laboratory experiment 3: standing waves and waveguides

14
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE320H1F / ECE357H1F EXPERIMENT 3: STANDING WAVES AND WAVEGUIDES 1. Introduction The purpose of this laboratory is to introduce you to a few types of planar transmission lines and waveguides and to illustrate how practical measurements of the fields produced by these transmission lines can be carried out. Using knowledge of transmission lines, you will physically measure the standing wave pattern produced by a terminated transmission line and waveguide. The lab will also introduce wave propagation characteristics within waveguides, which are discussed later in the course. You will need the following equipment and components to complete this lab: Agilent vector network analyzer Calibration kit (short, open, load, through) 2 SMA cables Lab 3 linear translator station Ruler Computer running MATLAB or Microsoft Excel, for recording data One unknown load The main component in this lab is a linear translator, which is shown in Figure 1. It consists of a moveable stage, which is suspended on three rods above a printed circuit board (PCB). The stage is moved back and forth using a stepper motor, with controls situated on a small board at one end of the positioner. On the PCB, there are two circuits. The first is a microstrip line[1], and the second is a special structure called substrate integrated waveguide, which is a special form of rectangular waveguide [2]. On the translation stage, there are two coaxial probes, which measure the electric fields produced by these two structures independently. By measuring these fields as a function of position, useful information about the structures themselves, as well as the loads terminating them, can be determined. In this laboratory, the network analyzer is used as a tuned receiver whereby signals are transmitted out port 1 and received via port 2, for a set of different frequency points. Hence,

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Page 1: Laboratory Experiment 3: Standing Waves and Waveguides

 

UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering

ECE320H1F / ECE357H1F

EXPERIMENT 3: STANDING WAVES AND WAVEGUIDES 1. Introduction The purpose of this laboratory is to introduce you to a few types of planar transmission lines and waveguides and to illustrate how practical measurements of the fields produced by these transmission lines can be carried out. Using knowledge of transmission lines, you will physically measure the standing wave pattern produced by a terminated transmission line and waveguide. The lab will also introduce wave propagation characteristics within waveguides, which are discussed later in the course. You will need the following equipment and components to complete this lab:

• Agilent vector network analyzer • Calibration kit (short, open, load, through) • 2 SMA cables • Lab 3 linear translator station • Ruler • Computer running MATLAB or Microsoft Excel, for recording data • One unknown load

The main component in this lab is a linear translator, which is shown in Figure 1. It consists of a moveable stage, which is suspended on three rods above a printed circuit board (PCB). The stage is moved back and forth using a stepper motor, with controls situated on a small board at one end of the positioner. On the PCB, there are two circuits. The first is a microstrip line[1], and the second is a special structure called substrate integrated waveguide, which is a special form of rectangular waveguide [2]. On the translation stage, there are two coaxial probes, which measure the electric fields produced by these two structures independently. By measuring these fields as a function of position, useful information about the structures themselves, as well as the loads terminating them, can be determined. In this laboratory, the network analyzer is used as a tuned receiver whereby signals are transmitted out port 1 and received via port 2, for a set of different frequency points. Hence,

Page 2: Laboratory Experiment 3: Standing Waves and Waveguides

 

port 1 is used to feed a signal into the input of the structure of interest, while port 2 is connected to the coaxial probe in order to measure the field picked up by the probe. Essentially, the network analyzer is used to measure the transmission coefficient from port 1 to port 2, which will be directly proportional to the voltage standing wave pattern produced with each of the PCB structures.

Fig. 1 Linear Translator

2. References

1. F. T. Ulaby, E. Michielssen, and U. Ravaioli, Fundamentals of Applied Electromagnetics, 6th ed. Upper Saddle River, NJ: Prentice-Hall, 2010, pp. 61-64.

2. ——, Fundamentals of Applied Electromagnetics, 6th ed. Upper Saddle River, NJ: Prentice-Hall, 2010, ch. 8, pp. 381–390.

3. H. Uchimura, T. Takenoshita, and M. Fujii, “Development of a “laminated waveguide”,” IEEE Trans. Microwave Theory Tech., vol. 46, no. 12, pp. 2438–2443, Dec. 1998.

4. K. Wu, “Integration and interconnect techniques of planar and non-planar structures for microwave and millimeter-wave circuits–current status and future trend,” in Asia-Pacific Microwave Conf., Taipei, Taiwan, R.O.C., Dec. 3–6, 2001, pp. 411–415.

