MTS 7.4.4 The Measuring Line

The slotted measuring line Sampling the field in front of a short-circuit plate and a termination

Principles Various line types for electromagnetic waves and their technical function Lines fulfill various functions in radio fre- quency technology: (a) They serve in the transmission of radio fre- quency signals between remote locations. Examples: radio link between the transmit- ter and a remote system, broadband cable for signal distribution of a satellite re- ceiver system to the individual subscribers and underwater cables. Transmission lines and directional radio links in free space are frequently considered potential alternatives in this function as a medium for communi- cations transmission (b) They also serve as circuit elements (in- stead of and inductors) and in- terconnecting lines for the realization of passive circuits (e.g. transmis- sion-line filters). In general, transmission lines can be consid- ered as structures for guiding electromagnetic waves. For this function a multitude of various types are suitable. Fig. 5.1: Various line types for electromagnetic waves (1 = metal conductor, 2 = isolator). The coaxial and two-wire line depicted in Fig. Numbering from left to right. 5.1a) guides transverse electromagnetic waves (a) Coaxial and two-wire lines (TEM waves) and can also be used for any ar- (b) Planar line structures: Microstrip line, bitrarily low . coplanar line, slotted line In other transmission line types, however, the (c) Waveguide: Rectangular, circular and wave propagation is dependent on the condi- ridged tion that the cross-sectional dimensions of the (d) Dielectric waveguides: Round and rectangular-shaped surface waveguide; line are at least about as large as a half free- optical waveguide λ space ( 0/2). This leads to the result that in the range of above approx. 1 GHz there are a larger number of transmis- sion line types available than for low frequen- waveguides shown there are based on the prin- cies. This is because it is only at the higher ciple of total reflection of electromagnetic λ frequencies where the dimension 0/2 results in waves at the boundary surfaces between the “handy” cross-sections. insulators with “higher” to “lower” relative per- ε Figure 5.1 b) shows various planar transmission mittivity r. line types, which are particularly well-suited for the design of microwave integrated circuits Necessary fundamentals drawn from elemtary (MIC). transmission line theory Transmission lines can be realized without any For the description of line-bound wave propa- metal conductors, see Fig. 5.1 d). The dielectric gation we can begin our investigation with a

27 MTS 7.4.4 The Measuring Line

simple two-wire line on which the voltage and i(x,t) x current can be specified at any given location. u(x,t) Z Subsequently the results can be generally ap- (a) 0 plied as a model for any other homogenous d u(x,t) t t=d t transmission line. û Fig. 5.2 shows a homogenous transmission line x (b) t =0 structure represented symbolically by two dou- l û g i(x,t) ble lines. The electromagnetic state on this line Z 2 can be expressed by specifying the spatial and 0 t=d t time dependency of the instantaneous voltage (c) u(x,t) and current i (x,t). t =0 f |U(x)| First we will consider the case in which a single u(x) 2p wave propagates on the line in the +x direction. |U(x)| The following holds true with ω = 2π f as the an- p λ 0 û gular frequency and g as the guided wavelength (“g” = guide) and where the attenuation on the f |I(x)| (d) I(x) line is disregarded 2p p ⎛ ⎞ û x Z u( xt,) =⋅ û cos⎜ωπ ⋅− t 2 ⎟ (5.1) 0 0 ⎜ λ ⎟ ⎝ g ⎠ Fig. 5.2: Wave propagation on a line (a) Spatial dependency of the voltage u (x,t) The phase velocity resulting here is at two different time points (b) Spatial dependency of the current i (x,t) at two different time points ωλ⋅ (c) Spatial dependency of the magnitude |U (x)| = g =⋅λ and phase Φ (x) for the complex voltage vph f g (5.2) u 2π amplitude (phasor) (d) Spatial dependency of the magnitude |I(x)| and phase Φ (x) for the complex current If Z expresses the line's characteristic imped- I 0 amplitude (phasor). ance, then the following holds true for the corre- sponding voltage u (x,t) and current i (x, t)

