Radiofrequency measurements
Literature: Agilent ”Back to Basics”, Tutorials at the Agilent’s web site:
www.agilent.com
2.1 DC, AC, and RF signals. Impedance matching.
Why do we need radio frequencies (RF)?
Not only because of radio and TV. There are several other reasons:
- as we have seen from the previous lecture, the flicker noise is getting smaller at
higher frequencies, which allows more sensitive measurements,
- filtering and frequency conversion is much easier at RF,
- RF signals can be transmitted through the air in the form of electromagnetic
waves over very long distances,
- fast modulation of the RF carrier allows very fast data transfer,
- energy of RF quantum is proportional to the frequency, which makes much
easier detection of higher frequency radiation and because of that
- numerous methods of magnetic resonance (ESR, NMR) utilize radio and mm-
wave frequency reaching very high sensitivity and resolution,
- most accurate clocks use transitions between internal atomic levels in the
frequency range between RF and optics,
- and many other reasons…
If we apply a constant voltage to the transmission line, after some short time its all
parts will be at the same potential, and a direct current (DC) will flow from the source to
the meter. What happens if we apply an alternating current (AC) to the conductor or the
transmission line? The answer depends on the frequency of applied signal. At low
frequencies where the wavelength of the signals is much larger than the length of the
circuit conductor, AC travels down the wire same way as the DC, and voltage and current
are the same no matter where we measure along the wire. Figure 2.1 represents
schematically the electromagnetic spectrum with notations adopted for different
frequencies/wavelengths.
Fig. 2.1 Electromagnetic spectrum
At high frequencies however, the wavelength of signals of interest are comparable to
or much smaller than the length of conductors. In this case power transmission can be
best thought in terms of traveling waves. In this case we have to take into account a
characteristic impedance of the transmission line. For a short piece of line having series
resistance R and inductance L, parallel conductance G and capacitance C, the
characteristic impedance Z if found according the formulae:
R i L
Z (2.1),
G i C
is in general a complex number, depending of the signal frequency. In many cases the
reactive resistance R and conductance are much smaller than the active values, which
gives frequency independent real value of impedance of the transmission line
L
Z (2.2)
C
When the transmission line is terminated to its characteristic impedance Z0, no
reflections appear and maximum power is transmitted to the load.
Zs = Zo
Zo
When the termination is not Z0, the portion of the signal which is not absorbed by the
load will reflect back toward the source. This creates a condition, where the envelope
voltage along the transmission line varies with position. In case when the end of the
transmission line is shorted (no voltage across the short), the reflected voltage must be
equal in voltage amplitude to the incident wave, and be 180 degrees out of phase with it.
For the open end of the transmission line total current at the end must be zero. This
implies that the current in the reflected wave has to be 180 degrees out of phase of the
incident current and equal in amplitude. For both the short and open cases, a standing-
wave pattern will be set up on the transmission line:
Zs = Zo
The peaks and zeroes of the short and open line will be shifted in position along the
line with respect to each other, and the peak voltage will be twice of the signal voltage.
If the line is terminated with the impedance with some resistor, say 25 ohm, part of
the power will be reflected, part absorbed in the load. Then the valleys of the standing
wave pattern will no longer be zero, and the peaks will be less than that of the short/open
case.
Zs = Zo
ZL = 25
As we se from the examples above, at radio frequencies (RF) devices behave
differently. The transfer of RF power from the source to the load is characterized by the
reflection coefficient:
V Z Z0
reflected L ei (2.3)
Vincident Z L Z 0
The standing wave pattern formed as result of interference of the incident and
reflected waves is typically characterized by the Voltage Standing Wave Ratio (VSWR):
V 1
VSWR m ax (2.4)
Vm in 1
Since some part of the power is reflected from the source, some is transmitted, we
also define the transmission parameters, transmission coefficient T:
V
T transmitted ei , (2.5)
Vincident
and Gain of the device under test (DUT):
Gain [db]= 20 log (2.6)
V Incident V Transmitted
DUT
V Reflected
So far we considered the simplest case of the two port device, when the RF signal
sent to one port can be partially reflected or transferred further. However there are more
complicated RF devices, which have more than two ports. In this case ist is more
convenient to characterize them with the use of so-called scattering or S-parameters. For
the two port device considered above S parameters are defined according to the following
notations:
Incident S 21 Transmitted
a1
S11 b2
Reflected DUT
S22
Port 1 Port 2 Reflected
b1
a2
Transmitted S12 Incident
b1 = S11a1 + S12 a 2
b 2 = S21 a1 + S22 a 2
Here we have the incident (a1 and a2) and output (b1 and b2) voltages of the traveling
waves applied to the corresponding ports. S11 and S21 are determined by measuring the
magnitude and phase of the incident, reflected and transmitted voltage signals when the
output is terminated in a perfect Z0 (a load that equals the characteristic impedance of the
test system).
