Download unit- 5 microwave measurements slotted line vswr

UNIT- 5
MICROWAVE MEASUREMENTS
A collection of techniques particularly suited for development of devices and monitoring of
systems where physical size of components varies from a significant fraction of an
electromagnetic wavelength to many wavelengths.
Virtually all microwave devices are coupled together with a transmission line having a uniform
cross section. The concept of traveling electromagnetic waves on that transmission line is
fundamental to the understanding of microwave measurements.
At any reference plane in a transmission line there are considered to exist two independent
traveling electromagnetic waves moving in opposite directions. One is called the forward or
incident wave, and the other the reverse or reflected wave. The electromagnetic wave is guided
by the transmission line and is composed of electric and magnetic fields with associated electric
currents and voltages. Any one of these parameters can be used in considering the traveling
waves, but the measurements in the early development of microwave technology made
principally on the voltage waves led to the custom of referring only to voltage. One parameter in
very common use is the voltage reflection coefficient Γ, which is related to the incident, Vi, and
reflected, Vr voltage waves by Eq. (1). (1)
SLOTTED LINE VSWR MEASUREMENT
A slotted line to measure voltage standing wave ratio. You might turn up such an instrument if
you work in a lab that is more than 25 years old. Basically it is a coax line with a slot down one
side where a probe can be moved longitudinally to measure varying electric field strength. The
probe has a detector that converts RF energy to DC voltage, so you can measure peaks and
valleys using an voltmeter. For circuits that were extremely mismatched (or open or short
circuited), the peaks and valleys are the most noticeable. The ratio of the peak voltage to the
valley voltage was the most directly calculated piece of data you can get with a slotted line...
hence "voltage standing wave ratio".
Using a slotted line, you could also measure an unknown frequency by measuring the distance
between the voltage peaks and noting that the distance is 1/2 wavelength.
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Let's look at a slotted line measurement. Suppose you recorded the detected voltage along a 25
centimeter slotted line. The data might look like this:
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In this case, the peak voltages are about 1.3 volts and the nulls are about 0.7 volts. That's a
standing wave ratio of 1.85:1.
What else can you tell from this measurement? You can measure the frequency of the source (if
it was unknown), if you know the dielectric constant of the slotted line (1 if it is air dielectric).
The distance between nulls is a half wavelength. You should always measure between the nulls,
not the peaks, because they are much sharper and easier to discriminate in distance (in spite of
the above graph where they look the same!) From the plot, if the X-axis is in centimeters, we
could estimate a wavelength of 6.25 cm (four wavelengths in 25 centimeters). Just divide that
into the speed of light (30,000,000,000 cm/s) and you will get an answer of 4.8 GHz.
STANDING WAVE RATIO MEASUREMET
In telecommunications, standing wave ratio (SWR) is the ratio of the amplitude of a partial
standing wave at an antinode (maximum) to the amplitude at an adjacent node (minimum), in an
electrical transmission line.
The SWR is usually defined as a voltage ratio called the VSWR, for voltage standing wave ratio.
For example, the VSWR value 1.2:1 denotes a maximum standing wave amplitude that is 1.2
times greater than the minimum standing wave value. It is also possible to define the SWR in
terms of current, resulting in the ISWR, which has the same numerical value. The power
standing wave ratio (PSWR) is defined as the square of the VSWR.
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RELATIONSHIP TO THE REFLECTION COEFFICIENT
The voltage component of a standing wave in a uniform transmission line consists of the forward
wave (with amplitude Vf) superimposed on the reflected wave (with amplitude Vr).
Reflections occur as a result of discontinuities, such as an imperfection in an otherwise uniform
transmission line, or when a transmission line is terminated with other than its characteristic
impedance. The reflection coefficient Γ is defined thus:
Γ is a complex number that describes both the magnitude and the phase shift of the reflection.
The simplest cases, when the imaginary part of Γ is zero, are:
Γ = − 1: maximum negative reflection, when the line is short-circuited,
Γ = 0: no reflection, when the line is perfectly matched,
Γ = + 1: maximum positive reflection, when the line is open-circuited.
For the calculation of VSWR, only the magnitude of Γ, denoted by ρ, is of interest. Therefore,
we define
ρ=|Γ|.
At some points along the line the two waves interfere constructively, and the resulting amplitude
Vmax is the sum of their amplitudes:
At other points, the waves interfere destructively, and the resulting amplitude
difference between their amplitudes:
Vmin
is the
The voltage standing wave ratio is then equal to:
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As ρ, the magnitude of Γ, always falls in the range [0,1], the VSWR is always ≥ +1.
The SWR can also be defined as the ratio of the maximum amplitude of the electric field strength
RETURN LOSS
In telecommunications, return loss or reflection loss is the loss of signal power resulting from
the reflection caused at a discontinuity in a transmission line or optical fiber. This discontinuity
can be a mismatch with the terminating load or with a device inserted in the line. It is usually
expressed as a ratio in decibels (dB);
where RL(dB) is the return loss in dB, Pi is the incident power and Pr is the reflected
power.