ux( ,t ) i( xt, ) = (5.3) φπ=−x =− ⋅ and u ()x 2 § x (5.6) Z0 λ g In the description of time-harmonic processes and as such the following is true for the current you can also make use of complex amplitudes. As such an AC voltage of the form Ix( ) = Ix( ) ⋅exp[] jφ ( x) (5.7) u (t) = û cos (ωt + φ) can be expressed by a cor- I responding complex amplitude U = û exp ( jφ) where and the following applies û u(t) = Re {U · exp (j ω t)} (5.4) ( ) = φφ( ) = ( ) =− ⋅ Ix und Iuxx§x Z Complex numbers are now specially marked 0 (5.8) by an underline. Equations (5.1) and (5.3) are given complex numbers with the form Figures 5.2 c) and d) show the spatial depend- ency of the magnitudes |U(x)| and |I(x)| as well U( xUx) = ( ) ⋅exp[] jφ ( x) (5.5) Φ Φ u as the phases u (x) and I (x) of the complex voltage and current amplitude. In equations where Ux( ) = û (5.6) and (5.8) the following phase constant has been introduced

28 MTS 7.4.4 The Measuring Line

π ω § ==2 (5.9) λ g vph as a second parameter to express line-bound wave propagation.

Description of the field in front of a short- circuit If there is a short at the end of the transmission line (x = 0) (see Fig. 5.3, above), the wave can be described from the superposition of one component travelling to the load u+(x, t) and one reflected component travelling back from the load u–(x, t). As the total voltage at x = 0 must be identical to zero (short-circuit!), it follows that:

u (x, t) = u+ (x, t) + u–(x, t) = û cos (ωt – ßx) –û cos(ωt + ßx) = 2 · û · sin (ωt) · sin (ßx) (5.10) The current of the reflected wave is related to u– by virtue of (–Z0) and thus it follows (see Fig. 5.3 b) that Fig. 5.3: on a line shorted on one end: a) Voltage characteristic u+(x, t) of the wave travelling to the load and u–(x, t) of the 2⋅û reflected wave travelling back at the time i(xt , )= ⋅⋅cos(ωβt)⋅⋅ cos( x) point t = T/8. Z 0 b) i+(x,t) and i–(x,t) at t = T/8. (5.11) c) Characteristic of the total voltage u (x, t) = u+ (x, t) + u– (x, t) The current and voltage now have distinct re- at 4 different points in time d) i (x, t) = i+ (x, t) + i–(x, t) sponses with respect to time and space. We are e) Spatial dependency of the magnitude |U (x)| Φ dealing with a standing wave [see Fig. 5.3 c) and phase u (x) of the complex and d)]. The nodes and maxima of the u and i voltage amplitude f) Spatial dependency of the magnitude |Ι(x)| characteristic are at locations unchanged with Φ and phase I(x) of the complex current respect to time. The nodes of the voltage are amplitude. λ shifted by g/4 with respect to the nodes of the current. Moreover, there is a phase shift of + π/2 between the voltage and current. This rectangular waveguide as shown in Figure 5.4 means that u exhibits its extreme values exactly is a special form of waveguide. This rectangu- when i (for each x) is zero and vice versa. If lar waveguide is the object of a series of experi- these relationships are represented for the mag- ments and should therefore be considered more nitude and the phase of the complex amplitude, closely in theoretical terms. This is also be- then we obtain the results depicted in Figures cause many of the results found for rectangular 5.3 e) and f). waveguides also apply to other forms of waveguides (e.g. circular waveguide). If you consider the zig-zag reflection of a plane Wave propagation in a rectangular uniform wave between the side surfaces (dis- waveguide tance a) of a waveguide, you obtain a critical Generally a waveguide is a “hollow tube” in frequency called the cut-off frequency, as of which electromagnetic waves can propagate. A which wave propagation becomes possible

