contentshft.uni-duisburg-essen.de/lehre/kdk/protected_kdk/... · 2012-04-19 ·...

Fakultat fur Ingenieurwissenschaften Universitat Duisburg−Essen

H F T

FachgebietHochfrequenztechnikProf. Dr.−Ing. K. Solbach

.. ....

Abteilung Elektrotechnik undInformationstechnik

Komponenten fur die drahtlose Kommunikation

Experiment No. 1 Version: April 19, 2012

Nonlinear Properties of Two-Ports

Last name: First name: Matr.-No.:

Tutor: Date:

Contents

1 Literature 2

2 Introduction 2

3 Theoretical Background 43.1 Nonlinear Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Dynamic Range of Nonlinear Two-Ports . . . . . . . . . . . . . . . . . . 10

3.2.1 Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Measurements and Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 Decibel Scale for Power and Voltage . . . . . . . . . . . . . . . . . 133.3.2 Deriving Taylor-Series Coefficients from Measurement . . . . . . . 143.3.3 Slope in Logarithmic Plots . . . . . . . . . . . . . . . . . . . . . . 17

4 Homework 18

5 Description of Instruments 205.1 Spectrum Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1.1 Main Setting Parameters . . . . . . . . . . . . . . . . . . . . . . . 215.1.2 R&S FS300 Spectrum Analyzer . . . . . . . . . . . . . . . . . . . 22

5.2 Signal Combiner Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Performing the experiment 266.1 Check of Test Signals and Spectrum Analyzer . . . . . . . . . . . . . . . 266.2 Tests of Transistor Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.2.1 The Third-Order Intermodulation . . . . . . . . . . . . . . . . . . 286.2.2 Blocking-Effect Demonstration . . . . . . . . . . . . . . . . . . . . 316.2.3 Mixing demonstration . . . . . . . . . . . . . . . . . . . . . . . . 336.2.4 The 1dB-Compression Power . . . . . . . . . . . . . . . . . . . . . 366.2.5 Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . 38

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Komponenten-Praktikum Nonlinear Properties of Two-Ports

1 Literature

a) D. Pozar: Microwave and RF Design of Wireless SystemsNew York: John Wiley & Sons, 2001, chapter 3.7

b) Agilent Technologies: Application Note 150,“Agilent Spectrum Analysis Basics”http://cp.literature.agilent.com/litweb/pdf/5952-0292.pdf

2 Introduction

Two-port circuits can be subdivided into passive and active circuits. Passive circuitsmay be linear or nonlinear circuits and active circuits are necessarily nonlinear. Typicalexamples of passive linear circuits are impedance matching circuits or filter circuits,while the classical active circuit is the two-port amplifier, employing a transistor as theactive element. As will become clear later, even active circuits may be assumed to belinear circuits to a good approximation, if very small signal amplitudes are considered;this is what we know as the ”small-signal approximation” of purely sinusoidal signals(using complex phasor description) in the analysis of electrical networks.

Our idealization of a linear two-port response to a signal at the input port is a signal atthe output-port that is delayed by some delay-time τ and is undistorted otherwise apartfrom some amplitude scaling: An amplifier should normally scale-up, i.e. increase theamplitude, while passive circuits tend to reduce.

Note that a scaling-up may be accomplished also by a simple (passive) trans-former, if only output voltage is compared to input voltage; yet this goes along witha scaling-down of currents so that power is held constant, if it is not reduced due to losses.

For an ideal linear voltage amplifier we may have output voltage v2 and input voltage u1and voltage gain factor gu, yielding

u2(t) = guu1(t− τ). (1)

In actual circuits we find three principal deviations from this ideal model:We have on the one hand additional random signals added to our expected (and intended)output signal, see the experiment no. 2 on Noise Characterization of Two-Port Networks.On the other hand, we find the output signal distorted with respect to the (original) inputsignal. This distortion can be divided into two classes, (a) the linear distortion and (b)the nonlinear distortion.

(a) The linear distortion is due to energy storage elements within the circuits, leadingto frequency-dependent transfer characteristics. The best example for this propertyis a frequency-filter, which attenuates or passes and shifts phase of certain partsof the spectrum of the input signal. The resultant signal may be restored to theoriginal form, if we apply an ”inverse” filter circuit, which amplifies what has beenattenuated before and turns the phase shift back where signal phase was shifted.This kind of operation is termed ”linear”. If we test for linear distortion, a sinusoidalsignal at the input has to be found at the output as a pure sinusoidal with onlyamplitude scaling and some phase shift. The term ”linear” also indicates that thesuperposition theorem applies: If we supply two separate signals to the input ofthe two-port, we find the output signal to consist of the sum of two output signals,

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where each is the response to one of the input signals. In particular, if we apply twosinusoidal signals at the input, we expect two sinusoids to appear at the output,each of the same frequency as the respective input signal and amplitude-scaled andphase shifted in the same way as if each were the only signal applied to the two-port.

(b) Contrary to the linear distortion in any circuit containing inductive and capacitivecircuit elements we find additional nonlinear distortion in circuits containingnonlinear circuit elements, e.g. diodes and transistors, which are characterizedby curved voltage-to-current characteristics. If we apply a sinusoidal voltageat the input of a nonlinear circuit we find a distorted, non-sinusoidal outputsignal, which by Fourier-analysis can be decomposed into a sinusoidal com-ponent at the original frequency and many harmonics of this fundamental.If we apply simultaneously two sinusoids at the input, we find the distortedoutput signal composed of fundamentals and harmonics of both signals plusadditional sinusoidal signals of combination frequencies of the two input signalfrequencies. These so-called “products” increase in relative magnitude withincreasing input signal magnitude and thus limit the usefulness of a two-port forhigh amplitude operation. In radio frequency communication systems, nonlineardistortion products can create serious interference problems as will be seen later on.

Note that the nonlinear distortion effect has long been known from audio ampli-fication where we can hear the non-fidelity in the reproduced acoustic signal froman amplifier/loudspeaker at too high volume setting.

