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Tutorial Note # 1
Important RF & MW Parametersfor Broadband Communication

At RF and microwave frequencies scatteringparameters are very common.
This is the general definition of the Sparameters for an N-port with different
characteristic impedances.
The requirement that all other input voltages
have to be zero also means that there are nore-reflections on any port.

If our device is a two port, and if we have thesame characteristic impedance on input andoutput, the definition of the s- parameters isvery easy.
Please note that the requirement that all other
input voltages have to be 0 also means that
there are no signals reflected on any port!

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Frequency response (amplitude and phaseresponse) is the Fourier transform of theimpulse response; the impulse response is thederivative of the step response.
Shown here is the effect that different
frequency responses have on a pulse:
The black and the red trace have similar
bandwidth, but the red trace has a steeper rolloff, therefore the step response shows moreringing.
The blue trace has the same shape than theblack, but less bandwidth, therefore the risetime is slower.
Keep in mind that not only the bandwidth isimportant for good pulse response, but also a
linear phase response.

It might not be obvious at a first glance why
a linear phase response is important. Lets
have a look why this is especially importantfor broadband data signals:
The square wave signal in our example isrepresented by the following equation:
U ( t ) = 1 ⋅ sin( ω ⋅ t ) +1⋅ sin( 3 ⋅ ω ⋅ t ) +311⋅ sin( 5 ⋅ ω ⋅ t ) + ⋅ sin( 7 ⋅ ω ⋅ t ) +7
511+ ⋅ sin( 9 ⋅ ω ⋅ t ) +⋅ sin( 11 ⋅ ω ⋅ t )911+

The corresponding spectrum is also shown.

If we send this signal over an idealtransmission line, we will observe a delay ofthe signal at the output of the line. The delay
will be: Length of line / Speed of propagation
Another way of specifying the delay is thephase:with:

ϕLengthline=360 ° Wavelength
Speed Pr opagationcWavelength ==Frequencyf

we get:ϕ (l , f ) = 360 ° ⋅ Lengthline ⋅ fc

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Another approach to understand the phase is
if we look at the sine wave signals thatmake up our square wave: Since the delay of
an ideal transmission line is the same foreach sine wave, the phase has to be different
for each frequency. This will lead to thesame formula:ϕ (l , f ) = 360 ° ⋅ Lengthline ⋅ fc

To characterize a signal in the frequency
domain completely we have therefore tospecify the amplitude and phase (or real andimaginary part) of each spectral component
or frequency.
In our example, the fundamental frequency isrotated by 40°, the 3rd harmonic has thus aphase rotation of 120°, the 5th harmonic of200°.

The phase response of our ideal transmission line
is linear, therefore we do not observe any
distortion of the transmitted signal.
The time domain signal would be represented by
the following equation:
1⋅ sin( 3 ⋅ ω ⋅ t − 3 ⋅ ϕ ) +311+ ⋅ sin( 5 ⋅ ω ⋅ t − 5 ⋅ ϕ ) + ⋅ sin( 7 ⋅ ω ⋅ t − 7 ⋅ ϕ ) +5711+ ⋅ sin( 9 ⋅ ω ⋅ t − 9 ⋅ ϕ ) +⋅ sin( 11 ⋅ ω ⋅ t − 11 ⋅ ϕ )911U ( t ) = 1 ⋅ sin( ω ⋅ t − ϕ ) +4

Let us have a look what will happen if we transmit a signal through a system with non-linear
phase response: The two signals on the right have the same spectral content, however thephase of the spectral components relative to each other are different. In the upper right picturethe phase of the spectral components is linear.
The distortion of the signal in the lower right is only due to the non linear phase relation of thespectral components, the signal in this example is defined by the equation:11⋅ sin( 3 ⋅ ω ⋅ t − 3 ⋅ ϕ + 36 ° ) + ⋅ sin( 5 ⋅ ω ⋅ t − 5 ⋅ ϕ − 36 ° ) +351
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1+ ⋅ sin( 7 ⋅ ω ⋅ t − 7 ⋅ ϕ ) + ⋅ sin( 9 ⋅ ω ⋅ t − 9 ⋅ ϕ − 36 ° ) +⋅ sin( 11 ⋅ ω ⋅ t − 11 ⋅ ϕ − 72 ° )7911U ( t ) = 1 ⋅ sin( ω ⋅ t − ϕ ) +It is important to mention that a non linear phase response not only causes distortion but also
contributes to jitter.
Now that we know why linear phase response is important we can introduce two new terms:
Deviation from linear phase and group delay:5At high frequencies, even short devices causea phase rotation by many degrees, so theslope of the phase response will be very
steep. Therefore it will be difficult to judge
whether the phase response is linear or not.

One solution to this problem is to subtract alinear phase response from the measured
phase response. The remaining non-linear
part can then be displayed and analyzed withhigh resolution.