Although, the ratio of voltages is used in the definitions of the quantities above, it is
more conventional to measure them in logarithmic units, as in the latter equation. Note
that multiplying the logarithm by 20 is equivalent to calculating the ratio of powers
(voltage squared). This is another convention adopted in the RF devices and
measurements: all gains and losses of the amplifiers, attenuators and other devices are
referred to the ratios of powers. The power of RF signals is usually measured in special
units called dbm, which is the ratio to 1 mW expressed in decibels:
Measured power
1 dbm 10 log
1 mW
For example, -30 dbm=10-3 mW=1 µW.
2.2 Frequency conversion. Mixing, heterodyning.
These terms describe the process by which two signals (typically at different
frequencies) are applied to a non-linear device (diode) to obtain a third signal. A basic
mixer has an RF (ωR) and an LO (local oscillator at ωL) inputs.
Fig. 2.2.1 Frequency conversion by a typical mixer.
The voltages applied to these ports VRsin ωR t + VLsin ωL t are summed at the diode. An
I-V curve of the diode can be represented as
I a0 a1V a2V 2 ... anV n (2.2.1)
From the quadratic term we obtain
VR2sin2ωR t + VLsin2 ωL t + 2VRVLsin ωR t sin ωL t (2.2.2)
The last term of the latter equation is converted as
2VRVLsin ωR t sin ωL t = VRVL[cos(ωR- ωL)t + cos(ωR- ωL)t], (2.2.3)
i.e. contains sum and difference of the frequencies applied to the inputs. Usually, the
diodes are chosen so that the higher order terms do not contribute much to the output, and
there are basically only frequencies ωR, ωL and ωR± ωL present at the output. Usually the
LO signal is very strong compared to the RF, and the latter component (ωR) can be
neglected at the output. Finally we end up with the carrier, or LO frequency ω L and two
sidebands ωR± ωL. A band pass filter installed after the mixer allows selection of the
upper or lower sidebands (+ or - sign). Thus the third frequency is derived from RF and
LO. Mixers are distinguished as up converters, when the final frequency is larger than the
signal RF or down converters, if the final frequency is smaller than ωR. Main
characteristics of the mixer are the Conversion Loss and Local Oscillator power when the
conversion loss is saturated. Conversion Loss is defined as the ratio of the power at the
mixer output to the incident power at RF port expressed in db. The losses decrease when
the local oscillator power is increased until saturation is reached. A good mixer may have
5-6db conversion loss, reached at 4-6 dbm of LO power.
2.3 RF instruments: Oscilloscope, Spectrum and Network Analyzer, RF Sources.
Traditionally, when you want to look at an electrical signal, you use an
oscilloscope to see how the signal varies with time. This is very important information:
however, it doesn't give you the full picture. From the basic theory we know that any
signal varying in time can be represented as a sum of sine waves. This is done by the
Fourier transformation, which gives us the signal as a function of frequency (see
Fig.2.3.1). To fully understand the performance of your device/system, we also need to
analyze the signal(s) in the frequency- domain. The spectrum analyzer gives us a
graphical representation of the signal in the frequency domain, i.e. works in a same way
as the oscilloscope for the time domain.
Figure 2.3.1 shows a signal in both the time and the frequency domains. In the
time domain, all frequency components of the signal are summed together and displayed.
In the frequency domain, complex signals (that is, signals composed of more than one
frequency) are separated into their frequency components and the level at each frequency
is displayed.
Fig. 2.3.1 a) Graphical representation of time and frequency domains. b) signals in the time
domain recorded by oscilloscope, b) frequency domain signal recovered by a spectrum analyzer.
Frequency domain measurements have several distinct advantages. For example,
let's say you're looking at a signal on an oscilloscope that appears to be a pure sine wave.
A pure sine wave has no harmonic distortion. If you look at the signal on a spectrum
analyzer, you may find that your signal is actually made up of several frequencies. What
was not discernible on the oscilloscope becomes very apparent on the spectrum analyzer.
Fig. 2.3.2. An example of a signals recorded by an oscilloscope (left) and same signal converted into the
frequency domain by the spectrum analyzer (right).
Modern oscilloscopes become very fast. Typical digitizing, or sampling rates
often exceed 10 Gsamples/s, implying accurate measurement of RF signals in GHz range.