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3 Theoretical Background

3.1 Nonlinear Signal Analysis

For the description of characteristics of electronic circuit elements, using the transistoras an example, in small-signal network analysis we normally assume that the RF signals(alternating current, a.c.) be very small compared to the bias (direct current, d.c.)voltages and currents of the electronic element at its operating point. Thus, it isacceptable to approximate the curved characteristic of solid state elements by a linearcharacteristic, e.g. a small-signal conductance due to the first derivative (slope) of thecurved diode characteristic at its operating point. More precisely, we need a nonlineardescription, which can be given using the ”Taylor Power Series Approximation”, asknown from mathematics:

We approximate a nonlinear voltage/current characteristic or the voltage-to-voltagetransfer characteristic of a two-port containing a non-linear element by a Taylor seriesexpansion which comprises zero-order (bias-) terms, the first order term (linear or small-signal) plus second and additional higher order terms:

y(x) = y0 + a1(x− x0) + a2(x− x0)2 + a3(x− x0)3 + a4(x− x0)4 + ... (2)

where y(x) is the output quantity (e.g. voltage at the output port of an amplifier), y0 isthe output operating point (e.g. the d.c. bias voltage at the output port of an amplifier),x0 is the input operating point (e.g. the d.c. bias voltage at the input of an amplifier),x is the input variable (e.g. the a.c. voltage at the input of an amplifier) and ai are thecoefficients of the Taylor series (Fig. 1).

x

y

x0

y0

operating point

outputsignal

input signal

Fig. 1: Transfer Characteristic of an Amplifier

As long as the term (x − x0) is small, the higher-power terms of the series expansionare even smaller and thus in many cases may be neglected, so that the simple small-signal approximation is justified. However, if we look closer, we find that the curvatureof nonlinear transfer characteristics is the cause of many unpleasant, degrading effectsbut can also be employed to create the important frequency translation functions in RFsystems. In the following, we analyze such behavior by truncating the Taylor series afterthe third-order term and assume the input signal x to be composed of two sinusoidal time-varying signals of angular frequency ω1 and ω2 and amplitudes A1 and A2, respectively. Ithas to be cautioned here that truncation of the power-series will limit the accuracy of the

4


approximation and that modeling the nonlinear behavior of two-ports by above nonlineartransfer characteristic in principle is only an approximation: This is so, since circuitresponses also depend on the slope of time-domain signals (or the frequency), which partlyis due to the existence of energy storage elements (and linear distortion); nevertheless,experience shows that good agreement of the theoretical model and measured results isachieved for many practical circuits and in particular, the broadband-type of amplifiersused in this lab.

3.1.1 Analysis

Consider a Taylor series of 3rd order and a two-tone input signal

x(t) = A1 cos(ω1t) + A2 cos(ω2t)

and bias x0 = 0.

The output signal y(t) as a function of time is

y(t) = y0 + a1 · x(t) + a2 · x2(t) + a3 · x3(t)= y0 + a1 [A1 cos(ω1t) + A2 cos(ω2t)]

+ a2 [A1 cos(ω1t) + A2 cos(ω2t)]2

+ a3 [A1 cos(ω1t) + A2 cos(ω2t)]3 .

(3)

Using

cos2 α = 12(1 + cos 2α)

cos3 α = 14(cos 3α + 3 cosα)

cosα · cos β = 12

[cos(α− β) + cos(α + β)]

(a+ b)2 = a2 + b2 + 2ab

(a+ b)3 = a3 + b3 + 3ab2 + 3a2b

we can first solve the binomials of second and third powers of our two-tone signals

y(t) = y0

+ a1A1 cosω1t + a1A2 cosω2t+ a2A

21 cos2 ω1t + a2A

22 cos2 ω2t + 2a2A1A2 cosω1t · cosω2t

+ a3A31 cos3 ω1t + a3A

32 cos3 ω2t + 3a3A1A

22 cosω1t · cos2 ω2t

+ 3a3A21A2 cos2 ω1t · cosω2t

(4)

and can go on with resolving the resultant terms (products) with respect to frequencies

5


y(t) =y0 + 1

2a2A

21 + 1

2a2A

22 : direct current (d.c.)

+ cosω1t ·[a1A1 + 3

4a3A

31 + 3

2a3A1A

22

]: fundamental wave (ω1)

+ cosω2t ·[a1A2 + 3

4a3A

32 + 3

2a3A

21A2

]: fundamental wave (ω2)

+ cos 2ω1t ·[12a2A

21

]+ cos 2ω2t ·

[12a2A

22

]: second harmonic (2ω1, 2ω2)

+ cos 3ω1t ·[14a3A

31

]+ cos 3ω2t ·

[14a3A

32

]: third harmonic (3ω1, 3ω2)

+ cos((ω1 − ω2)t

)· [a2A1A2] + cos

((ω1 + ω2)t

)· [a2A1A2] : mixing products

(sum- and difference-frequencies)

+ cos((2ω1 − ω2)t

)·[34a3A

21A2

]+ cos

((2ω2 − ω1)t

)·[34a3A1A

22

]: intermodulation in-band

+ cos((2ω1 + ω2)t

)·[34a3A

21A2

]+ cos

((2ω2 + ω1)t

)·[34a3A1A

22

]: intermodulation out-of-band

Fig. 2: Output Spectrum of Second and Third-Order Two-Tone Intermodu-lation Products, Assuming ω1 < ω2

We learn from this table of terms:An input signal x(t) to a transfer characteristic of third order (and with x0 = 0) createsan output signal

y(t) = y0 + a1 · x(t) + a2 · x2(t) + a3 · x3(t) (5)

which consists of several types of products:

a) Direct Current Products

y(ω = 0) = y0 +a2 (A2

1 + A22)

2(6)

⇒ The d.c. bias point y0 of a two-port will be offset, increasing with the square of theinput signals. This is exploited, e.g. in detector circuits, where a.c. voltage is convertedto d.c. voltage

b) Fundamental or 1st Harmonic Products

y(ω1) = A1

(a1 + 3a3

A21 + 2A2

2

4

)(7)

⇒ The factor a1 is the ”linear” amplification factor of the two-port, but with increasingamplitude of a desired signal A1, the second term can become large and reduce theamplification (because the third-order term a3 < 0 as seen later in chapter 3.3.2). Thismeans, the output signal becomes ”compressed”. Such compression also takes place if

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we have a second strong signal present (note amplitude A2 in the second term).

