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EM Theory
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FACULTY OF ENGINEERING
LAB SHEET
ELECTROMAGNETIC THEORY
EMG2016
TRIMESTER I (2012/2013)
MW1 – MICROWAVE FREQUENCY AND SWR MEASUREMENTS
*Note: On-the-spot evaluation may be carried out during or at the end of the
experiment. Students are advised to read through this lab sheet before doing
experiment. Your performance, teamwork effort, and learning attitude will count
towards the marks.
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EXPERIMENT MW1:
MICROWAVE FREQUENCY AND SWR MEASUREMENTS
OBJECTIVES:
1.
2.
3.
4.
5.
To trace out the standing wave pattern developed along a waveguide.
To make direct SWR measurements using an SWR meter.
To make indirect SWR measurements using the Double Minimum Method.
To make a direct frequency measurement using a cavity wavemeter.
To make an indirect frequency measurement via a waveguide wavelength
measurement.
APPARATUS:
Klystron Source
Klystron Power Supply
Cavity Wavemeter
Isolator
Slotted-line probe
SWR meter
Variable attenuator
Short-circuit plane
BNC coax cable x 1
INTRODUCTION:
At radio frequencies (RF) and microwave frequencies, signals must be viewed
as waves instead of currents and voltages. One of the reasons is the wavelength
becomes very short that the current at one point of a long conductor is not the same as
that at another point. As a result, the ordinary Circuit Theory can no longer be used for
the analysis of transmission lines that have a physical dimension greater than 1/10 of
the signal wavelength. This kind of problems can be solved using Electromagnetic
Theory.
The electromagnetic field at any point on a transmission line may be
considered as the sum of two traveling waves: the incident wave propagates from the
generator and the reflected wave from the load. These give rise to a standing wave
along the line due to constructive and destructive interference, as shown in Fig. 1. The
electromagnetic field strength varies periodically with distance. The maximum field
strength Emax is found where the two waves add in phase, and the minimum Emin
where the two waves add in opposite phase. The Voltage Standing Wave Ratio
(VSWR or just SWR) is defined as the ratio of the maximum to the minimum field
strength along the line, as given in Eq. (1). It is a parameter used to describe the size
of standing wave or the strength of reflected wave relative to that of incident wave.
The SWR will be higher (i.e. larger peaks and nulls in the standing wave pattern) if
there is more wave power reflected from the load end. The magnitude and phase of the
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reflected wave are related to the load impedance which is generally a complex value.
Thus, the SWR and the positions of constructive and destructive interference depend
upon the load impedance.
Power
Emax2
Emin2
Distance
g / 2
Fig. 1: Standing wave pattern
For a waveguide system, the SWR can be measured using a slotted-line
method. A short length of wire, acting as a probe, protrudes into the waveguide
through a longitudinal slot to couple to the electric field confined within the
waveguide. A small fraction of power is picked up and detected by a diode detector
connected to the other end of the probe. By sliding the probe along the longitudinal
slot, the maximum power Pmax and minimum power Pmin can be measured. The SWR
is then given by
SWR 
E max
Pmax

Pmin
E min
or
20 log 10 SWR  Pmax (dBm)  Pmin (dBm)
(1)
(2)
In normal operation, the diode detector gives a voltage which is proportional to
the electromagnetic wave power (or square of the electric field strength). This is the
case if the power is small enough to allow the diode detector to work in its “squarelaw” region. The probe must not protrude too deep into the waveguide so that the
power picked up by the probe is small. This is also necessary to ensure that the field
distribution in the waveguide is not significantly disturbed.
An SWR meter can be used to give direct readings of SWR. A simple one
consists of a power meter with SWR scales on it. One way to establish an SWR scale
on a power meter is to make Pmax a known constant using a variable gain amplifier. A
convenient constant for this purpose is the full scale value of the power meter. With
Pmax made constant, there is a one-to-one conversion between Pmin and SWR, and
SWR values can be marked on the power meter in place of the power values. Fig. 2
shows an example SWR meter. There is a power scale and an SWR scale on the meter.
To use the SWR scale, one has to first set Pmax to the full scale value, 1 W in the
example, with the aid of a variable gain amplifier. Then, Pmin is measured. The SWR
will be
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SWR 
Pmax

Pmin
4
1
(3)
Pmin
Note that this is exactly the relationship between the power and SWR scales. In other
words, the SWR can be read directly from the meter at the point of Pmin, no further
division is necessary.