29 MTS 7.4.4 The Measuring Line

cc == fc (5.12) λ c,0 2a

For the phase velocity a relationship is yielded c υ = (5.13) ph ⎛ λ ⎞ 2 0 1−⎜ ⎟ ⎝ 2a ⎠ Fig. 5.4: Rectangular waveguide λ A calculation of the guided wavelength g and the phase constant ß amounts to The field strength is maximum in the middle of the waveguide at x = a/2, and it holds true that v λ λ ==ph 0 g (5.14) 2 f ⎛ λ ⎞ 2 − 0 E = ⋅ U 1 ⎜ ⎟ y, max ⋅ 0 ⎝ 2a ⎠ ab

whereby U0 constitutes an arbitrarily intro- duced voltage. π π ⎛ λ ⎞ 2 22 0 § ==1 −⎜ ⎟ (5.15) Its corresponding magnetic field has both a λ λ ⎝ 2a ⎠ g0 transverse component Hx(x, z) as well as a lon- gitudinal component Hz(x, t). For more information regarding the derivation The transverse component Hx has the same of these relationships we wish to refer to the phase as Ey and also the same spatial depend- specified sources in the bibliography. ency, and so it follows that: Another approach used for describing propa- gation originates from the solution of Maxwell's ⎛ ⎞ § 2 π − equations with the boundary conditions exist- H (,)xz =− ⋅ ⋅U ⋅sin⎜ xe⎟ jßz x ωµ ab 0 ⎝ a ⎠ ing at the metal surfaces taken into account. But 0 do not be alarmed. Here only the result of this (5.17) analysis is reproduced for the fundamental At every cross-sectional point z = const. the ra- mode (TE10): The electrical field has only one Cartesian com- tio of Ey to (–Hx) is equal and given by the char- ponent. It points in the y-direction and so it fol- acteristic impedance lows that: Exz(,) ωµ120⋅ π = y ==0 2 ⎛ π ⎞ Z0 ( ) = ⋅⋅ ⋅ − jßz (− ) § 2 E y xz,U sin⎜ xe⎟ Hxzx (,) ⎛ λ ⎞ ab⋅ 0 ⎝ a ⎠ 1− ⎜ 0 ⎟ ⎝ 2⋅a ⎠ (5.16) (5.18) It is evident that Ey is not only dependent on the propagation coordinate z, but also on the trans- verse coordinate x. As the tangential compo- The following applies for the longitudinal com- ponent H : nents of the electrical field must vanish at the z surface of (ideal) conductors, it is true that π ⎛ π ⎞ ( ) =⋅1 2 ⋅ − jßz H z xz, j ωµ U 0 cos⎜ xe⎟ Ey (x = 0, z) = Ey (x = a, z) = 0. a 0 ab ⎝ a ⎠ (5.19)

30 MTS 7.4.4 The Measuring Line

E-plane (a) T-plane E-Plane T-plane

λ g 2

Fig. 5.6: Existence zones for various wavetypes capable → –E .. H of propagating in the rectangular waveguide (a = 2.25 b).

(b) Slotted measuring line As will be demonstrated with particular care in I experiment 6, a reflection at the end of the I transmission line has the effect that maxima II and minima are formed in the spatial distribu- tion of the field strength along the line. Based → ..H on the ratio of the amplitude values (maximum/ I II minimum) and the locations of the maxima and minima you can draw conclusions as to the magnitude and phase of the reflection coeffi- Fig. 5.5: Field pattern of the fundamental mode cient. (TE10-wave) in the rectangular waveguide (a) E and T fields If to this purpose you wish to measure the dis- (b) Current density distribution on the tribution of the field strength along the trans- metal walls mission line (slotted line), the following must be taken into consideration: (a) You should interfere as little as possible with the electromagnetic field in the waveguide. This requirement is met, if a π “narrow” slit is added in the center of the Thus it is phase-shifted by /2→ with →respect to the transverse components of E and H . In con- wide section of the waveguide (see trast to these it assumes at x = 0 and x = a its Fig. 5.7 and compare to Fig. 5.5 below). maximum value (in terms of magnitude) at the metal sidewalls. The spatial dependencies of the field components reproduced in Equations (5.16) to (5.19) are shown in Fig. 5.5. The wave type under consideration till now is also re- ferred to as “TE10-mode”. Here the TE stands for transverse electric and “1” means that the number of halfwaves in the x-direction is 1 while the index 0 means that the field is con- stant in the y-direction. At higher frequencies higher wave modes, namely TEmn-waves and TMmn-waves are capable of propagation. According to Fig. 5.6 the TE20 mode is capable of propagation starting from a frequency of Fig. 5.7: Principle of the slotted measuring line (exploded f = 2 f , when there is a side ratio of view) c2 c,TE10 1 Slit in the waveguide a/b = 2.25 so that a “frequency octave” is avail- 2 Field probe = short rod-type antenna able for single-mode operation. 3 Detector diode