Since the signals are obtained in the digital format, it’s easy to make any transformations
with them, like Fast Fourier Transformation (FFT), which transfers the signal from time
to the frequency domain and recovers the signal spectrum. This way the oscilloscope can
perform as a Spectrum Analyzer. High quality Oscilloscopes have a powerful processor
and a hard drive inside, with Windows or Linux operating systems, connected to the local
or global network. This allows easy and flexible programming and remote control of the
measurement from any Internet access point.
Using the oscilloscope as a Spectrum Analyzer is limited by its input bandwidth
and rarely exceeds 1-2 GHz. To go beyond these limits and also improving the speed and
sensitivity of the spectral measurement one uses a Swept Spectrum Analyzer.
Commercially available Spectrum analyzers may cover frequency range from 3 Hz to
over 300 GHz. The mixing and heterodyning technique is the essence of these devices.
Typical scheme is presented in Fig. 2.3.2. Input signal is mixed with the local oscillator
signal at the mixer followed by a narrow band filter. The bandwidth of the filter defines
the resolution bandwidth of the Spectrum Analyzer. The local oscillator frequency is
swept through a desired region, so that at the mixer+filter output only certain selected
frequencies are detected. A good spectrum analyzer may measure signals from -170dbm
to 30dbm and have the resolution bandwidth from several Hz to several GHz. Spectrum
Analyzers are widely used in the communication and computer technologies. They are
however, quite expensive, the price of a good device may easily go above 20,000€.
Fig. 2.3.2 Typical scheme of the spectrum analyzer.
Spectrum Analyzers work with the signals generated by scientific apparatus, or
emitted by some sources which we are studying. Sometimes we need to test and
characterize passive devices, which are not capable of generating their own radiation, but
having the function of manipulating the RF signals. These components can be amplifiers,
attenuators, filters, cables, resonators etc. To test them we provide a stimulus, or
excitation (see Fig. 1.1), and measure the response of the device under test, i.e. reflected
or transmitted signal. This can be expressed in terms of the S-parameters, discussed in
section 2.1. The device suitable for these purposes is called a Network Analyzer.
Generalized block diagram of a Network Analyzer is presented in Fig. 2.3.3.
Incident Transmitted
DUT
Reflected
SOURCE
SIGNAL
SEPARATION
INCIDENT REFLECTED TRANSMITTED
(R) (A) (B)
RECEIVER / DETECTOR
PROCESSOR / DISPLAY
Fig.2.3.3. Generalized Network Analyzer block diagram
RF signal from the Source is divided to two parts. Half of it is sent to the DUT,
the other half to the receiver. The latter is called Reference (R). Signals transmitted (B)
and reflected (A) from DUT are also directed to the receiver/detector, where they are
compared with the reference signal R. The source frequency is changed or swept and the
display shows the ratio of the reflected (A/R) or transmitted (B/R) signals to the reference
as a function of frequency. This gives the frequency dependence of the DUT S-
parameters. The other option is to keep the frequency constant and change the power
applied to the DUT. This way we obtain dynamic characteristics of the DUT. For
example we can find out how linear is an amplifier versus the strength of applied signal,
or at what values of input powers it gets saturated.
RF sources used in scientific measurements and industry are distinguished as
Continuous Wave (CW), Swept, and Function or arbitrary waveform generators.
CW sources are designed to generate signals with constant frequency, which
however, can be set by the user. The main requirement and characteristic feature of the
CW source is its frequency and phase stability and spectral purity. This is usually
achieved by the technique of the frequency synthesis, and such devices are called
Frequency Synthesizers. Such device contains a complicated chain of frequency
multipliers/dividers and filters, which convert the signal from some reference source to
any desired value. The other option is to use voltage controlled oscillator and a phase
locked loop (PLL) technique to lock its frequency to that of the Reference. The reference
source is basically any fixed frequency (usually 1 or 10 MHz) high stability oscillator.
The simplest are quartz crystal based oscillators with accurate temperature stabilization.
They may have short term relative frequency stability better than 10-10, and long term
drifts below 10-9. Sometimes even better stability is required, which may be obtained
from international frequency standards, e.g. Rubidium or Cesium atomic clocks. The
reference signals from these clocks are maintained by the National Institute of Standards
and Technology (NIST) and transmitted via the air in the northern hemisphere, like a
signals from radio stations. These signals can be captured by special receivers and
converted to 10 MHz Reference, which is then used to phase lock or synthesize any other
frequency. Short/ long term relative frequency stabilities currently available from these
RF transmitters are 10-12 and 10-10.