Two lines or devices are well matched if the return loss is high. A high return loss is therefore
desirable as it results in a lower insertion loss. Return loss may be given a minus sign, see below.
where RL'(dB) is the negative of RL(dB).
Caution is required when discussing increasing or decreasing return loss since these terms
strictly have the opposite meaning when return loss is defined as a negative quantity.
RETURN LOSS MEASUREMENT
Return loss is a measure of VSWR (Voltage Standing Wave Ratio), expressed in decibels (db).
The return-loss is caused due to impedance mismatch between two or more circuits. For a simple
cable assembly, there will be a mismatch where the connector is mated with the cable. There
may be an impedance mismatch caused by nick or cuts in a cable. At microwave frequencies, the
material properties as well as the dimensions of the cable or connector plays important role in
determining the impedance match or mismatch. A high value of return-loss denotes better quality
of the system under test (or device under test). For example, a cable with a return loss of 21 db
is better than another similar cable with a return loss of 14 db, and so on.
Equipment required to measure return-loss of a co-axial cable:
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A co-axial cable is chosen to measure the return-loss for study purpose. Typically, for a device
or a system, return-loss is measured at the input or at the output. The following equipment are
used to measure the return loss of a co-axial cable at microwave frequencies:
a.
b.
c.
d.
e.
f.
g.
Frequency source
Network Analyzer (either a scalar network analyzer or a vector network analyzer)
Detector with calibration source.
Reflection bridge
Co-axial Short
Cable under test (this could be any device under test)
A 10dB attenuator (optional, but recommended).
VSWR Meter Operation.
VSWR Measuring Equipment and Techniques
Equipment Setup and Characteristics: A typical setup of equipment used in VSWR
measurements is shown in Figure1. The signal generator (Klystron, Gun diode, HP 8620) should
be amplitude modulated at the tuned frequency of the VSWR meter amplifier.
If the klystron of a signal generator is modulated at the grid or repeller, undesirable harmonics
and frequency modulation effects will be present. Square wave modulation reduces
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Detector Probe Penetration: A general rule in slotted line measurements is to use minimum
penetration of the detector sampling probe. The power picked up by the sampling probe causes a
distortion in the standing wave pattern. The effect on the pattern is greater as the probe penetration is increased and this may be explained by considering the probe as an admittance shunting
the line. The impedance in a standing wave pattern is greatest at a voltage maximum. A shunt
admittance such as the probe lowers these impedances, causing the measured SWR to be lower
than the true SWR. The maxima and minima are shifted from their natural position and this
effect is more pronounced at a voltage maximum than at a voltage minimum.
An exception to the minimum penetration rule occurs when it is desired to examine in
detail a voltage minimum in a high SWR measurement: For this, greater probe penetration can be
tolerated since the voltage minimum corresponds to a low impedance point in the line. This
applies only at the voltage minimum.
Use of Crystal Detectors: When a crystal detector with a matched load resistor is used, the
VSWR meter INPUT SELECTOR must be set to the XTAL 200K position to obtain accurate
square-law response. With an unloaded crystal, use the setting which gives maximum sensitivity,
usually the XTAL 200 Ohms position.
Locating Maximum or Minimum: It is more desirable to locate a voltage minimum than a
voltage maximum, since the former results in less probe loading on the line. With a low SWR,
the voltage minimum is fairly broad, and locating the exact point of the minimum is difficult
with a single measurement. A more accurate procedure is to note the probe carriage position at
two equal output readings on either side of the minimum, then average the two readings.
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Errors from Signal Sources: Signal sources may introduce some undesirable characteristics in
slotted line measurement techniques. When possible, the effect of RF harmonics, incidental FM
signals, and spurious signals should be minimized.
RF harmonics: RF harmonics are a common cause of errors in SWR measurements. These
harmonics are generally excessive only with a coaxial signal output since this type of output
often passes harmonics more efficiently than the fundamental. With coaxial systems it is
common for the harmonic standing wave pattern to be equal to or greater than the fundamental
standing wave. Thus the slotted line probe may be erroneously tuned to a second harmonic. Even
if a probe is tuned properly, the presence of harmonics may still couple sufficient energy into the
probe to give wrong readings. To minimize the possible effects from harmonics, a signal
generator used as the source should be rated low in harmonic content. Suitable low pass filters
may be inserted between the signal source and the other equipment to reduce harmonic effects.
SPURIOUS SIGNALS: In tuning a klystron, ordinarily the repeller voltage is adjusted for
maximum power at a desired frequency. Then if square wave modulation of the klystron is added
without regard to the operating voltages, the repeller voltage excursions will be incorrect. On
square wave peaks the repeller voltage will not stabilize at maximum power output but will
swing between two points which may not
lie in the desired, or even the same mode. The result is closely-spaced frequencies and the
typical, result on a standing wave pattern is illustrated in Figure 3. When square wave
modulation is used, it is necessary to adjust the repeller voltage and the modulating voltage
so the modulating peaks will lie at the maximum of the desired repeller mode and the modulating
troughs do not lie in a mode. It is also desirable that the rising and falling portions of the
modulating voltage do not pass through undesired modes.