0.4
0.6
0.2
0.8
1.0
0
1.6
1.3
2.2
Power (W)
SWR
1.1

1.0
Fig. 2: An example SWR meter
For the case where the SWR is high (for example, SWR > 3), the null points of
the standing wave pattern along the waveguide may have very low field strength. The
probe depth can be increased so that a sufficiently high meter deflection can be
obtained at the minimum power point. However, this will deform the field when the
probe is moved to the maximum power point. Also, the large electric field strength at
the maximum point may cause the diode detector to operate out of the “square-law”
region. The above problem can be overcome using the “Double Minimum Method”.
In this method, one measures the distance between the points where the detector
output is double the minimum field strength, as illustrated in Fig 3. It can be shown
that the SWR is given by:
SWR 
E max
1
 1
2
E min
sin d / g


Power
d
Emax2
2Emin2
Emin2
3 dB
Distance
g / 2
d1
d2
Fig. 3: The Double Minimum Method
(4)
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A wavelength is defined as the distance between two successive points with
the same phase on a waveform. An electromagnetic wave is reflected from one
waveguide’s wall to another as it propagates along a waveguide As a result, the
apparent longitudinal wavelength is different from the free-space wavelength, as
shown in Fig. 4. The relationship between the apparent longitudinal wavelength or
simply waveguide wavelength g and the free space wavelength o is given by:
1

2
o

1

2
g

1
(5)
c2
where c is the cutoff wavelength for the particular mode of propagation. For an
electromagnetic wave with freespace wavelength equals to c, g will be infinitely
long, which means that no field variation occurs along the waveguide. A wave with
freespace wavelength equals or larger than c would not be able to propagate through
the waveguide (in the particular mode of propagation) and therefore c is called the
cutoff wavelength (for the particular mode of propagation). For the usual TE10
dominant mode operation of a rectangular waveguide, c = 2a with ‘a’ being the
broad-face dimension of the waveguide.
waveguide’s wall
equiphase
wavefront
o
plane wave
propagation
g
the longitudinal
direction
Fig. 4: Waveguide wavelength
Eq. (5) provides an indirect way of measuring the signal frequency. The
waveguide wavelength g can be measured as twice the distance between two
successive minimum points of the standing wave pattern. Using Eq. (5), the free space
wavelength can be calculated. The frequency f can be determined using the
relationship u = fo, where u is the speed of light in the waveguide medium. For an
air-filled waveguide, u = 3  108 m/s.
Frequency can also be measured using a variable size resonant cavity known
as cavity wavemeter. A resonant cavity is made of a rectangular or circular waveguide
with both ends short-circuited. A cavity wavemeter is coupled to a waveguide by a
small hole. Because the hole is small, the cavity wavemeter can normally absorb only
a tiny fraction of the energy propagating in the waveguide. At a certain cavity size, a
standing wave can be set up and sustained between the cavity walls. When this
happens, the cavity will absorb a larger fraction of power from the waveguide system,
which results in a progressive building up of the the standing wave within the cavity.
EM Theory
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This in turn causes a small dip in the power in the down-stream of the waveguide
system. In this condition, the cavity is said to be in resonance with the signal
frequency. The resonant frequency can be calculated by solving Maxwell’s equations
with the dimensions of the cavity defining the boundary conditions. Varying the cavity
size will alter the resonant frequency. Some cavities are marked in frequency for
direct measurement reading. Others come with calibration charts so that the frequency
can be read off the charts. It is important to note that resonant frequency is not unique.
There are numerous resonance modes, each gives rise to a different resonant
frequency. A solution to this problem would be to use only the dominant resonant
mode (lowest resonant frequency), to search for the resonance from the smallest cavity
size and gradually increasing its size, and to read from the first resonance occurrence.
In the following experiments, you will trace out the standing wave pattern
developed along a waveguide. You will make direct and indirect SWR measurements
using an SWR meter and the Double Minimum Method, respectively. Also, you will
make direct and indirect measurements of the signal frequency using a cavity
wavemeter and Eq. (5), respectively.
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PREPARATION
 Conduct background study about the experiment, e.g. read the lab sheet.
 Bring log book for recording observations and results.