Swept source function is to provide a variable frequency stimulus at relatively
high power. It allows rapid changing of the frequency and therefore can not provide same
frequency stability as the CW source. Main requirements for this device are the linearity
and the speed of the sweep.
Function or arbitrary waveform generators are designated to provide periodic
signals of any form. This can be pulsed, triangular (saw-tooth shape) or, in principle
anything. For the pulsed generators the main characteristic is the raise/fall time of the
pulse. This can be shorter than 10 ps for high quality pulsed generators. Modern pulsed/
arbitrary waveform generators utilize digital signal processing, and sometimes are
produced in the form of plug-in card for the PC.
2.4. RF components
Connectors and adapters. Three major types of 50Ω connectors are used in the RF
instruments: BNC, SMA, and N-type. SMA connectors are the most compact, and are
used for highest frequencies up to 18 GHz. The connectors of this type are usually gold
plated and have highest quality. BNC is commonly used for lower frequencies, up to 2
GHz. N type has much larger size and is used when larger RF powers are required.
Adapters allow connecting the cables and devices having different types of connectors.
Connectors are characterized by their VSWR, which is normally between 1.05-1.1
depending on the frequency and type.
Amplifiers. The function of an amplifier is to increase the signal. Remember that the input
noise, which can be considered as a signal, is amplified by the same amount, therefore an
amplifier can not increase signal-to-noise ratio. Noise factor is always larger than 1. The
amplifiers are characterized by the operating frequency (band), gain, maximum power at
the input and/or output, noise figure, VSWR and DC power consumed. By the gain and
input power they are classified as low noise, low power, medium power, medium high,
and high power. Usually, the noise figure increases with the power available from the
amplifier. Special types of amplifier represent pulsed amplifiers. They have very broad
band, starting from DC up to hundreds of MHz.
Attenuators are the devices with negative (in log terms) gain. They are used to control the
signal strength. Attenuators may be fixed value, and variable. The latter can be digitally
controlled using the computer. Attenuators main
characteristics are the VSWR and accuracy of their
attenuation. In very high power applications some 1
special, actively cooled types can be used.
Detectors usually incorporate some semiconductor based 2
diodes, which convert RF into the low frequency, nearly
DC band. Usually detectors require a DC bias.
Power splitters. These devices are designed to split or
combine the RF power from one to two other ports. A
special kind of RF splitter is the directional coupler. One 3
can inject a portion of RF power into the transmission
line using the directional coupler. Directional couplers Fig. 2.3.4. Three port
are used for the measurements of the S parameters, circulator. If the rf power
especially the reflection coefficients. Circulator is a circulates along the arrows, the
losses are small. Large
special type of the power splitter where the power sent to attenuation is provided in
the port 1 goes with very small losses to port 2, same reverse direction
from port 2 to port 3, ... from port N-1 to N, from port N
to port 1. But if we send it in reverse direction, from port N to N-1 - it is strongly
attenuated. See an example in Fig. 2.3.4. If the matched load is connected to the port 2,
then all power sent to port 1 will be absorbed by the device., while the power sent from
port 3 to 1 will be transmitted with no losses. Such devices are called isolators. They are
very useful after powerful rf sources, oscillators, since they provide a high degree
isolation of the load from the source. Source is connected to the port 3 and load to the
port 1. Then all reflections from the load will be absorbed by the matched load in the port
2.
Filters. Filters can be distinguished as low pass, high pass band pass and resonant.
Characterizing the band of the filter usually the so-called 3-db loss point is used. It is the
frequency at which the power losses of the transmitted signal reach 3 db. There is a large
variety of filters differing in the shape of the attenuation versus frequency curve, and on
the type of operation. Most of the filters operate in reflective mode, i.e. they reject the
incoming signal outside the transmission band.
Mixers and frequency multipliers transform the signals in the frequency domain.
Operation of a mixer has been considered above. Its main characteristics are the
conversion loss, which is defined as the ratio of the power at the output to the input
power. Frequency multipliers generate harmonics of the input signal. They may have
filters incorporated to suppress unwanted harmonics, and amplifiers to amplify
selectively the frequencies of interest.
Matching Pads and Impedance Transformers are designated to provide matching of the
impedances between the devices which can not have 50 Ω impedance due to some
principal physical limitations, like some of the custom sensors may have very small or
very large impedance.
Terminations provide a fixed load with a certain impedance, usually equal to 50 Ω. They
are needed, e.g. for the measurements of the S-parameters of multi-port devices.