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LOW SWR (1:1 to 10:1) :
When measuring SWR in the low range, proceed as follows:
Turn VSWR Meter POWER switch to AC.
a- Set INPUT SELECTOR for type detector being used. The XTAL (crystal) 200K
position gives the proper load on crystal detectors for optimum square law detection.
b- Connect detector cable to INPUT.
c- Set GAIN and VERNIER controls to about 1/2 maximum.
d- Set RANGE switch to 40 db or 50 db, EXPAND switch to NORM (normal) position.
Adjust probe penetration to obtain an on-scale reading.
e- Peak the VSWR Meter (i.e. move needle to right) by adjusting the modulation
frequency of the signal source, if adjustable. Frequency peaking is also achieved by
adjusting the FREQ (frequency) control on the front panel of the VSWR meter. Reduce
probe penetration to keep needle on scale.
f- Peak the VSWR meter by tuning the probe detector, if tunable. Reduce probe
penetration to keep needle on scale.
g- Peak the VSWR meter by moving the probe along the line, reducing probe penetration
to keep needle on scale.
h- Adjust GAIN and VERNIER of VSWR meter and/or output power from signal source
to obtain a SWR reading of exactly 1.0.
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i- Move the probe along the line to a voltage minimum (needle moves to left). Do not
retune probe or detector.
j- If the indicated SWR is between 1 and 3.2, greater accuracy in a reading is possible if
the EXPAND function is used. For a SWR between 1 and 1.3, a SWR scale is provided
for a direct reading when the EXPAND switch is changed to the 0.0 position. If the SWR
is between 1.3 and 3.2, the EXPAND db scale may be used to obtain an accurate db
reading for conversion to SWR using Figure 4. For SWR between 3.2 and 10, a scale on
the meter provides the reading. Refer to Figure 6 and obtain the SWR reading as follows:
1- For a reading between 1 and 1.3 on the 1 to 4 SWR scale after step 10
above, change the EXPAND switch to 0.0. This SWR segment is then
expanded to full scale and the reading is taken from the 1 to 1.3 DB
scale.
2- If needle is between 1.3 and 3.2 after step j, note the reading on the 0
to 10 DB scale and change EXPAND switch to a position which
normalizes the beginning of the proper 2.5 db segment. For example, if
the reading is approximately 8.4 db, then change EXPAND switch to
7.5. The reading is then the sum of the needle indication on the 0 to 2.5
DB scale and the EXPAND switch setting (for example, 0.95 + 7.5 =
8.45 db). Locate this db reading on Figure 4 and note the corresponding
SWR value (8.45 db is read as SWR of 2.66).
3- After step j, if the needle is to the left of 3.2 (possibly to the left, offscale) on the 1 to 4 SWR scale, change the RANGE switch to the next
10 DB position above the initial setting. The SWR is then indicated
directly on the 3.2 to 10 db scale.
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High SWR (Above 10:1):
A reading is possible on the VSWR meter for SWR greater than 10. If
the indication is more than 10 after step j (3), change the RANGE switch to the next
10 db position (now two steps, or 20 db, above the initial setting). Read the SWR on
the 1 to 4 SWR scale and multiply the reading by ten. Here, again, a db reading may
be converted to SWR using Figure 5. The reading used for converting is the amount
of db above the initial RANGE switch setting in step 3.
Panel Descriptions -VSWR Meter
1- AC AND CHARGE: Indicator lights when POWER switch is in AC or BATTERY
CHARGE position.
2- POWER: This switch selects the desired power source to the Model 415D.
3- INPUT: BNC connector for an input from the signal source.
4- INPUT SELECTOR: The XTAL 200K or the 200 Ohms positions take an input from a
crystal detector matched for one of these impedances.
5- BIAS CHECK: Used only when a bolometer is used in place of the crystal diode.
6- FREQ: This adjustment allows variation of the center frequency of the tuned amplifier
by 5% when using 1000 cps.
7- BANDWIDTH: This adjustment changes the bandwidth: 15 cps to 100 cps.
8- RANGE: Settings of this switch provide different signal input attenuation values.
Steps are indicated in 10-db increments.
9- EXPAND: Settings of this switch allow full scale expansion of 2.5 db portions of the
10-db meter scale.
10- GAIN: This variable control allows an initial setting of a reference on the meter.
11- VERNIER: This variable control provides a fine adjustment of the initial reference.
12- METER MECHANICAL ADJUSTMENT: Adjustment screw for exact setting of
meter needle.
Here are the equations that convert between VSWR, reflection coefficient and return loss (as
well as mismatch loss which we will cover later):
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Calculating VSWR from impedance mismatches
The mismatch of a load ZL to a source Z0 results in a reflection coefficient of:
=(ZL-Z0)/(ZL+Z0)
Note that the load can be a complex (real and imaginary) impedance. If you can't remember in
which order the numerator is subtracted (did we just say "ZL-Z0" or Z0-ZL"?), you can always
figure it out by remembering that a short circuit (ZL=0) is on the left side of the Smith chart
(angle = -180 degrees) which means =-1 in this case, which means that the minus sign belongs
in front of Z0.