31 MTS 7.4.4 The Measuring Line

Fig. 5.8: Experiment setup

(b) The characteristic impedance Z0 (see Required equipment Equation [5.19]) is slightly altered by this 1 Basic unit 737 021 modification. This can be compensated for 1 Gunn oscillator 737 01 by slightly increasing the waveguide 1 Diaphragm with slit width a in the region of the slit. 2 x 15 mm 90° 737 22 (c) A short probe (“electrical dipole”) as 1 Slotted measuring line 737 111 shown in Fig. 5.7 supplies a voltage to its 1 Coax detector 737 03 output which is proportional to the trans- 1 Short-circuit plate, from verse component |Ey| of the electrical field accessories 737 29 strength. Thus behind the detector probe 1 Waveguide termination 737 14 (square-law rectification) you obtain a 1 Set of thumb screws (2 each) 737 399 voltage Additionally required equipment 2 1 Oscilloscope (optional) 575 29 U =⋅KE (5.20) D y 1 XY recorder (optional) 575 663 2 Stand bases 301 21 m 2 2 Supports for waveguide Here K is a constant with the dimension V components 737 15 1 Stand rod 0.25 m 301 26 2 Coaxial cables with BNC/BNC Low reflection waveguide termination plugs, 2 m 501 022 By inserting a wedge-shaped absorber material the power of the incident wave is nearly com- Recommended pletely absorbed thus suppressing any reflec- 1 PIN modulator 737 05 tion almost totally. 1 Isolator 737 06

32 MTS 7.4.4 The Measuring Line

Experiment procedure 9 V (Tip: first increase it to 10 V and then Note: adjust it back to the desired value), set the If you are using a PIN modulator and isolator modulation switch to GUNN-INT. the experiment setup in Fig. 5.8 is supple- 2.3 Place the slotted measuring line probe at mented as explained in the preface. one end position. Now slowly push the slide to the other end position and at the 1. Set up the experiment arrangement in ac- same time search for the maximum detec- cordance with Fig. 5.8 (perhaps modify tor voltage (always adjusting the gain to a according to Fig. 0.5) suitable level). Calibrate the maximum Note: value to “0” dB using the “ZERO” con- The diaphragm slit serves as a frequency- troller. The maximum position is desig- selective component (filter) to improve the nated x0 (read off the scale and note down) spectral purity of the guided wave. and functions as the reference position. 2. Measurement with short-circuit plate 2.4 Now shift the position of the probe in 2.1 Attach the short-circuit plate at the open 2 mm steps (in the direction of the end po-

end of the slotted measuring line. sition which is farthest removed from x0), 2.2 Set the range switch V/dB of the SWR me- in other words to the corresponding posi-

ter to the most insensitive range. tions |xn – x0| = n · 2 mm. Enter the values Set the Gunn voltage to between 8 V and measured for the detector voltage into the second column of Table 5.1. Table 5.1

Display in dB ⎛ ⎞ Probe position − U( xx− ) 2π Ux( x0 ) 0 ⋅− =⋅ cos⎜ xx0 ⎟ xx− in mm 20 log ⎜ λ ⎟ 0 ⎝ g ⎠ Umax Umax 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Minimum at

Distance of the Minima λ g =______mm ∆x / mm=______

33 MTS 7.4.4 The Measuring Line

Table 5.2

Display in dB Probe position − − Ux( x0 ) U( xx0 ) − =⋅20 log xx0 in mm Umax Umax 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