 Bring scientific calculator.
PROCEDURE:
CAUTION 1: The RF power levels in the following experiments are not
harmful, but a human eye may be damaged by low level of radiation. Do not
look into the waveguide at any time when the equipment is on.
CAUTION 2: Klystron tube get extremely hot when it is operated and must
not be handled by hand.
A) Initial Setup (30 minutes)
The experimental setup (see Fig. 5) is rather simple. It mainly consists of a source
(Klystron source), a transmission line (slotted-line probe), and a variable load
(variable attenuator terminated in a short-circuit). An isolator is used to protect the
source from reflected power. The cavity wavemeter is for direct frequency
measurement. Further details about the components are given in the Appendix. Note
that the SWR meter used in the setup is a power meter with SWR scales on it and it
has a built-in variable gain amplifier. Power increases from the left to the right side of
the scales.
A primary objective of this initial setup is to tune the components for maximum signal
strength at the SWR meter. This includes tuning the Klystron source’s repeller voltage
for maximum output power, the tuning gear and the source’s modulation frequency
and amplitude for maximum reception at the SWR meter.
A1. Set up the equipment as shown in Fig 5. (already done by the lab technician but
do verify). Caution: Make sure the SWR meter input setting is ‘XTAL
LOW’.
SWR meter
Cavity
Wavemeter
Variable
attenuator
Klystron
Source
Isolator
Slotted-line
Probe
Fig 5: Setup for slotted-line measurement.
short-circuit
plane
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A2. Set the slotted-line probe’s depth to 0.5 mm (preset by the lab technician).
A3. Set the attenuator to > 2.8 mm (> 20 dB return loss).
A4. Make sure that the SWR meter is in the normal deflection mode (set by the
smaller dial at and on top of the range-gain knob).
A5. Switch on the Klystron power supply (with internal modulation active) and
SWR meter.
A6. Adjust the SWR meter’s gain until a significant deflection appears on the meter.
Note that the SWR meter has 3 gain knobs: range-gain, coarse vernier gain and
fine vernier gain.
A7. Adjust the repeller voltage knob for maximum reception at the SWR meter*
A8. Adjust the tuning gear of the slotted-line probe for maximum reception at the
SWR meter*.
A9. Adjust the modulation frequency and amplitude for maximum reception at the
SWR meter*.
A10. At this stage, the SWR meter should give a good deflection at gain settings of
30 to 40 dB. If necessary, increase the slotted-line probe’s depth to achieve the
stated condition. Make sure that a full deflection can be obtained by adjusting
the vernier gain knob.
A11. Make sure that the cavity wavemeter dial is set to maximum clockwise.
* As one is tuning towards the maximum, the SWR meter will indicate that power is
increasing. As one’s tuning reaches the maximum and then moves away from the
maximum, the SWR meter will indicate that power reaches a maximum and then
decreases. Adjust the SWR meter’s gain if necessary to have a decent deflection on
the meter and to avoid overdriving the meter.
Note:
 Do not disturb the repeller voltage, slotted-line probe’s depth, and tuning gear
beyond this point. Else, you might have to repeat the above.
 For slotted-line measurements below, make your measurements at the central part
of the waveguide; avoid the two ends of the waveguide.
 All measurement readings have to be in the correct precision. Please make it a
habit.
 Make your own tables of observations. Make it a habit too.
Evaluation A: (cognitive – knowledge, level 1) [10 marks]
i. Please make sure you know all components of the setup and their functions. You
may consult the lab instructor if necessary.
ii. Request for evaluation when you are ready.
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B) Trace Out The Standing Wave Pattern (40 minues)
B1. Set the attenuator’s micrometer setting to 1.1 mm (this corresponds to about 6
dB return loss, i.e. reflected power is 6 dB below the incident power).
B2. Move the probe along the slotted-line waveguide to locate the point of
maximum power. Henceforth, for convenience, it will be referred to as the
maximum point (or minimum point for minimum power).
B3. Adjust the gain of the SWR meter so that its reading is 0 dB on the bottom scale
(i.e. full scale deflection). Do not overdrive the SWR meter.
B4. Move the probe along the slotted-line waveguide to measure and record the
relative power at numerous points over a 50 mm distance along the waveguide.