⇒ Special Case: Single-Tone Operation (A2 = 0)

linear range

compression

saturation

region

y

A10

Fig. 3: Compression in Single-Tone Operation

For a single-tone operation, we find that the compression effect increases as the thirdpower of the input signal amplitude and relies on the third-order term of the Taylorseries.

yfund.(t) =[a1A1 + 3

4a3A

31

]︸︷︷︸y(ω1)

cosω1t (8)

With increasing amplitude A1 we first see a compression (reduced gain) as a3 < 0. Ateven higher levels, complete saturation with constant output signal level takes placein real systems! However, this is outside our present mathematical model, which isonly able to describe a limited amount of compression due to the third-order term (Fig. 3).

⇒ Special Case: Two-Tone Operation

A2

A1

interferingsignalwith AM

RF filter width

RF signal

1 2

Fig. 4: Radio Reception

The second term in y(ω1) contains the effect of a second signal on the first signal by thecontribution from A2

2:

yCM(t) =3

2· a3 · A1 · A2

2 · cosω1t (9)

If the second signal is a carrier with constant amplitude, the effect is to increase com-pression, i.e. to reduce the output from the first signal; we speak of ”blocking” of the first

7


signal by the second one (Fig. 4). If the second signal is an amplitude modulated carrier(e.g. in radio broadcast reception), we have

A2 = A2 · cos(ωNFt) (10)

and the blocking effect varies in the same way as the modulation of the second signal; inradio broadcast reception we can hear the program of a second radio station overlaid tothe program of our tuned-in station. This ”cross-modulation” is also a third-order effect,since it increases with desired signal strength times the square of the second (unwanted)signal strength and it relies on the third-order term of the Taylor series.

c) Higher-Order Harmonic Signal Generation in Case of Single-Tone orTwo-Tone Operation

The curvature of nonlinear active elements transfer characteristics is often used in orderto produce signals of higher frequency by multiplication of the fundamental frequency byan integer number. Such multiplier circuits need to employ frequency filters to filter-outthe desired harmonic signal from the output signal.

Second harmonics increase as the square of the input signal amplitudes and rely on thequadratic term in the Taylor series:

yn =a22A2

1 cos(2ω1t)

yn =a22A2

2 cos(2ω2t)

n = 2 (11)

Third harmonics increase as the third power of the input signals and rely on the third-order term of the Taylor series:

yn =a34A3

1 cos(3ω1t)

yn =a34A3

2 cos(3ω2t)

n = 3 (12)

d) Mixing Products

The second-order term of the Taylor series is responsible for the creation of signals atthe non-harmonic difference- and sum-frequencies of the two tones. This is exploited inRF-systems for the translation of signals from one frequency to another frequency and iscalled ”mixing”. E.g. in radio receivers, we use to translate the received broadcast signalat ω1 to a lower frequency in our receiver (the ”Intermediate Frequency” or IF) by mixingwith a sinusoidal signal at a frequency ω2 which is offset by the intermediate frequency|ω1 − ω2| =IF:

8


→ Difference frequency: yIF(t) = a2A1A2 cos((ω1 − ω2)t

)In typical transmitter systems, we translate the low-frequency information signal atω1 up into a higher radio frequency (RF) by mixing with a sinusoidal signal of highfrequency ω2 and extracting the sum frequency for further amplification and transmission:

→ Sum frequency: ySF(t) = a2A1A2 cos((ω1 + ω2)t

)In certain case we produce the second sinusoidal signal locally (local oscillator, LO)and introduce it at high signal level to the nonlinear circuit in order to yield ashigh as possible level of the mixing product due to the factor A2 in both mixingproduct terms above. We can also be sure that the frequency-translated signal willbe a true replica of the original signal A1 cos(ω1t) with any phase- and amplitude-modulation preserved, since A1 appears only in its first power (”linear”) and the local os-cillator signal is described by a constant amplitude A2 and constant angular frequency ω2.

In a practical mixing circuit, we again need to employ a filter circuit in order to select thedesired mixing product from the output signal of the nonlinear circuit. As an example,the principle circuit diagram of a mixer circuit using a field-effect transistor shows theRF-signal fed to the gate, the LO-signal fed to the source and the IF-signal tapped fromthe drain terminals (Fig. 5). The nonlinear element in the circuit is the approx. quadraticrelation between drain current and gate-source voltage uGS. In a typical application forVHF broadcast reception, we would receive a station at fRF = 100MHz, operate thelocal oscillator at fLO = 110.7MHz and produce an intermediate frequency signal atfIF = 10.7MHz, which is amplified and demodulated in a following stage.

Intermediatefrequency signal

LocalOscillator

Am

plitu

de

IF

local oscillatorRF-band

!1

!2

2!1

2!2

!1!

2

fRF

fLO

fIF

uGS

!2!

1

Fig. 5: Mixer Circuit With a Field-Effect Transistor (FET)

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e) Intermodulation Products

The intermodulation products

yIM(t) =3

4a3A

21A2 cos

((2ω1 − ω2)t

)yIM(t) =

3

4a3A1A

22 cos

((2ω2 − ω1)t

) (13)

are created at non-harmonic frequencies (2ω1 − ω2) and (2ω2 − ω1) which may be closeto both of our two tones ω1 and ω2, if ω1 and ω2 are close. Therefore, in many cases it ishardly possible to filter out these products and we have to live with them although theymay cause interference to our system. The amplitudes of intermodulation products maybe very small, if both A1 and A2 are small, but, as both grow, the products A2

1A2 orA1A

22 grow much steeper than each of the two input signals alone. The intermodulation

products are third-order products, as their creation relies on the third-order term ofthe Taylor series and since their amplitudes are determined by a third power of signalamplitudes.