The magnitude of the reflection coefficient is given by:
=mag(
)
For cases where ZL is a real number,
=abs((ZL-Z0)/(ZL+Z0))
Note that "abs" means "absolute value" here. VSWR can be calculated from the magnitude of the
reflection coefficient:
VSWR=(1+ )/(1- )
For cases where ZL is real, with a little algebra you'll see there are two cases for VSWR,
calculated from load impedance:
For ZL<Z0: VSWR=Z0/ZL
For ZL>Z0: VSWR=ZL/Z0
Just remember to divide the larger impedance by the smaller impedance, because VSWR is
always greater than 1.
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POWER MEASUREMENT AT RF AT MICROWAVE
FREQUENCIES
This application note reviews the fundamentals of power measurements from DC to
MicroWave. Also covered is how test equipment, circuit, and coaxial cable interact and
influence the accuracy of power measurement.
A required increase in microwave power is expensive whether it be the output from a laboratory
signal generator, the power output from a power amplifier on a satellite, or the cooking energy
from a microwave oven. To minimize this expense, absolute power must be measured. Most
techniques involve conversion of the microwave energy to heat energy which, in turn, causes a
temperature rise in a physical body. This temperature rise is measured and is approximately
proportional to the power dissipated. The whole device can be calibrated by reference to lowfrequency electrical standards and application of appropriate corrections. The power sensors are
simple and can be made to have a very broad frequency response. A power meter can be
connected directly to the output of a generator to measure available power PA, or a directional
coupler may be used to permit measurement of a small fraction of the power actually delivered to
the load.
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In DC circuits, power measurement is relatively easy. For example, if the voltage V across and
current I through a resistance R is measured, the power is easily calculated by using the
following.
P
E2
R
I 2R
In an AC circuit containing inductance or capacitance, power measure becomes complex
because the voltage and current are out of phase. Power is calculated by using the following.
P=e
i
= e i cos
Where is the phase angle difference between e and i, and the magnitudes of e and i are the
RMS values.
At RF/MW frequencies power measurement presents more of a challenge. Instruments used to
measure voltage or current cause de-tuning of the RF/MW circuit or coaxial transmission line. A
RF voltmeter would add pico-farads of capacitance and a current measurement would add nanohenerys of inductance to the circuit or transmission line thereby making measurements
inaccurate. However, power causes the same amount of resistive component heating regardless
of source frequency (DC, 60 Hz AC, or RF/MW). Accurate power measurement of RF/MW
circuits (amplifiers, oscillators, filters, etc) is possible because of the output-matching network
used to couple the circuit to a load or coaxial cable (usually 50 or 75 ohms). Voltage and current
vary with the length of a lossless coaxial cable, but power is not a function of length of a lossless
coaxial cable.
Thermistor Mount
Power measurement at RF/MW frequencies is accomplished by connecting a load capable of
responding to the applied RF/MW power. One such RF/MW load is a small bead thermistor that
responds with a change in resistance when RF/MW power is applied. A diagram of a RF/MW
thermistor mount is shown below.
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Figure 1, RF/MW Thermistor Mount
The thermistors in the RF Bridge are biased so that their individual resistance is 100 ohms
thereby presenting a 50-ohm load to the RF/MW power because of the surface mount capacitor
connected between the DC bias line and ground. The surface mount capacitor is an open circuit
to DC, but at RF/MW frequencies the value of the surface mount capacitor is chosen so that a
low impedance (short circuit) is presented to the RF/MW signal.
Refer to Figure 2 for an explanation of the DC Bridge operation.
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Figure 2, Power Meter Block Diagram
When RF/MW power is applied to the thermistor mount through the coaxial connector,
additional thermistor heating is created and the resistance of both thermistors is lowered. To
bring the bridge back into balance, an equal amount of DC power is automatically subtracted by
the bridge and displayed on the power meter indicator as a power measurement.
Since ambient temperature changes will cause the thermistors to change resistance and create
zero drift, temperature compensation is required. A separate bridge and thermistors are
incorporated into the measurement system to respond to temperature only. With no RF/MW
power applied to the thermistor mount, the term (Vc-Vrf) = 0 over ambient temperature
variations and the meter remains zeroed.
Schottky Barrier Diode Detector
Schottky barrier diodes are also used to measure RF/MW power, but do not use the heating
effect of the applied power as an indicator. RF/MW diode detectors depend on the non-linear
diode junction to generate a DC voltage when a RF/MW signal is applied. A diagram of a circuit
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used to measure RF/MW power is shown in Figure 3. The detector diode is biased at
approximately 27 uA to lower video resistance to a point where it can drive a DC load.
Figure 3, Power Measurement using a Schottky Barrier Diode
The graph of Figure 4 shows the square law and linear region of a typical Schottky Barrier
Diode. In the square law region Vo = KPin with input RF/MW power levels between -60 and
-20 dBm. Above about -20 dBm input power level, the diode operates as a rectifier and is in the
linear region.
