the simulated and manufactured circuit matches well, with an average error of 3.27% for reflection. The phase difference, however, shows a significant difference of almost 3deg at each point. The shape of the phase is different as well, comparing the graphs of the measured phase and the simulated phase. From the tolerance analysis in Table 1, it seems that phase is most drastically affected by dielectric thickness and mode suppression via location. The shift upward of 3deg could be attributed to a slightly thicker Rogers dielectric, and even an offset of the holes. Overall, the results show much promise in the design. Even though the phase is not exactly what was simulated, the shape of the phase matches well with the simulated result. The next step would be to analyze the characteristics of the actual model compared to the HFSS result, and try to have the HFSS simulation match the actual component. The variations in the HFSS model to make it look like the manufactured model can be inversely applied to create an offset and a more desired result upon manufacture.