Example: Communications Transmitter

Am

pli

tude

Am

pli

tude

additional

sidebandescarrier

side band

1 2

22 1

21 2(a)

(b)

Fig. 6: Modulated RF-Signal. (a) Single-Sideband, (b) Distorted Modulationand Broadened Spectrum

A carrier signal ω1 and a modulation sideband ω2 are amplified by a transmitter poweramplifier before radiation through the antenna. Due to intermodulation, the spectrum ofthe transmit signal after the power amplification is broadened (additional, false modu-lation sidebands). This creates interference in the neighboring channels, which normallyare reserved for other users, services or programs.

3.2 Dynamic Range of Nonlinear Two-Ports

As mentioned above, intermodulation products may be very close to our desired signalsand thus may result in serious interference. On the other hand, such interference signalsincrease at a steeper slope than the original signals: In measurements it is usual to applythe two tones at the same level A1 = A2 = A , so that

yIM(t) =3

4· a3A3 cos

((2ω1 − ω2)t

). (14)

We recognize from this, that the intermodulation product increases with the thirdpower of the input signal amplitude. On a logarithmic scale, the slope of the transfer

10


characteristic (yIM versus A) is thus three, while the fundamental output signal increasesonly proportionally to the input signal (yfund. versus A), i.e. by the first power; thus,the slope for the fundamental signal component is one. When we plot the transfercharacteristic, we usually plot output power Pout in decibels (dBm) versus input powerPin in decibels (dBm) of the various signals, which does not change the above mentionedslopes.

Note: The quasi-unit of dBm indicates that before taking the logarithm we normalizethe signal powers P to the power of P0 = 1mW!

3.2.1 Dynamic Rangeslop

e=1

slop

e=3

DGain

IP3 Intercept Point

N0

Df

Pin (dBm)

Po

ut(d

Bm

)

1dB

Fig. 7: Transfer Characteristic for Fundamental and Third-Order Signals ofa Single Test Signal

We recognize (Fig. 7) the straight line under 45 (slope = 1) as the transfer characteristicof the fundamental signals which relates the output signal power to the input signalpower by a constant offset of 30dB, the ”linear” amplification of this two-port.

We also realize that the output power compresses to lower levels than the linear curvedemands: There is a certain point, where the output power Pout stays below the ”linear”power by exactly 1dB. This output power P1dB is often mentioned in data sheets ofamplifiers as the 1dB-compression power and marks the upper limit of useful outputpower available from the amplifier for many practical applications, or we call it theupper limit of the dynamic range.

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At the lower left corner of the chart we come to lower levels of power (input and output)where the transfer characteristic crosses the noise floor. Noise power level depends onthe bandwidth B of measurement and can be calculated by:

N0[dBm] = 10dB · log (G · F · kT0 ·B)

= 10dB · log

(G · F · 4 · 10−21 Ws ·B

[1

s

])= 10dB · log

(G · F · 4 · 10−18 mWs ·B

[1

s

])= G[dB] + F [dB]− 174[ dBm] + 10dB · log(B/Hz)

where G is the amplification gain of the two-port, F is the noise figure, T0 the referencetemperature of 290 K and B is the system bandwidth (see experiment no. 2 ”NoiseCharacterization of Two-Port Networks”).

In practical RF systems applications signals carry a certain modulation bandwidth tocontain the transmitted information (so this fixes the minimum system bandwidth) andusually require a certain signal-to-noise ratio (S/N), e.g. a factor of 10 or 10dB, in orderthat the transmitted information is received without too many errors. As a standard,we assume a minimum (S/N) of 1 (or 0dB) which is equivalent to assuming the noisefloor to be the lower end of the system dynamic range.

If we process two-tone signals, the upper limit of the dynamic range is no longer solelydefined by the compression effect, but in many applications it is rather defined by thecreation of undesired intermodulation signals: We recognize a certain input signal levelwhere the intermodulation transfer characteristic (slope = 3) crosses the noise floor. Thismarks the level of signals that we can process without noticing the existence of inter-modulation signals, because their level stays below the noise. Thus, our spectrum seems”spurious-free” below this limit and the range from the lower limit of the dynamic rangeof signal power to this point is called the ”Spurious-Free Dynamic Range” (SFDR = Df).

Repeatedly some definitions (see also Fig. 7):

Dynamic Range D:Dynamic range of power between the noise floor and the 1dB-compression point:

D[dB] = P1dB[dBm]−N0[dBm]. (15)

Spurious Free Dynamic Range Df :Dynamic range without intermodulation products above the noise floor:

Df [dB] =2

3(IP3[dBm]−N0[dBm]) . (16)

Intercept Point of Third-Order IP3:Power level marking the cross-over of the linear and the third-order transfer characteris-tics, where the intermodulation product levels would be equal to the fundamental signallevels. As can be seen, this does not happen in reality because also the intermodulationproducts compress at high signal levels. The ”Intercept Point of Third-Order”, some-times called ”Third Order Intercept” (TOI) is the most common single value describing

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the large-signal handling capabilities of nonlinear components, like amplifiers, and canrange around a few dBm (mW) in small-signal receiver stages. Cell phone base stationtransmitters are very critically limited with respect to the broadening of the transmitspectrum, so that intermodulation generation must be suppressed to a minimum. Thisrequirement can be fulfilled by operating a transmitter of many 100W output power ca-pability (thus very high IP3 of the order of 1kW) and reducing the drive signal powerto such a level that only 10% of the power capability is actually realized as generatedoutput power to the antenna; this mode of operation is called ”back-off” . Since the back-off mode is very expensive in equipment cost and supply power consumption, today weemploy linearization schemes that improve the linearity of amplifiers by means of extraequipment, e.g. by pre-distortion or distortion cancellation.