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Figure 4, Schottky Barrier Diode square law and linear region
To remove the DC component from the measurement and provide temperature compensation,
an additional diode of the same type as the detector is needed. With the same bias and physical
proximity, the DC level on each diode should be approximately equal. A matched diode pair is
preferred. The differential configuration shown in Figure 5 can be used to measure RF/MW
power in the -60 to -20 dbm range providing the DVM is capable of displaying a fraction of a
mille-volt.
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Figure 5, Differential RF/MW Power Measurement
A coaxial attenuator can be used to extend the upper power limit on any of the power sensors
described in this application note.
dBm
dBm (sometimes dBmW) is an abbreviation for the power ratio in decibels (dB) of the measured
power referenced to one milliwatt (mW). It is used in radio, microwave and fiber optic networks
as a convenient measure of absolute power because of its capability to express both very large
and very small values in a short form. Compare dBW, which is referenced to one watt
(1000 mW).
Since it is referenced to the watt, it is an absolute unit, used when measuring absolute power. By
comparison, the decibel (dB) is a dimensionless unit, used for quantifying the ratio between two
values, such as signal-to-noise ratio.
Unit conversions
Zero dBm equals one milliwatt. A 3 dB increase represents roughly doubling the power, which
means that 3 dBm equals roughly 2 mW. For a 3 dB decrease, the power is reduced by about one
half, making −3 dBm equal to about 0.5 milliwatt. To express an arbitrary power P as x dBm, or
go in the other direction, the following equations may be used:
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Impedance Measurement
The voltage reflection coefficient Γ is related to the impedance terminating the transmission line
and to the impedance of the line itself. If a wave is launched to travel in only one direction on a
uniform reflectionless transmission line of infinite length, there will be no reflected wave. The
input impedance of this infinitely long transmission line is defined as its characteristic
impedance Z0. An arbitrary length of transmission line terminated in an impedance Z0 will also
have an input impedance Z0. If the transmission line is terminated in the arbitrary complex
impedance load ZL, the complex voltage reflection coefficient ΓL at the termination is given by
Eq. (2).
(2)
Even when there is no unique expression for ZL and Z0 such as in the case of hollow
uniconductor waveguides, the voltage reflection coefficient Γ has a value because it is simply a
voltage ratio. In general, the measurement of microwave impedance is the measurement of Γ.
Both amplitude and phase of Γ can be measured by direct probing of the voltage standing wave
set up along a transmission line by the two opposed traveling waves, but this is a slow technique.
Directional couplers have been used for many years to perform much faster swept frequency
measurement of the magnitude of Γ, and more recently the use of automatic network analyzers
under computer control has made possible rapid, accurate measurements of amplitude and phase
of Γ over very broad frequency ranges.
Scattering coefficients Measurement
While the measurement of absolute power is important, there are many more occasions which
require the measurement of relative power which is equivalent to the magnitude of voltage ratio
and is related to attenuation. Also there arises frequently the need to measure the relative phase
of two voltages. Measurement systems having this capability are referred to as vector network
analyzers, and they are used to measure scattering coefficients of multi-port devices. The concept
of scattering coefficients is an extension of the voltage reflection coefficient applied to devices
having more than one port. The most simple is a two-port. Its characteristics can be specified
completely in terms of a 2 × 2 scattering matrix, the coefficients of which are indicated in the
illustration. The incident voltage at the reference plane of each port is defined as a, and the
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reflected voltage is b. Voltages a and b are related by matrix equation (3), where (Snm) is the
scattering matrix of the junction. Writing Eq. (3) out for a two-port device gives Eqs. (4)
(3)
(4)
(5)
and (5). Examination of Eq. (4) shows, for example, that S11 is the voltage reflection coefficient
looking into port 1 if port 2 is terminated with a Z0 load (a2 = 0).
.
A two-port inserted between a load and a generator
McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.
Dielectric Constant Measurement of a solid using waveguide
The relative static permittivity (or static relative permittivity) of a material under given
conditions is a measure of the extent to which it concentrates electrostatic lines of flux. It is the
ratio of the amount of stored electrical energy when a voltage is applied, relative to the
permittivity of a vacuum. The relative static permittivity is the same as the relative permittivity
evaluated for a frequency of zero.
The static relative permittivity is a special case of the more general relative permittivity. The
latter is denoted εr(ω) (sometimes κ or K) and is defined as
where ε(ω) is the complex frequency-dependent absolute permittivity of the material, and ε0 is
the electric constant. The former is simply the latter evaluated at the limit ω → 0:
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where εs is the static absolute permittivity.
Other terms for the relative static permittivity are the dielectric constant, or relative dielectric
constant, or static dielectric constant. These terms, while they remain very common, are
ambiguous and have been deprecated by some standards organizations.[6][7] The reason for the
potential ambiguity is twofold. First, some older authors used "dielectric constant" or "absolute
dielectric constant" for the absolute permittivity ε rather than the relative permittivity.[8] Second,
while in most modern usage "dielectric constant" refers to a relative permittivity[7][9], it may be
either the static or the frequency-dependent relative permittivity depending on context.