You can conclude the measurement after 3.4 Position the probe in 2-mm steps in the exceeding the position of the 2nd maxi- same locations as in Experiment 2.4 (see mum (Display/dB ≈ 0). also Table 5.1), and enter the display val- 2.5 Determine the distance ∆x/mm between ues in Table 5.2. two minima and enter the result in the last Note: line of Table 5.1. (Tip: By successively If you have an XY recorder, a digital stor- increasing the gain (V/dB) in the minima, age oscilloscope or a CASSY interface at you can determine the positions more pre- your disposal you can also directly record cisely). the measurement curve. For this purpose the slotted measuring line is equipped with 3. Measurement with reflection-free an integrated displacement sensor. waveguide termination. Ž Connect the IN socket of the slotted meas-

3.1 The short-circuit plate is replaced by the uring line to the X output (supplying |UG|) “reflection-free” waveguide termination. of the basic unit (or with an external power 3.2 Slide the probe along the slotted measur- supply with 10 V DC). ing line over the entire range and note Ž Connect the X socket of the slotted meas- down by which value (in dB) there is de- uring line to the X input of the XY re- viation in the characteristic, i.e. by which corder (oscilloscope or CASSY). value the level (∆a) has dropped in com- Ž Connect the Y input of the XY recorder parison to the short-circuit experiment. (oscilloscope or CASSY) to the AMP-OUT 3.3 As in point 2.3 move the probe over the socket of the basic unit. entire range and search for the location of the maximum detector voltage and again The AMP-OUT socket supplies a linear output re-calibrate this value to 0 dB. signal whereby 0 V is about –20 dB and approx. 4.5 V corresponds to 0 dB.

34 MTS 7.4.4 The Measuring Line

Furthermore, it is important to point out that the the value indicated in the SWR meter dis- integrated displacement sensor is not com- play in accordance with the expression pletely backlash-free (it does possess hyster- 10 · log(UD / UD,ref). From this you obtain esis), for that reason the curve should only be the relationship plotted once in one direction. − Display Ux( x0 ) Questions =⋅20 log 1. Based on the distance between two dB U max neighboring zero points (distance to the = ( ) minima) ∆x/mm (see Table 5.1) determine where UUxmax 0 the wavelength λ of the guided wave in g 6. According to Fig. 5.3 e) the voltage (field mm. strength) responds accordingly. λ 2. Using the value for g determined above − ⎛ ⎞ U( xx0 ) 2π under point 1 with the waveguide width of =⋅−cos⎜ xx⎟ ⎜ 0 ⎟ a = 22.9 mm taken into consideration, de- U ⎝ λ ⎠ λ max g termine the free-space wavelength 0 and frequency f of the guided wave. [Note: In Equation 5.10 the sinusoidal term is de- 3. What is the mathematical value resulting fined as sin(ßx). However, here it is per-

for the phase velocity vph and the phase mitted to convert to the cosine term be- constant ß according to the Equations cause it only concerns a phase-shift and a

(5.13) and (5.15). maximum is assumed to exist at x – x0 = 0]. The values resulting from this equation are

4. What is the cut-off frequency fc of the TE10 entered into column 4 of Table 5.1 and mode in the given rectangular waveguide, then they are compared to the values in

and from which frequency is the TE20 column 3. Discuss any possible causes if wave capable of propagation? For this you notice regular deviations between the also refer to Fig. 5.6. values in the two last columns.

5. In addition to the values from the measur- 7. In conjunction with the values determined ing instrument displays determined experimentally under point 3.4 from the through experimentation in experiment measuring instrument display, calculate 2.4 calculate the ratio of the respective the ratio of the voltage (field strength) to voltages (field strength levels) to the max- the voltage maximum at the respective lo- imum value and enter these in column 3 of cation of the slotted measuring line. Enter Table 5.1. the values into the 3rd column of Ta- Note: ble 5.2. As described in Experiment 2 the detector voltage is proportional to the square of the 8. Discuss the value ∆ obtained under experi- received field strength (or here the voltage ment point 3.2. from the transmission line model accord- By what magnitude in dB do the measured ing to Fig. 5.2). You take the logarithm of values from subpoint 3.4 vary?

35