Do record both the relative power and position. Do record enough points to
capture variation in the relative power. Relative power can be read from the
SWR meter as the negative of that given by the bottom scale, e.g. relative power
is -10 dB if the bottom scale reads 10 dB. Obviously, the SWR meter gain
should not be disturbed throughout the measurement.
B5. Sketch a plot of relative powers versus positions. Yes, sketch now.
Evaluation B: (psychomotor – complex overt response, level 5) [20 marks]
i. Tabulate your measurements, sketch the plot and analyze the results.
ii. Be ready to demonstrate how the measurements were done.
iii. Request for evaluation when you are ready.
C) Measure SWR Using an SWR Meter (30 minutes)
C1. Set the attenuator’s micrometer setting to 1.5 mm (about 12 dB return loss).
C2. Move the probe along the slotted-line waveguide to locate the maximum point.
C3. Adjust the gain of the SWR meter so that it has full scale deflection at the
maximum point (remember? 0 dB on the bottom scale).
C4. Move the probe to locate the minimum point. Evaluate the SWR from the SWR
meter**. [10 marks]
C5. Set the attenuator to 1.1 mm (about 6 dB return loss). Repeat steps C2. to C4.
[10 marks]
C6. Set the attenuator to 0.5 mm (about 2 dB return loss). Repeat steps C2. to C4.
[10 marks]
** There are 5 scales on the SWR meter. From the top, the first two are meant for the
expand mode (Not used. Remember? We use normal mode). Third and fourth are
linear scales. The third scale is used when the maximum and minimum are
observed using the same gain setting. The fourth scale is used when the minimum
is observed using a range-gain 10 dB higher than that of the maximum. The fifth
or the bottom scale is a log scale (dB). Difference in range-gain between the
maximum and minimum should be added to the reading on the log scale to obtain
the SWR in dB.
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Evaluation C: (psychomotor – complex overt response, level 5) [20 marks]
i. Tabulate your measurements and analyze them.
ii. Be ready to demonstrate how the measurements were done.
iii. Ask for evaluation when you are ready.
D) Measure SWR Using The Double Minimum Method (40 minutes)
D1. Set the attenuator to 0 mm (about 0 dB return loss).
D2. Move the probe along the waveguide to locate two successive minimum points.
Adjust the SWR meter gain if necessary to have a decent deflection at the
minimum points. Record the positions of these two points (x1 and x2).
Determine the waveguide wavelength g (see Introduction). [10 marks]
D3. Set the attenuator to 1.1 mm (about 6 dB return loss).
D4. Move the probe along the waveguide to locate the minimum point.
D5. Adjust the SWR meter gain to obtain a reading of 3 dB on the bottom scale (the
power is 3dB below that at 0 dB).
D6. Move the probe to the left of the minimum point until the detected power
increases to 0 dB. Record the probe position, d1. You see, this point has power 3
dB more than the minimum. [5 marks]
D7. Move the probe to the right of the minimum point. Record the position d2 where
the meter reading is 0 dB. [5 marks]
D8. Evaluate the SWR using Eq. (4). [10 marks]
D9. Set the attenuator to 0.5 mm (about 2 dB return loss). Repeat steps D4. to D8. .
[20 marks]
Evaluation D: (psychomotor – complex overt response, level 5) [30 marks]
i. Tabulate your measurements and analyze them.
ii. Be ready to demonstrate how the measurements were done.
iii. Ask for evaluation when you are ready.
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E) Frequency Measurement (20 minutes)
E1. Using the waveguide wavelength g from the above experiment, calculate the
free-space wavelength o using Eq. (5). The inner dimensions of the waveguide
is given as a=2.2870cm and b=1.0160cm. Calculate also the frequency fo.
E2. Set the attenuator to > 2.8 mm (> 20 dB return loss).
E3. Adjust the SWR meter gain to have a decent deflection on the meter.
E4. Starting with the cavity wavemeter dial set to maximum clockwise, unscrew the
dial slowly. Initially there will be little effect on the SWR meter reading. At one
point, cavity resonance causes a noticeable dip*** in signal power (so is the
meter reading). Read the resonant frequency fo’ of the cavity wavemeter at this
condition.
***
In some cases, the power dip is small. Though small, the dip is noticeable if
monitored closely. A quick approach to locate the dip is to rotate the
wavemeter at moderate speed while closely monitoring the meter deflection.
When a suspect dip is found, rotate the wavemeter forward and backward
about the suspect dip to verify.