Relative permittivity is a dimensionless complex number. By definition, the linear relative
permittivity of vacuum is equal to 1[9], that is ε = ε0, although there are theoretical nonlinear
quantum effects in vacuum that have been predicted at high field strengths (but not yet
observed).[10]
The static relative permittivity of a medium is related to its static electric susceptibility, χe, as
εr(ω) = 1 + χe
The relative static permittivity, εr, can be measured for static electric fields as follows: first the
capacitance of a test capacitor, C0, is measured with vacuum between its plates. Then, using the
same capacitor and distance between its plates the capacitance Cx with a dielectric between the
plates is measured. The relative dielectric constant can be then calculated as
For time-variant electromagnetic fields, this quantity becomes frequency dependent and in
general is called relative permittivity.
Lossy medium
Again, similar as for absolute permittivity, relative permittivity for lossy materials can be
formulated in terms of "optical conductivity" σ (units S/m, siemens per meter) as ([12], eq.(11.61),
p. 479):
or, expanding the angular frequency ω = 2πc/λ and the electric constant ε0 = 1/(µ0c2), as:
where λ is the wavelength, c is the speed of light in vacuum and k = µ0c/2π ≈ 60.0 S−1 is a
constant (units reciprocal of siemens, such that σλk = εr" remains unitless).
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MICROWAVE FREQUENCY MEASUREMENT
Microwave frequency can be measured by either electronic or mechanical techniques.
Frequency counters or high frequency heterodyne systems can be used. Here the unknown
frequency is compared with harmonics of a known lower frequency by use of a low frequency
generator, a harmonic generator and a mixer. Accuracy of the measurement is limited by the
accuracy and stability of the reference source.
Mechanical methods require a tunable resonator such as an absorption wavemeter, which has a
known relation between a physical dimension and frequency.
Wave meter for measuring in the Ku band
In a laboratory setting, Lecher lines can be used to directly measure the wavelength on a
transmission line made of parallel wires, the frequency can then be calculated. A similar
technique is to use a slotted waveguide or slotted coaxial line to directly measure the
wavelength. These devices consist of a probe introduced into the line through a longitudinal
slot, so that the probe is free to travel up and down the line. Slotted lines are primarily
intended for measurement of the voltage standing wave ratio on the line. However,
provided a standing wave is present, they may also be used to measure the distance
between the nodes, which is equal to half the wavelength. Precision of this method is limited
by the determination of the nodal locations.
Counters and pre-scalers for direct frequency measurement in terms of a quartz crystal reference
oscillator are often used at lower frequencies, but they give up currently at frequencies above
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about 10GHz. An alternative is to measure the wavelength of microwaves and calculate the
frequency from the relationship (frequency) times (wavelength) = wave velocity. Of course, the
direct frequency counter will give a far more accurate indication of frequency. For many
purposes the 1% accuracy of a wavelength measurement suffices. A resonant cavity made from
waveguide with a sliding short can be used to measure frequency to a precision and potential
accuracy of 1/Q of the cavity, where Q is the quality factor often in the range 1000-10,000 for
practical cavities.
"Precision" and "accuracy".
Precision is governed by the fineness of graduations on a scale, or the "tolerance" with which a
reading can be made. For example, on an ordinary plastic ruler the graduations may be 1/2mm at
their finest, and this represents the limiting precision.
Accuracy is governed by whether the graduations on the scale have been correctly drawn with
respect to the original standard. For example, our plastic ruler may have been put into boiling
water and stretched by 1 part in 20. The measurements on this ruler may be precise to 1/2mm,
but in a 10 cm measurement they will be inaccurate by 10/20 cm or 5mm, ten times as much.
In a cavity wavemeter, the precision is set by the cavity Q factor which sets the width of the
resonance. The accuracy depends on the calibration, or even how the scale has been forced by
previous users winding down the micrometer against the end stop...
Wavelength measurement.
Wavelength is measured by means of signal strength sampling probes which are moved in the
direction of wave propagation by means of a sliding carriage and vernier distance scale. The
signal strength varies because of interference between forward and backward propagating waves;
this gives rise to a standing wave pattern with minima spaced 1/2 wavelength.
At a frequency of 10 GHz the wavelength in free space is 3 cm. Half a wavelength is 15mm and
a vernier scale may measure this to a precision of 1/20mm. The expected precision of
measurement is therefore 1 part in 300 or about 0.33%
The location of a maximum is less precise than the location of a minimum; the indicating signal
strength meter can be set to have a gain such that the null is very sharply determined. In practice
one would average the position of two points of equal signal strength either side of the null; and
one would also average the readings taken with the carriage moving in positive and negative
directions to eliminate backlash errors.
Multiple readings with error averaging can reduce the random errors by a further factor of 3 for a
run of 10 measurements.
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Signal strength measurement
The 10 GHz microwave signal in the waveguide is "chopped" by the PIN modulator at a
frequency of 1kHz (audio) and the square wave which does this is provided by the bench power
supply.