3.3 Measurements and Evaluation

3.3.1 Decibel Scale for Power and Voltage

Nonlinear two-ports can be modeled by their nonlinear transfer characteristic using theTaylor series expansion of output signal voltage versus input signal voltage. However,it is customary to measure the power of signals (instead of voltage) and express at alogarithmic scale in dBm:

A sinusoidal signal is measured as peak-voltage U over a load resistor R to give absorbedpower P

P =1

2· |U |

2

R(17)

where the power P is in Watt, voltage |U | in Volt and the load resistor R in Ω.

The power is normalized to a power of P0 = 1 mW and given as dBm by:

P [dBm] = 10dB · log

(P

P0

)= 10dB · log

(P

1mW

)(18)

while the voltage |U0| from P0 = 1mW of power is

|U0| =√

2 · P0 ·R =√

2mW ·R (19)

i.e., for the standard system impedance of R = 50Ω we have:

|U0| = 0.316V. (20)

Instead of giving the power in decibel over P0 = 1mW, we could also give voltage indecibel over |U0| = 0.316V (50 Ω-impedance system)

|U |[dB] = 20dB · log

(|U ||U0|

)= 20dB · log

(|U |

0.316 V

)(21)

which leads us to exactly the same decibel-numbers as for power, when we express powerby voltage squared:

P [dBm] = 10dB· log

(P

1 mW

)= 10dB· log

(|U |2

(0.316 V)2

)= 20dB·log

(|U |

0.316 V

). (22)

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3.3.2 Deriving Taylor-Series Coefficients from Measurement

a) When we measure the power ratio of a fundamental tone, with levels far below thecompression level, we know this as the ”linear” power gain

G[dB] = 10dB · log

(Pout

Pin

). (23)

We can determine the ratio of voltages from this by

G[dB] = 20dB · log|U |out|U |in

(24)

and the Taylor series coefficient a1 becomes

a1 =|U |out|U |in

= 10G[dB]/20 dB with [a1] = 1. (25)

b) We can evaluate the measured 1 dB-compression output power P1dB by inspectionof the full fundamental wave output signal formula based on the Taylor series andusing only a single tone x(t) = A cosωt

yfund.(t) = cos(ωt) ·[a1A+

3

4a3A

3

]. (26)

The condition for 1dB-compression is that the output fundamental signal amplitudeis 1dB smaller than it would be if there were no compression.

At 1dB-reduction means a factor of

10−1/20 = 0.891 ≈ 0.9 (or 90%) (27)

so that

a1A+3

4a3A

3︸︷︷︸linear term plus compression term

= 0.891 · a1A︸︷︷︸only linear term

(28)

or

a1(1− 0.891) +3

4a3A

2 = 0 (29)

or

a3a1

=−0.109

34A2

. (30)

This shows, that a3 will be negative if a1 is positive!

14


With the linear gain G given, we know a1 and from the output power P1dB wecalculate the uncompressed power Punc. to:

Punc.[dBm] = P1dB[dBm] + 1 dB. (31)

From Punc. and the linear gain G we conclude on the input signal powerPin = Punc.[dBm]−G[dB] and on the signal voltage squared

A2 = (|U0|1)2 · 10P1dB[dBm]+1dB−G[dB]

10 dB (32)

since voltage |U | squared is proportional to power P .

c) We can determine the second-order coefficient a2 of the voltage Taylor series from asuitable measurement, e.g. the second order harmonic generation from a single-toneexcitation. The appropriate expression from our table of products is:

y2(t) = cos(2ωt)

[1

2a2A

2

]. (33)

For the determination of a2 we need a measurement of the second harmonic signalvoltage amplitude at the output ∼ 1

2a2A

2 at a given input signal voltage ∼ A.If both signal powers are measured in Watts and assuming negligible frequencyvariation of the two-port (flat frequency response, no filter included)

Pout =1

2·(12a2A

2)2

Rand

Pin =1

2· A

2

Rwith R = 50Ω

we can solve for a2:

a2 =

√2R · Pout

R · Pin

with [a2] =1

V(34)

15


d) The intercept point of third order IP3 can be constructed as the meeting pointof the fundamental wave line of slope = 1 and the third-order products line ofslope = 3 in a logarithmic plot. We need at least two different voltages for thetwo input tones to construct each of the two lines; more measurement points canimprove the accuracy of the graphical construction.

From the intercept point of third order IP3 we may derive the third-order coefficienta3 in the following way:

The condition for the intercept point of third order IP3 is that the ”linear” funda-mental output signal voltage is of equal magnitude as the output intermodulationsignal voltage when we use equal amplitudes two-tones of amplitude A:

a1AIP3 = −3

4a3A

3IP3

or

a3 = −4

3

a1A2

IP3

(35)

where AIP3 is the signal input voltage producing the output power of the interceptpoint of third order IP3 by amplification due to a1

PIP3 =1

2· (a1AIP3)

2

R. (36)

Note that we use a negative sign in order to compensate that the coefficient a3 < 0.

We now solve for a3:

a3 = −2

3

a31PIP3 ·R

with [a3] =1

V2. (37)

Note: Should the intercept point of third order IP3 be given in dBm, we need toconvert to power in Watt:

PIP3 = 10IP3[dBm]

10 dB · 10−3 = 10IP3[dBm]

10 dB−3

16


3.3.3 Slope in Logarithmic Plots

When we plot the output power Pout in dBm versus the input power Pin in dBm we applythe logarithm to the square of the voltages at output and input:

Pout [dBm] = 20dB · log

(|Uout|

0.316 V

), (38)

Pin [dBm] = 20dB · log

(|U in|

0.316 V

). (39)

For the linear term in the Taylor series we find Uout = a1 · |U in| and


(a1 ·

|U in|0.316 V

)= C1 + 20dB · log

(|U in|

0.316 V

). (40)

For the quadratic term in the Taylor series we find |Uout| = a2 · |U in|2 and


[a2 · 0.316V

(|U in|

0.316 V

)2]

= C2 + 20dB · log

(|U in|

0.316 V

)2

= C2 + 2 · 20dB · log

(|U in|

0.316 V

), (41)

and for the cubic term in an analogue way

Pout [dBm] = C3 + 3 · 20dB · log

(|U in|

0.316 V

). (42)

Introducing (39) into (40) - (42) gives

Pout,linear[dBm] = C1 + 1 · Pin[dBm],

Pout,quadr.[dBm] = C2 + 2 · Pin[dBm],

Pout,cubic[dBm] = C3 + 3 · Pin[dBm]

where C1, C2, C3 are constants.