Evaluation E: (psychomotor – complex overt response, level 5) [20 marks]
i. Tabulate your measurements and analyze them.
ii. Be ready to demonstrate how the measurements were done.
iii. Ask for evaluation when you are ready.
Evaluation F: (cognitive – analysis, level 4) [100 marks]
i. Consider that Experiment D was conducted to demonstrate the ability of the
Discussions
Double Minimum Method to measure SWR. Write a lab report for Experiment D.
The lab report
shouldmeasurements
include
1. Compare
the SWR
you obtained from the SWR meter and that
introduction
section
to explain the problems of using an SWR meter
froma.theAn
Double
Minimum
Method.
the need
for an
alternative
method.
[20 marks](see Appendix). Discuss
2. Estimateandreturn
losses
from
the SWR
measurements
b. Athe
formulation
explain
Double with
Minimum
Method
and its
whether
estimatedsection
return tolosses
are the
consistent
the given
attenuator
settings.advantages over using an SWR meter. [20 marks]
c. An
experimental
setup
to explainMethod
how the
experiment
was
3. Discuss
whether
you can use
the section
Double Minimum
to measure
the SWR
conducted.
[20setting.
marks]
at 1.5 mm
attenuator
d. A the
results
and discussions
section
4. Compare
frequency
measurements
fo and to
fo’ .present and discuss the results.
the results
with inthose
meter, i.e. those
of
5. DiscussCompare
the sensitivity
of error
fo tousing
error anin SWR
slotted-line-probe
position
Experiment C. [30 marks]
measurements.
A conclusion
section to conclude the experiment. [10 marks]
6. And,e.other
relevant discussions.
ii. Do not copy anything from this lab sheet. Plagiarism is strictly prohibited.
iii. Submit the lab report to the lab technician within 14 days from the day the
experiment was conducted. Late submissions are subjected to penalty.
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APPENDIX
Isolator
An isolator is a non-reciprocal device, which attenuates waves propagating in one
direction and on the other hand causes negligible attenuation for waves propagating in
the opposite direction. Such a device is, therefore useful in transferring power from a
microwave generator to the load with negligible loss and at the same time ensuring all
reflections from the load end are completely absorbed by the time they reach the
generator. This is important to ensure that the frequency stability and output power
level are not affected by the load variation.
The isolator must be connected in the proper direction. An arrow, which indicates the
forward (lossless) direction, usually comes with the label on the device.
Variable Attenuator
An attenuator can be constructed by placing a resistive card (made of lossy material)
at the centre of a waveguide section at which the electric field strength is the highest
for TE10 mode. A fraction of the microwave will be absorbed, with the remaining
power transmitted. The resistive card is tapered at both ends to maintain a low VSWR.
A variable attenuation can be realised by varying the position of the resistive card
across the waveguide broad dimension. If the device is terminated with a short-circuit
plane, a variable load impendence can be emulated for which the return loss is equal
to twice the attenuation level (in dB). For example, if attenuation is 3 dB, then
Return loss   20 log   6dB
where  is the reflection coefficient
  0.50 .
And,
SWR 
1   1  0.50

 3.01
1   1  0.50
wave propagation
Tapered resistive card
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Cavity Wavemeter
A cavity resonator is a rectangular or circular waveguide with both ends shortcircuited. A given resonator can have an infinite number of resonant modes, and each
mode corresponds to a definite resonant frequency. The mode having the lowest
resonant frequency is known as the dominant mode. When the frequency of the wave
signal is equal to a resonant frequency, a standing wave will be set up in the cavity.
high-order
mode
moving
plunger
dominant
mode
The cavity resonator can be used as a frequency meter. A moving plunger in the cavity
gives a variable resonant frequency. If the cavity is coupled to a waveguide system, a
small amount of energy will be extracted through the coupling hole only if the
resonant frequency is tuned to the frequency of the waves. At other frequencies, it will
have no effect. An absorption-type cavity wavemeter acts by absorbing a small
amount of the microwave power at its resonant frequency causing a small dip in the
indicated power in the waveguide system. The transmission-type wavemeter acts as a
high-Q bandpass filter. The power extracted by the cavity is coupled to an output
waveguide or detector. Calibration against some standard frequencies is usually
performed and a calibration chart or a direct-reading scale can be produced to relate
the cavity size to resonant frequency.