The detector diodes in the mounts on the wavemeter and slotted line rectify and filter this 10GHz
AM signal and return a 1kHz square wave which you can observe directly on the oscilloscope.
They are actually being used as "envelope detectors" as is the detector diode in your AM radio.
The VSWR indicator is a 1kHz tuned audio amplifier with 70dB dynamic range at least, and a
calibrated attenuator sets its gain. The meter measures the size of the audio signal at 1kHz.
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An X-band slotted line
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Another example of an X-band slotted line.
SWR tuned amplifier meter and indicator
Since the detectors are "square law" their output voltage is proportional to the square of the
microwave signal voltage. Regarded as a linear meter then, the VSWR indicator gives a
deflection proportional to the POWER of the microwave signal (V*V/Zo). That is the reason for
the curious calibration on the VSWR scales.
Half scale deflection on the VSWR meter therefore represents a microwave voltage of 1/sqrt(2)
or 0.707 of that corresponding to full scale deflection.
Moreover, the VSWR meter is calibrated "backwards" in that one sets the voltage maximum at
full scale deflection, then reads the VSWR from the voltage minimum. Thus the calibration point
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at half scale deflection is actually 1/0.707 or 1.414 VSWR. Check this. At 1/10 of full scale
deflection the VSWR calibration point is sqrt(10) or 3.16. At this point one increases the gain by
a factor of 10 with the main attenuator adjustment, and reads the VSWR scale from 3.16 to 10 on
the other half of the VSWR scale. Get a demonstrator to show you how if this isn't yet clear.
Note that the gain dB scales and the attenuator on the VSWR indicator correspond to POWER of
the microwave signal, not to POWER of the 1kHz audio input.
Measurements of impedance and reflection coefficient.
A visit to your favourite microwave book shows that a measurement of the standing wave ratio
alone is sufficient to determine the magnitude, or modulus, of the complex reflection coefficient.
In turn this gives the return loss from a load directly. The standing wave ratio may be measured
directly using a travelling signal strength probe in a slotted line. The slot in waveguide is cut so
that it does not cut any of the current flow in the inside surface of the guide wall. It therefore
does not disturb the field pattern and does not radiate and contribute to the loss. In the X band
waveguide slotted lines in our lab, there is a ferrite fringing collar which additionally confines
the energy to the guide.
To determine the phase of the reflection coefficient we need to find out the position of a standing
wave minimum with respect to a "reference plane". The procedure is as follows:First, measure the guide wavelength, and record it with its associated accuracy estimate.
Second, find the position of a standing wave minimum for the load being measured, in terms of
the arbitrary scale graduations of the vernier scale.
Third, replace the load with a short to establish a reference plane at the load position, and
measure the closest minimum (which will be a deep null) in terms of the arbitrary scale
graduations of the vernier scale. Express the distance between the measurement for the load and
the short as a fraction of a guide wavelength, and note if the short measurement has moved
"towards the generator" or "towards the load". The distance will always be less than 1/4 guide
wavelength towards the nearest minimum.
Fourth, locate the r > 1 line on the SMITH chart and set your dividers so that they are on the
centre of the chart at one end, and on the measured VSWR at the other along the r > 1 line. (That
is, if VSWR = 1.7, find the value r = 1.7).
Fifth, locate the short circuit point on the SMITH chart at which r = 0, and x = 0, and count
round towards the generator or load the fraction of a guide wavelength determined by the
position of the minimum.
Well done. If you plot the point out from the centre of the SMITH chart a distance "VSWR" and
round as indicated you will be able to read off the normalised load impedance in terms of the line
or guide characteristic impedance. The fraction of distance out from centre to rim of the SMITH
chart represents the modulus of the reflection coefficient [mod(gamma)] and the angle round
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from the r>1 line in degrees represents the phase angle of the reflection coefficient
[arg(gamma)].
Microwave waveguide benches
These demonstration benches introduce the novice student to the essentials of the behaviour of
microwaves in the laboratory. The wavelength is convenient at the operating frequency in X
band (8-12 GHz approx) The waveguide used is WG90, so called because the principal
waveguide dimension is 0.900 inches, 900 "thou" or "mils" depending on whether you are using
British or American parlance. The guide wavelength at 10 GHz in WG90 is notionally 3.98 cm
(the free space wavelength is 3 cm) so the standing wave pattern repeats at a distance of about 2
cm.
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An X-band waveguide bench.
Another X-band waveguide bench, used for transmitting.
The benches include an attenuator, and an isolator. Both of these help to stop the reflected power
from reaching the oscillator and pulling the frequency of the cavity and Gunn diode off tune
when the load impedance is varied.
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An isolator, made from a magnet and ferrite-loaded waveguide.
There is a dual directional coupler, arranged as a pair of crossed waveguides, which samples
some of the forward wave power and couples it to a calibrated cavity wavemeter for measuring
the oscillator frequency. Taken together with a measurement of guide wavelength, we have then
two independent checks on the oscillator frequency. There is also a PIN modulator which chops
the 10GHz signal at a frequency of 1KHz square wave.