Plotting these relations as Pout[dBm] vs. Pin[dBm] shows:

a) The output power Pout in dBm is a linear function of the input power Pin in dBmwith a slope of 1(1dB increase in output power per 1dB increase in input power or 1dB/dB),

b) a slope of 2dB/dB for the quadratic relationship and

c) a slope of 3dB/dB for the cubic relationship.

17


4 Homework

(In preparation for the lab!)

4.1 A transistor band-pass amplifier exhibits

(a) linear distortion only,

(b) nonlinear distortion only,

(c) both.

Check, which answer is right!

4.2 A two-tone signal at frequency f1 = 11MHz and frequency f2 = 12MHz excites anonlinear two-port. What are the frequencies f of

(a) the second and third harmonics,

(b) the mixing products and

(c) the intermodulation products?

Which of these products can pass a band-pass filter of bandwidth B = 4MHz,centered at frequency f0 = 11.5MHz?

4.3 A short-wave receiver is troubled by nonlinear distortion at the presence of verystrong local transmitters. Instead of just receiving the really existing radio stationsignals, our receiver produces additional signals, which interfere with real signalsfrom the antenna.

There is only one way to discriminate the real from the ”spurious” signals:

We use a step-attenuator at the antenna port to reduce the incident signal levelsand thereby reduce nonlinear distortion. Our signal strength-meter indicates thefollowing changes in the level of three different signals when we switch-in a 10dBattenuation:

Signal Signal level0 dB attenuator 10 dB attenuator

# 1 -30 dBm -40 dBm# 2 -60 dBm -80 dBm# 3 -45 dBm -75 dBm

(a) Determine which signal is a fundamental signal and which are ”spurious” sig-nals, either a third-order intermodulation product or a second-order mixingproduct.

(b) What is the improvement in intermodulation suppression (ratio of fundamentaland intermodulation product in dB) that we achieve per dB of attenuation?

18


4.4 A broadband transistor amplifier is excited by a single-tone generator. We measurea ”linear” gain of G = 20dB for the fundamental wave and a second harmonicproduct output power of Pout = 1mW at an input power of Pin = 1mW when weuse generator and load impedance of R = 50Ω.

(a) Determine the Taylor series coefficients a1 and a2.

(b) What will be the second harmonic output power Pout at an input power ofPin = 2mW?

4.5 A transistor amplifier is characterized by a system bandwidth of B = 1MHz, a noisefigure of F = 10dB and by the following two-tone measurements (power levels pertone or product).

Pin Pout

fundamentals intermodulation-10 dBm 10 dBm -35 dBm0 dBm 20 dBm -5 dBm

(a) Determine the intercept point of third-order output level IP3 in dBm.

(b) Determine the spurious free dynamic range SFDR in dB.

19


5 Description of Instruments

5.1 Spectrum Analyzer

In our experiment we use a spectrum analyzer for the measurement of single-tone andtwo-tone signal levels at the input and output of our nonlinear two-ports. The principleof operating of a spectrum analyzer is explained based on Fig. 8.

Signal In

Attenuator RF-Filter Mixer IF- Amp.

IF-Filter

Detector

Low-pass Filter

Display

Local Oscillator

Sweep Generator

f

Fig. 8: Principle Block Diagram of Spectrum Analyzer

A signal from the input terminal has to pass a bandpass filter before being mixed downto an intermediate frequency fIF and being amplified, detected and displayed. TheRF-filter and the local oscillator are tuned simultaneously in such a way that the signalcan mixed to a fixed frequency fIF.

Before detecting of the frequency translated signal by a detector circuit, the IF-signalhas to pass through a filter bank of selectable bandwidth B; the selected bandwidthdefines the system bandwidth with respect to the noise floor.

Since in many situations the signal frequency is unknown in the beginning, the RF-filterand LO-frequency can be varied using the sawtooth sweep generator: In sweep mode,the receiver system continuously searches a predetermined frequency band (selectablefrom the front panel) at a selectable repetition rate. If there is just one signal withinthe search range we observe a single peak at the display; using a marker we are able todetermine the precise frequency of the signal. If there are several signals, there will beseveral peaks displayed, with the lowest frequency at left and the highest frequency atright; in other words: The horizontal (x-) line is the frequency axis.

The vertical excursion of the peaks indicates the amplitude or power of the signals. Usinga logarithmic scale, we may choose a resolution of 10dB per division on the screen. Thismeans, the highest peak is the strongest signal and precise values can be read-out usingthe marker. Without any signal at the input we observe the noise floor at the lower endof the display.Note that the noise floor varies with the selected bandwidth B!

20


One problem connected with the frequency sweeps is that in each sweep a signal iscaught only for a short moment, depending on the sweep rate, the RF sweep width andthe IF bandwidth. We must carefully select suitable combinations of the adjustmentsin order to allow the signal to build-up through the narrow-band IF-filter (minimumtime roughly τ = 1/B) in order to allow precise measurement results. Modern spectrumanalyzers indicate a warning to the user if incorrect settings are chosen or select validsettings automatically.

5.1.1 Main Setting Parameters

Spectrum analyzers usually provide the following elementary setting parameters:

• Frequency Display RangeThe frequency range to be displayed can be set by the start and stop frequency(that is the minimum and maximum frequency to be displayed), or by the center

frequency and the span centered about the center frequency.