5. F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 1, pp. 66–73, Jan. 2005.

Page 3: Laboratory Experiment 3: Standing Waves and Waveguides

 

3. Background 3.1 Microstrip Line A microstrip line consists of a narrow strip of conductor, suspended over a large conducting ground plane, usually by a dielectric substrate, as shown in Figure 2(a). The geometry of the microstrip line impacts its characteristic impedance. By varying the strip width w, substrate height h, and substrate dielectric constant rε , one can realize a wide range of characteristic impedances for the application at hand. When the line is excited by placing the strip at an electric potential relative the ground plane, the electric and magnetic field lines shown in Figure 2(b) result. Clearly, in addition to the “parallel plate” type field structure created directly beneath the strip, fringing fields are also created, the bulk of which propagate within the air region above the transmission line. Since the fields are not confined to purely the air region or the dielectric region, the wave velocity of the line is somewhere between that of the speed of light in air ( 8103×=c m/s) and that in the substrate (

rc ε/ ). The wave velocity is usually described using an effective dielectric constant effε , so that

the speed of

Fig. 2 Microstrip line (courtesy of Ulaby, Fig. 2-10)

Page 4: Laboratory Experiment 3: Standing Waves and Waveguides

 

light along the microstrip line is effc ε/ . The effective dielectric constant is dependent on the

geometry of the line and the substrate parameters, just as the characteristic impedance is. There relationships are well known and best captured in empirical formulas [1]. The effective dielectric constant is found using

xyrr

eff s

⎟⎠

⎞⎜⎝

⎛+⎟

⎞⎜⎝

⎛ −+

+=

10121

21 εε

ε

where s=w/h and

)107.11ln(05.043.0107.3ln02.01

39.0

56.0

344

244

05.0

ss

ssy

xr

r

−−

+++⎟⎟⎠

⎞⎜⎜⎝

+

−++=

⎥⎦

⎤⎢⎣

+

−=

εε

The characteristic impedance of a microstrip line can be found using empircal formulas as well, given by

2

20

67.30

41)62(6ln60

⎟⎠

⎞⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−+=

st

sseZt

eff

π

ε

The line in this lab has been fabricated on a substrate called FR-4, which is a commonly used material for making printed circuit boards. FR-4 has a dielectric constant of 4:4 nominally. The line on the PCB in this lab has been designed on a h = 1.5 mm substrate so that it is matched to the system impedance of the network analyzer instrument, which is Z0 = 50 . If you look carefully at the coaxial probe above the microstrip line, you will see that it consists of a short segment of wire extending from the coaxial connector. This little wire forms a small dipole antenna which is sensitive to a vertically-oriented electric field. When placed in proximity to the microstrip line, this probe senses the vertically-oriented fringing field produced by the microstrip line in the air region above the line. Note that if the microstrip line did not produce any fringing fields, this type of measurement would not be possible. Since the electric field is proportional to the voltage produced between the two conductors, the measurement of this field can be used to determine the voltage standing wave pattern along the line.

Page 5: Laboratory Experiment 3: Standing Waves and Waveguides

 

3.1.1  Transmission  Line  Measurement  and  Voltage  Standing  Waves    

The   voltage   on   a   transmission   line   is   a   combination   of   a   forward   travelling  wave  )0(1Ve zjβ−  and  a  backward  travelling  wave   )0(2Ve zjβ .    The  amplitude  of  the  voltage  on  the  line  

is  then  given  by    

)0(1)0()0()0()()()( 212121 Γ+=+=+= +− zjzjzj eVVeVezVzVzv βββ

As  shown  in  Fig.  3  the  amplitude   )(zv  varies  between  a  maximum  value  of    

{ } πβ nzVV 22)0(at located )0(1)0( max1max =+Γ∠Γ+=

and a minimum value of

{ } πβ )12(2)0(at located )0(1)0( min1min +=+Γ∠Γ−= nzVV The ratio

VSWRVV

=Γ−

Γ+=11

min

max

is  known  as  the  voltage  standing  wave  ratio.  Note  that   Γ    is  a  constant  along  the  line  and  we  find      

11

+−

=ΓVSWRVSWR

To  determine  the  phase  of   )0(Γ at  the  load  as  shown  in  Fig.  7,  let  the  first  voltage  minimum  be  located  at   minmin dz −= .    Thus        

λπ

πβπβπ minminmin

422)0( ddz +=+=−=Γ∠  

 Thus  by  measuring  the  VSWR  and  the  position  of  a  voltage  minimum,  the  complex  

reflection  coefficient   )0(Γ at  the  input  port  of  a  transmission  line  can  be  determined.  The  input  impedance  can  be  derived  from  the   )(zΓ to   )(zZin transformation.    