Another solution is to determine thederivative of the phase versus frequency.
This leads to the group delay: The groupdelay is defined as : T G = - dφ/dω = - dφ/2πdf
As a network analyzer measures at discrete
frequencies the group delay will beapproximated by: -Δφ/2πΔf, the term Δf iscalled the aperture. It is very important thatyou use the same aperture if you compare
measurements, as a bigger aperture willsmooth the group delay and lead to “better”
results!

6VSWR.... Voltage Standing Wave Ratio
The original method of measuring impedanceor match was to use a slotted line, where adetector measured the voltage maximum andthe voltage minimum (as a function of theposition) of the standing wave caused by any
mismatch on the output of the line.
Although today a vector network analyzer
can measure the reflection coefficient
directly, the VSWR is still a very commonspecification.

If a line is terminated with a device having
an s11 that is not zero, a part of the incident
wave will be reflected. If the source has an
impedance that is different than thecharacteristic impedance of the line, a part of
the reflected signal will be re-reflected.
Incident waves and re-reflected waves will
add as vectors and this will cause ripple in
the frequency domain.

Also in the time domain, reflections cause
disturbances: If we send a pulse down theline the re- reflection will add to the original
pulse. We will observe overshoot and ringing
on the pulse. Shown in this slide is the pulse
response for three different terminations:i)
As a short can not absorb energy,
the entire signal is reflected.
ii)
As the voltage across a short iszero, the reflection cancels theincident signal, i.e. the reflection
has a relative phase of 180° to
the incident signal.
iii)
With a Z0 termination, theincident signal is absorbed in theload and no reflections occur.
If we terminate with an open, the reflection
will be in phase with the incident signal,
therefore the voltage across the open will
double.

7There are many parameters characterizing
match or mismatch: Impedance, reflection
coefficient, S11, VSWR and return loss.
Shown in this slide is the relation between
these parameters. Keep in mind that
impedance and reflection coefficient (S 11) arevectors, whereas return loss and VSWR are
scalar parameters.
It is sometimes confusing as the parameters
have of course different values for match andmismatch:
ParameterZΓVSWR
arMatch Short
Z000-1
1∞
∞
0

Open

∞

+1∞
0

Since the s-parameters are only valid for thelinear region of the device - or for non-linear
devices only for a particular input power the nonlinear behavior has to be specified aswell.
A common specification is the 1dBcompression point. This compression point is
the output power at which the output power
is one dB less than the output power of an
ideal (linear) amplifier with the same gain.
Normally amplifiers are used in their linear
region to avoid distortion of the signal, but
for some applications the non-linear behavior
is helpful: The limiting characteristic of anamplifier can be used to clip overshoot andringing of data signals. In this case evenhigher compression (2dB or 3dB) is used.
Another specification of active components
and systems is the noise figure: The noise
figure is a measure of how much noise isadded by e.g. an amplifier or in other words
how much the signal to noise ratio of theinput signal is degraded at the output.
If the input is terminated with 50 Ω, the noiseat the output will be:-174dBm/Hz +10log10(BW) + G/dB + NF/dB

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As mentioned the noise figure is the ratio of
the input- and output- signal to noise ratio:
F =(S/N)Input / (S/N)Output
and:NF/dB = 10 Log (F)As the noise power density of a 50 Ohm
resistor is given by: PNoise = kTB, where:
k ….Boltzman Konstant (1.38 10-23 J/K)
T….Absolute Temperature / °K
B.... Bandwidth / Hz
The output noise power can calculated to be:
PNoise/W = kTBGF
G…. Gain (in linear terms!)

The noise figure can be expressed in linear
terms or in dB.
Sometimes the noise temperature is usedinstead of the noise figure: The noisetemperature is the theoretical temperature ofthe input resistor that would give the same
noise power at the output of a noise free
network than the “real world” network.
The relation between F and T N is:
TN = (F-1)T0 ; where T0 = 290° K.

Another application for broadband amplifiers
is to use them as preamplifiers for receiversor spectrum analysers. How does thesensitivity improve when we use apreamplifier?
According to the cascaded noise equation thenoise figure of two cascaded systems is:

FSystem = F1 +F2 − 1
G1With F1 = 6.3 (108/10), F2 = 3162 (1035/10)
and Gain = 398 (1026/10) we get:FSystem = 14.25 and NFSystem = 11.5 dB
In our example the use of the pre-amplifier
would improve the sensitivity of thespectrum analyser from -139dBm/Hz to-162dBm/Hz.

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For broadband HEMT amplifiers the noise
figure varies with frequency. The noise
figure increases especially below 100 kHz

Note that the noise figure in the above
diagram the frequency scale is logarithmic,
when the frequency scale is linear the peak at
low frequencies is not that obvious.

As the noise output power of our broadband
amplifiers is far below one μW and as thesignal output power is normally between 10
mW and 100 mW, noise is not an issue when
amplifying broadband communication
signals.

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