• Level Display RangeThis range is set with the aid of the maximum level to be displayed (the reference

level) and is indicated by the top horizontal line on the CRT display. Theresolution/div defines the level distance between two horizontal grid lines onthe CRT display. The attenuation of an input RF attenuator depends on thereference level and will generally be adjusted automatically.

• Frequency ResolutionFor analyzers operating on the heterodyne principle, the frequency resolution is setvia the bandwidth of the IF filter. The frequency resolution is therefore referred toas the resolution bandwidth (RBW). A video filter defines the video bandwidth

(VBW). It is a first order lowpass configuration used to free the video signal fromnoise, and to smooth the trace that is subsequently displayed so that the display isstabilized.

• Sweep TimeThe time required to record the whole frequency spectrum that is of interest isdescribed as sweep time. This value will generally be adjusted automatically,since it is limited by the video bandwidth.

21


5.1.2 R&S FS300 Spectrum Analyzer

Fig. 9: R&S FS300 Spectrum Analyzer

The R&S FS300 spectrum analyzer (Fig. 9) is designed to analyze modern communi-cation signal waveforms from frequency f = 9kHz up to f = 3GHz in one sweep operation.

Operating the spectrum analyzer FS300 is quite simple because of many automaticsettings controlled by the instrument firmware. The instrument front panel is dividedinto a data field, where parameters can be adjusted by a tuning knob or by keyboardand a function field, where the main setting parameters are arranged.

For example: If you want to adjust the center frequency to f = 868MHz, press the CursorKeys until the freq/span key is highlighted and the associated menu is presented.

(Function area III at the right side of the display)

22


Fig. 10: R&S FS300 Spectrum Analyzer Display

Now, select the center function key and enter the frequency by numerical entry andconfirming by pressing the blue MHz-unit key (key pad at the right side). Other menusappear after selection using the Cursor key again. The ampt menu is used to set theRF input attenuator and the reference level, the mkr menu for setting the marker to thesignal peaks for read-out of the levels and frequencies and the bw/sweep menu for settingthe resolution bandwidth. You step in and out of the menus by activating the Cursor key.

We start the measurements by switching-on the instrument and set it to thepreset state (Factory) using the sys Key (F9)!

For most of our measurements, the Spectrum Analyzer should be set to:

FREQ/SPAN CENTER FREQ 868MHzSPAN 5MHz

AMPT RF ATTEN MANUAL 20dB → Enter

REF LEVEL -10dBm → Enter

MKR MARKER 1 PEAK orNEXT PEAK LEFT orNEXT PEAK RIGHT

BW/SWEEP RES BW MANUAL 10kHz → Enter

23


5.2 Signal Combiner Circuits

Spectrum Analyzer

Combiner

DUT Signal Generators

Fig. 11: Test Setup for a Two-Tone Analysis

In our tests we need to combine the signals from two signal generators to be fed intoour nonlinear Device Under Test (DUT). Since we want to investigate the creationof nonlinear products in our DUT, we need pure sinusoidal signals to feed into theDUT. Therefore we have to avoid to terminate the signal generators by unmatchedimpedances and have to avoid that the signal from our generator feeds into the othergenerator producing nonlinear products there. Hence, combining the signal of the twogenerators requires special 3-ports combiner circuits, which match the generator portsand attenuate signals traversing from one generator port to the other (see Fig. 11).

One option is to use a resistive power splitter/combiner which is a matched three-port with 6dB of insertion loss and 6dB of isolation of the two generators. Othercombiner/splitter circuits have been developed, e.g. the Wheatstone-bridge and theWilkinson-combiner, which give lower insertion loss and higher isolation but at the priceof restricted bandwidth.

In our case, the generators need more isolation and therefore we employ the specialpower combiner circuit of Fig. 12 with block diagram in Fig. 13, which provides twodouble nonreciprocal isolators connecting to the generators and a 3dB-hybrid directionalcoupler to combine the two signals; this circuit operates over a frequency range of aboutf = 800MHz to f = 1200MHz and affects an insertion loss of about 4dB and an isolationof about 50dB to the generator signals.

24


Fig. 12: Circuit for the Combination of two Signal Generators (GeneratorsInputs at the Left, Combined Signal Output at the Right)

G1

G2

Monitor

Output

3dB

C

50W

50W 50W

50W 50W

50W

Fig. 13: Power Combiner - Block Diagram

25


6 Performing the experiment

6.1 Check of Test Signals and Spectrum Analyzer

Before doing tests on our two-port device we need to check the signal purity and level afterthe combiner circuit. Our signal generators are the synthesizer circuits of our ME1000RF Training Kit (both the RX- and the TX-Boxes), Fig. 14, which we activate throughthe computer control board.

Fig. 14: Frequency Synthesizer Circuit With Coaxial Attenuator

In our measurement set-up, Fig. 15, we set one synthesizer (RX-Box) to a frequencyof f = 868MHz and the second one (TX-Box) to a frequency of f = 869MHz. We usetwo coaxial cables to connect the output ports of both synthesizers to the two inputports of the signal combiner with a 10dB coaxial SMA attenuator inserted to each of theconnecting lines.

The two signals from the combiner circuit travel through the variable attenuator andenter the RF input of the Spectrum Analyzer. We check the signals with the atten-uator set to a = 0dB (highest possible amplitudes at the input of our spectrum analyzer):

We should see two peaks with levels around P = −20dBm (use the marker for preciseread out!) at approximately the right frequencies (some inaccuracy is due to thefrequency synthesizers and some is due to the inaccuracy of the Spectrum Analyzer).Third-order intermodulation products to the sides (at f = 867MHz and f = 870MHz)should be invisible (they are below the noise floor!).

Note the input-power Pin at frequency f = 868MHz so that we can refer to this maximuminput power in later measurements!