   

Page 6: Laboratory Experiment 3: Standing Waves and Waveguides

 

   

Fig.  3    Voltage  standing  wave  on  a  transmission  line  

3.2 Waveguides and Substrate Integrated Waveguide (SIW) 3.2.1 Rectangular Waveguides The rectangular waveguide is a metallic ’pipe’ of rectangular cross section that can guide electromagnetic waves along the longitudinal z-axis, as shown in Figure 4. The advantages of the rectangular waveguide (compared to say planar transmission lines) are that it is a low-loss medium and can propagate high power signals. Due to these advantages, it is still in use for applications such as satellite communications and for making base-station channel filters.

Fig. 4 Rectangular waveguide

A rectangular waveguide supports an infinite number of transverse electric (TE) and transverse

Page 7: Laboratory Experiment 3: Standing Waves and Waveguides

 

magnetic (TM) modes for which the longitudinal electric and magnetic fields vanish, respectively. In this laboratory, we will experiment with the fundamental mode (lowest cutoff frequency) which is the TE10 mode (assuming a > b). Normally this is the mode that the rectangular waveguide is used in practice. The TE10 mode can be thought of as a superposition of two plane waves bouncing on the two vertical walls and with this understanding, we can write the electric field in the guide as

zjkzeayExE −⎟⎠

⎞⎜⎝

⎛=

πsin0⌢

Indeed this electric field expression satisfies the boundary conditions: it vanishes on the two vertical walls y = 0 and y = a as it should. Now we can calculate the corresponding magnetic field by the application of Faraday’s law yielding

zjkz

zjkzy

z

z

eay

aEjH

eayEkH

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=

⎟⎠

⎞⎜⎝

⎛=

ππωµ

πωµ

sin

sin

0

0

To gain further understanding of the fundamental mode, consider the bouncing diagram shown in Figure 5.

Fig. 5 Reflections within a rectangular waveguide.

As shown, the plane waves are directed along k where µεω=k and where ε and µ are the permittivity and permeability of the medium that fills the waveguide. From the bouncing diagram we can write 222

yz kkk += (consistency condition). At a given frequency, the bouncing

angle that the plane waves make with the z-axis is not arbitrary and is determined by the fact that

ak y

π= (to satisfy the boundary condition 0=xE at y = a). Therefore, we can determine the

propagation constant of the TE10 mode along the z direction as

Page 8: Laboratory Experiment 3: Standing Waves and Waveguides

 

gz a

kkλππ 22

2 =⎟⎠

⎞⎜⎝

⎛−=

where gλ is the guided wavelength (along z). From the above expression, we see that the propagation

constant zk vanishes when ak /π= . This defines the cutoff frequency:

µεaacfc

21

2==

where µε/1=c is the speed of light in the material that fills the waveguide. Note that for frequencies smaller than the cutoff frequency cff < the propagation constant becomes purely

imaginary thus leading to evanescent waves (attenuating) along the z direction zkze− . Therefore, the waveguide only propagates energy along the z axis for frequencies larger (or equal) than the cutoff frequency cff ≥ . This is understandable since low-frequency currents cannot be guided by a single conductor as there is no return path. Equivalently, the larger dimension of the guide must be greater than half

a dielectric wavelength to ensure operation above the cutoff frequency, i.e. 2λ

≥a . The

propagation constant can be written in terms of the cutoff frequency as

g

cz f

fkkλπ21

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

This equation reveals more clearly that at cutoff cff = , the propagation constant is zero, i.e.

0=zk . When cff < the propagation constant becomes imaginary leading to an evanescent wave along z. It can also be noted that as the frequency tends to infinity kkz → but in practice the upper operating frequency is limited due to the excitation of higher-order modes. This equation also reveals that unlike a transmission line, the rectangular waveguide is inherently dispersive since zk is a non-linear function of the frequency.

If the rectangular waveguide is terminated to a load (e.g. shorted or left open) then there will be reflections traveling along the negative z axis. In this case, standing waves will form along the z axis which will be characterized by the guided wavelength gλ (e.g. the distance between two

successive maxima will be 2/gλ ).