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Slotted-line Probe
The electromagnetic field at any point along a transmission line may be considered as
the sum of two travelling waves; the incident wave generated by the source, and the
returned wave reflected by the load. The interference between the two travelling
waves gives rise to a standing wave. The maximum field strength is found where the
two waves add in phase, and the minimum is encountered where the two waves add in
opposite phase. The distance between two successive minimum field points
corresponds to one half the guide wavelength g. The Voltage Standing Wave Ratio
(VSWR) can be determined by taking the ratio of the maximum field strength Emax
over the minimum Emin .
Variable shortcircuit tuner
slotted
waveguide
A
SWR
Meter
The electric field strength along the waveguide can be probed using a slotted-line
section. A short length of wire (electric probe) is inserted so as to couple to the
electric field in the interior of the waveguide. The probe is mounted on a carriage,
which slides along the slot. The field ‘picked up’ by this probe may be detected by a
diode detector. By moving the probe along the slot, a sampling of the E-field variation
can be performed. Usually the probe protrusion into the waveguide is adjustable so
that coupling is variable. The deeper the protrusion, the larger will be the detected
output. However, the shortest length of the probe is preferred because the deeper the
protrusion the greater will be the perturbation of the probe to the fields within the
waveguide. In some designs, a tunable stub assembly is cascaded with the probe so as
to maximise the detector output. In addition, this tuning also serves to minimise the
small loading effect (i.e. shunt capacitance and resistance) of the probe on the
transmission line circuit.
Microwave Diode Detector
When microwave voltage is applied across a diode (pint-contact diode or Schottkybarrier diode), a DC current is produced. If the microwave is AM modulated, the
output current will be the modulating signal with a frequency span from DC up to a
few MHz. The current is proportional to the microwave power. This is true if the
diode is operating in the ‘square law’ region of the non-linear I-V characteristic curve
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at which the incremental current I is proportional to the square of the incremental
voltage V across the diode. The applied signal level must be low enough to prevent
saturation. Otherwise, the diode will operate in the linear region and the current will
instead be proportional to the applied voltage. In order to obtain maximum output, the
impedance of the detector should be matched to the characteristic impedance of the
transmission line. In the case of a standing wave indicator, this is usually
accomplished by using a movable coaxial short circuit known as the stub the matching
tuner.
SWR Meter
The output from a diode detector is usually very low in amplitude. A high gain tuned
amplifier is incorporated in the SWR meter to amplify this detected signal prior to the
displaying meter. The bandpass response of the tuned amplifier shall be set at the
modulating signal frequency range. Tuning of the centre frequency and the bandwidth
may improve the signal selectivity.
Square law characteristic ( I o  KVs2 ) of the detector is usually assumed, and the SWR
meter scales are then calibrated such that the needle deflection is proportional to the
microwave power ( P  Vs2 R  K ' I o ). The gain of the amplifier may be adjusted
subject to the range of signal level being measured so that the reading is within the
meter scales.
Klystron Source
There are basically two types of Klystron tubes: reflex-type and single-transit-type.
The reflex Klystron is generally used in microwave oscillators. A highly focused beam
of electrons passing through the cavity gap will be accelerated or decelerated by the
induced current on the cavity. The alternate action of increasing and decreasing the
electron velocity will modulate the electron beam into varying dense and sparse of the
electrons referred to as electron bunches. This electron bunching mechanism is also
called intensity modulation or velocity modulation. At the exit chimney of the cavity
gap, a high negative electrostatic potential on the repeller will stop the movement of
the electrons, turn them around and send them back through the cavity gap. As the
repelled electrons re-enter the cavity, they give up their energy. Their volume charge
due to bunching will be phased such that it reinforces the next wave of bunched
electrons. As a result, this energy serves as a regenerative feedback to sustain the
oscillation at the resonant frequency of the cavity. This is the case if the repeller
voltage is set such that the electrons complete their travel through the gap, turn around,
and back through the gap in three-quarter cycle of the RF waveform. In other words,
the Klystron tube will oscillate only if the repeller voltage is properly set within
certain voltage ‘windows’. Switching the repeller voltage in and out of the operating
region will produce an over-modulated AM waves in the form of the square pulses.
The oscillation frequency may be altered by turning the compression set screw that
changes the cavity size.
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