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The PIN modulator, directional coupler, and part of the wavemeter scale.
The guide wavelength is an important property to be measured, and should not be changed
during the course of a series of measurements. A half guide wavelength (about 2 cm) represents
a plot of once round the SMITH chart. As remarked, we can determine the position of a
minimum to about 1/20mm precision, or about 1 degree of angle around the chart. That
represents 0.00125 lambda error in the phase plot on the SMITH chart.
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Network analyser
A network analyser makes measurements of complex reflection coefficients on 2-port
microwave networks. In addition, it can make measurements of the complex amplitude ratio
between the outgoing wave on one port and the incoming wave on the other. There are thus four
possible complex amplitude ratios which can be measured. If we designate the two ports 1 and 2
respectively, these ratios may be written s11 s12 s21 s22. These are the four "s-parameters" or
"scattering parameters" for the network. Together they may be assembled into a matrix called the
"s-matrix" or "scattering matrix".
The network analyser works on a different principle to the slotted line. It forms sums and
differences of the port currents and voltages, by using a cunning bridge arrangement. The phase
angles are found by using synchronous detection having in-phase and quadrature components.
From the measured voltage and currents it determines the incoming and outgoing wave
amplitudes. As we recall from elsewhere in the notes, V+ = (V + ZoI)/2 and V- = (V - ZoI)/2.
The Network analysers can be automated and controlled by computer, and make measurements
at a series of different frequencies derived from a computer controlled master oscillator. They
then plot the s-parameters against frequency, either on a SMITH chart or directly.
important experimental technique to the use of a network analyser lies in the calibration
procedure. It is usual to present the analyser with known scattering events, from matched
terminations and short circuits at known places. It can then adjust its presentation of s-parameters
for imperfections in the transmission lines connecting the analyser to the network, so that the
user never has to consider the errors directly providing he/she can trust the calibration procedure.
It is even possible to calibrate out the effects of intervening transmission components, such as
chip packages, and measure the "bare" s-parameters of a chip at reference planes on-chip.
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A network analyser SMITH chart plot of a dipole
The Measurement of Co-axial cable losses :
The measurement process consists of calibrating the test set-up for insertion and return-loss. If
you have dual channel network analyzer, both insertion and return losses can be measured
simultaneously. You can also measure insertion and return losses separately as is done here.
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INSERTION LOSS MEASUREMENT
Step 1. Set the sweep source to the required frequency range. Make sure that the
output of the sweep source is within the desired amplitude limits, otherwise, it
may saturate the detector head and any measurements taken would not be
accurate. You may use an attenuator at the output of the sweep source to mitigate
any problem that may arise due to mismatch between the cable under test and the
sweep source. It is recommended to use a 10 dB attenuator for this purpose.
For example, you can set the values as below:
Sweep frequency (measurement frequency): 100MHz – 2.3 GHz
Sweep power: 12 dBm
Note: Also, make sure that you are measuring same impedance. For example, if
the cable is 50Ohm, and the Sweep generator output is 75 Ohm, you need to use a
50 to 75 Ohm impedance matching device.
Step 2: Calibrate the test system by connecting as shown in the figure 1,
bypassing the cable under test. Calibration is nothing but setting a reference line
taking all stray measurement errors into consideration.
Step 3: Now you have done the equipment calibration, connect the cable as
shown in the figure 2, without disturbing any other parameters such as sweep
power output or the attenuator value. The trace in the network analyzer display
now shows the Insertion loss of the cable against the frequency.
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Return loss Measurement:
Step 1. The sweep source is already set during insertion loss measurement. You
may use an attenuator at the output of the sweep source to mitigate any problem
that may arise due to mismatch between the cable under test and the sweep
source. It is recommended to use a 10 dB attenuator for this purpose.
For example, you can set the values as below:
Sweep frequency (measurement frequency): 100MHz – 2.3 GHz
Sweep power: 12 dBm
Note: Also, make sure that you are measuring same impedance. For example, if
the cable is 50Ohm, and the Sweep generator output is 75 Ohm, you need to use a
50 to 75 Ohm impedance matching device.
Step 2: Calibrate the test system by connecting as shown in the figure 3,
bypassing the cable under test. Calibration is nothing but setting a reference line
taking all stray measurement errors into consideration. You need to short the
bridge port as shown in the figure. For better accuracy, Open/Short method is can
be used. In Open/Short method, you calibrate for the system by using both an
Open and a Short instead of only a Short used in this example.
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Step 3: Now you have done the equipment calibration, connect the cable as
shown in the figure 4, without disturbing any other parameters such as sweep
power output or the attenuator value. The end of the cable needs to be terminated
with a 50 Ohm termination.
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The Graph:
The trace in the network analyzer display in figure 5 shows the Insertion Loss and
Return Loss of the cable against the frequency. Note that the Insertion Loss is
typically low in the desired band of frequencies, and the Return Loss is high.
Typically, Insertion loss will of a fraction of a db (for co-axial cable) and the
Return loss is 10 dB or more.
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