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3.2.2 Substrate Integrated Waveguide (SIW) Waveguides offer a low-loss, shielded transmission medium. In traditional sense, they can be thought of as hollow metallic pipes having rectangular, circular or elliptical cross-section. These conventional waveguides are bulky; hence, they are not suitable for board-level integration. Recently, a substrate integrated waveguide (SIW) structure that is fully compatible with electronic system substrate has been introduced [3, 4]. The adaptation of waveguides to PCB technology along with its inherent low-loss bandpass characteristic has made them attractive structures for microwave and millimeter-wave applications.

Fig. 6 Substrate integrated waveguide.

An SIW is a rectangular waveguide configured within the circuit substrate, as depicted in Fig. 6. The dielectric layer is sandwiched between two solid conductor planes. Signal propagation is confined between the waveguide sidewalls, which are formed by rows of plated via holes. The first step in the design process is to determine the operating frequency range of the SIW. Its TE10 cutoff frequency ( cf ) can be expressed as

0021

εµεµ rreffc

Wf =

rε is the dielectric constant of the substrate material. The TE10 cutoff is usually chosen to be

around 15–20% below the lowest frequency of interest. Weff is the effective width of the SIW, which is related to the width of the waveguide, W (measured between via centers), by [5]

WD

sDWWeff

221.008.1 +−=

s is the center-to-center spacing between the rows of vias, and D is the diameter of the via. Since the sidewalls are formed by metallic vias, energy leakage can potentially occur if the vias are placed too far apart. As a rule of thumb, it is recommended to keep 10/min,gs λ< and D should

Page 10: Laboratory Experiment 3: Standing Waves and Waveguides

 

be as small as possible. gλ is the guided wavelength. Generally, s and D will be limited by

the PCB fabrication tolerances. We will now use the above design guideline to design an SIW operating at 1 GHz. The substrate material is FR-4 ( rε = 4.4). The TE10 cutoff of the SIW is then set at 0.85 GHz, which gives the initial value of Weff to be 84.1 mm. Subsequently, s and D are arbitrarily chosen to be 10 mm and 1 mm, respectively. From (13) and (14), cf and Weff are calculated to be 0.842 GHz and 84.9 mm, respectively. At 1 GHz, it is found that gλ = 265.4 mm, which gives s = 0.038 gλ .

4. Microstrip Line StandingWave Measurements Notation: In the following instructions, [Command] refers to a softkey on the VNA front panel while Command refers to an option appearing on the VNA touchscreen.

Before we conduct measurements, we must properly calibrate the network analyzer. In this way, when the two cables from the network analyzer are connected together, we will measure a transmission coefficient of unity (T = S21 = 1 = 0 dB). 4.1 Network Analyzer Setup and Calibration Procedure 1. Press [Start] and enter 800 MHz. Choose [Stop] to be 1.2 GHz. 2. Press [Sweep Setup] and select Points to be 801. 3. Press [Cal]àCorrection ONàCalibrateàEnhanced ResponseàSelect Ports 2-1. 4. Attach the open, short, load, and through to the network analyzer ports until the calibration is complete. 5. Check your calibration by connecting the SMA cables end-to-end, and ensuring that: (a) The input reflection coefficient is very low (< -20 dB) across the entire frequency band. (b) The transmission coefficient is nearly unity (0 dB) across the entire frequency band.

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4.2 Measurement of Microstrip Line Characteristics In this portion of the lab, you will be using the linear positioner to measure the fringing fields produced above the microstrip line, the magnitude of which are directly proportional to the voltage along the line. Hence, the positioner can be used to plot the voltage standing wave as a function of position. The stage is moved back and forth using the buttons on the control board. Pressing one of the two buttons moves the stage in a corresponding direction until the button is released. This can be used for fine control of the stage position. Conversely, depressing the buttons momentarily will result in the stage moving approximately 5 mm in the selected direction.

1. Measure the width of the transmission line w. Then use the formulas given in Section 2.1 to compute the characteristic impedance to confirm it is near 50Ω. Determine the effective dielectric constant of the line and the phase velocity of the line, assuming it is constant for all frequencies.

2. Attach the SMA cables and connect port 1 of the network analyzer to the input port of the microstrip line, feeding the SMA cable through the small hole at the end of the linear positioner. 3. Connect a short circuit to the end of the microstrip line. 4. Using the control buttons, move the stage so that the coaxial probe used to measure the microstrip line is positioned above the load end of the microstrip line. 5. Press [Meas] à S21. 6. Press [Format] and select Log Mag. 7. Press [MARKER] and set the marker to f = 1 GHz. Use [Scale]àAutoscale to see the curve. 8. At this point, the transmission coefficient (S21 measured by the network analyzer should be very small (less than 60 dB). 9. Move the stage in 5 mm increments away from the load, and plot the standing wave pattern over a minimum one one full wavelength at 1 GHz. Use the ruler supplied to make precise measurements of the probe location relative to the short-circuited end of the PCB.

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10. Using fine positioning of the probe, locate the maxima and minima to obtain more precise locations for the field readings at these positions. 11. Compute the VSWR and compare to that of a short circuit, bearing in mind that the field values reported by the network analyzer are on a power-dB scale. 12. Compute the wavelength and corresponding effective dielectric constant of the microstrip line. Compare the theoretical expectations, and comment on any differences.

4.3 Using Standing Wave Patterns for Load Calculations In this portion of the lab, you will measure the standing wave pattern produced when the microstrip line is terminated in an unknown load, and using analytical / Smith Chart techniques to determine the value of the unknown load.

1. Disconnect the short circuit used in the previous experiment and connect the unknown load to the end of the microstrip line. 2. Re-position the probe so that it is measuring the field directly above the PCB edge where the load is located. 3. Move the stage in 5 mm increments away from the load, and plot the standing wave pattern over a minimum one full wavelength at 1 GHz. Use the ruler supplied to make precise measurements of the probe location relative to the short-circuited end of the PCB. 4. Using fine positioning of the probe, locate the maxima and minima to obtain more precise locations for the field readings at these positions. 5. Using the techniques of Section 2.1.1, compute the impedance of the load based on the standing wave pattern you have measured. Bear in mind that the field values reported by the network analyzer are on a power-dB scale. The maximum and minimum magnitudes are used to find the VSWR (and hence, the magnitude of the impedance) and dmin yields the phase of the load impedance. 6. Disconnect the SMA connectors from the microstrip portion of the translator.

Page 13: Laboratory Experiment 3: Standing Waves and Waveguides

 

7. Connect the unknown load to port 1 of the network analyzer and measure its impedance. As in previous labs, the extra length of SMA cable must first be accounted for using the Port Extension process.

(a) Press [CAL] à Port Extension à “Extension ON” à Auto Port Extension à Select PortsàPort 1 ON. Now attach an Open and Short connection and press the corresponding screen buttons. (b) With the unknown load attached, press [Meas]àS11à[Format]àSmithàR+jX and select Reflection. (c) Finally, press [Marker] and set the marker to 1 GHz and record the impedance from by the network analyzer. Compare to your computed result and discuss any discrepancies.

4.4 SIW Standing Wave Measurements In this part of the lab, you will use the field measurement apparatus to measure the electric fields produced with a type of rectangular waveguide, which has been realized practically in the form of a substrate integrated waveguide (SIW). Sampling of the electric field is made possible by introducing a slot in the top (and bottom) of the waveguide, which is clearly visible on the PCB, and inserting a coaxial probe in the vicinity of the slot. Fields normally contained within the rectangular waveguide produce fringing effects in the vicinity of the slot, which can be detected in a similar way as the microstrip line fringing fields. However, structurally, the field configuration with the SIW is fundamentally different from the microstrip line, as will be made apparent in your measurements. One end of the SIW line is probe-fed by a coaxial line. The other end of the line is shorted out with a short circuit (ZL = 0). Hence, a standing wave is produced with a very high standing wave ratio along the length of the line, which is visible by measuring the fringing fields from the slot. The fringing fields are proportion to the values of the transverse electric fields within the SIW.

1. Connect port 1 of the network analyzer to the input port on the SIW. This is the only port on the SIW side of the PCB, which is a flange-mounted connector on top of the PCB. 2. Connect port 2 of the network analyzer to the coaxial probe on the SIW side of the probe stage. 3. Switch the network analyzer back to a log-magnitude view of the transmission coefficient, and move the marker to a frequency of 900 MHz.

Page 14: Laboratory Experiment 3: Standing Waves and Waveguides

 

4. Move the stage to the end of the slot farthest from the input connector. Note that the end of the slot is not at the load (short-circuit) point, but rather, some arbitrary point along the SIW. 5. Move the stage in 5 mm increments away from the starting position, and plot the standing wave pattern over a minimum one one full wavelength at 900 MHz. Use the ruler supplied to make precise measurements of the probe location relative to the short-circuited end of the PCB. 6. Using fine positioning of the probe, locate the maxima and minima to obtain more precise locations for the field readings at these positions. 7. Measure the standing wave ratio and determine the phase velocity pv and guide

wavelength gλ of the wave inside the waveguide. How does its value compare to the

speed of light for a TEM wave travelling within the substrate? Explain.