Pin = dBm

26


Fig. 15: Measurement Set-Up to Verify Low Distortions in the CombinedTest-Signals

Fig. 16: Cable Connections for the Measurement Set-Up

We should also find that the third harmonic of the signal generators is below the noisefloor when we set the center frequency to f = 3 x 868MHz = 2604MHz for a moment(Don’t forget to reset to frequency f = 868MHz afterwards!). This means, wehave a clean combined signal with little distortion which is worthy of feeding into ourDUT.Otherwise: If the input signal already shows considerable distortion, the harmonics andintermodulation products will be amplified by the DUT and it becomes impossible toseparate the contribution of the DUT to the harmonics and products at its output.

27


6.2 Tests of Transistor Amplifier

6.2.1 The Third-Order Intermodulation

We extend our measurement set-up by inserting our Device Under Test (DUT), the LowNoise Amplifier (LNA) of the ME1000 RF Training Kit (RX-Box), Fig. 17.

Fig. 17: Amplifier (LNA) as DUT for Third-Order Intermodulation

Fig. 18: Measurement Set-Up and DUT for Third-Order Intermodulation

(a) Switch-on the amplifier using the computer control board.

(b) The input power Pin of the amplifier now is the same as we measured before!

Increase the attenuation a of the variable attenuator in two steps from a = 0dB toa = 6dB and read-out from the Spectrum Analyzer display the amplitudes of thetwo fundamentals at frequency f = 868MHz and frequency f = 869MHz as wellas of the two intermodulation products at frequency f = 867MHz and frequencyf = 870MHz. Take the average of both pairs of amplitudes than enter in table andplot in Fig. 19.

28


(c) Increase the attenuation a of the variable attenuator in 10dB-steps from a = 10dBto a = 30dB and read-out from the Spectrum Analyzer display the amplitudes ofthe two fundamentals at frequency f = 868MHz and frequency f = 869MHz. Takethe average of both pairs of amplitudes, enter in table and plot in Fig. 19 as pointsPout versus Pin for the fundamental signals.

a/dB 0 3 6 10 20 30

Pin/dBm

Pout/dBm

PIM/dBm

From the ratio of the output power Pout and the input power Pin determine theamplifier gain G:

G = dB

(d) Draw the best fit straight lines through the measured points (linear regression)and check the slope of these lines and compare to theory:

Slopefundamental = dB(output)/dB(input)

SlopeIMproduct = dB(output)/dB(input)

(e) Extend the fundamental line and the 3rd order line and determine the interceptpoint IP3:

IP3(output) = dBm

(f) Enter the noise floor as displayed by the Spectrum Analyzer and indicate theSpurious Free Dynamic Range Df in the graph and give the numerical value:

Noise floor = dBm

Df = dB

29


−80 −70 −60 −50 −40 −30 −20 −10 0−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Pin (dBm)

Pout(dBm)

Fig. 19: Plot of Output Power Pout Versus Input Power Pin for Fundamentaland IM-Products

30


6.2.2 Blocking-Effect Demonstration

We now insert an additional amplifier, the Power Amplifier of the TX-Box, to boost thesignal magnitude of our 868MHz-signal and insert an additional RF Band-Pass Filter(V1.01) of the TX-Box to this signal to suppress higher harmonics, see Fig. 20 andFig. 21.

Fig. 20: Connections of the Different Units between the TX- and RX-Boxes

Fig. 21: Set-Up for the Measurement of the Blocking-Effect

The variable attenuator is removed and placed after the filter to attenuate only thissignal, but not the second signal. The second synthesizer is shifted to frequencyf = 870MHz (TX-Box).

The Spectrum Analyzer now displays signals at different levels. Set the Analyzer to ahigher reference level:

• AMPT→ REF LEVEL→ 0dBm

and set the marker to the f = 870MHz-signal

• MKR→ MARKER1→ PEAK→ NEXT PEAK RIGHT.

31


We assume that the second signal at frequency f = 870MHz is the signal that we want toreceive in a receiver. The first signal is an interfering signal which can block the receptionof our wanted signal:

(a) Start with the measurement of the signal strength of the wanted signal when theinterfering signal is highly attenuated by setting the attenuator to a = 30dB.This is approximately the same as if no attenuated signal were present. Nowincrease the interferer strength by reducing the attenuation a until the wantedsignal strength reduces by 1dB - the interferer signal strength in this situation isthe 1dB-blocking level P1dBblock at the amplifier input:

a1dBblock = dB

P1dBblock = dBm

(b) The blocking turns into cross-modulation when the interfering signal is amplitudemodulated. You can demonstrate this by switching in and out the attenuator10dB-step.

Note: With a 10dB amplitude modulation depth in a AM broadcast system wewould hear the second station!

32


6.2.3 Mixing demonstration

Remove the fixed attenuator between the 870MHz-signal generator and the combiner(see Fig.21) - this signal is now 10dB stronger than before! However, we are interestedin the mixing effect between the two signals which creates mixing products at frequencyf = 2MHz and frequency f = 1738MHz at the output of our DUT, Fig. 22.

X

Fig. 22: Set-Up for the Mixing Demonstration

Tune the Spectrum Analyzer to the f = 2MHz difference frequency product.

• FREQ/SPAN→ CENTER→ 2MHz

and set the marker to the signal

• MKR→ MARKER1→ PEAK

In our measurement set-up, the 870MHz-signal amplitude is constant while only theother 868MHz-signal can be varied in amplitude. The theoretical mixing productamplitude depends on the product of both amplitudes; the theoretical dependence canbe checked by variation of the first signal amplitude by stepping-down the attenuationfrom a = 20dB to a = 0dB with observing the mixing power P2MHz:

Mixing product at 20dB attenuation

P2MHz,20dB = dBm

(a) Normalize the mixing signal amplitude P2MHz to the amplitude at the a = 20dBattenuator setting and plot this point as 0dB rel. power Rel.Power over a = 20dBattenuator setting.

Rel.Power = P2MHz/P2MHz,20dB

Rel.Power/dB = P2MHz/dBm− P2MHz,20dB/dBm

33


(b) Measure the mixing signal amplitudes P2MHz at attenuator increments of a = 5dBand plot the points in Fig.23.

a/dB 20 15 10 5 0

P2MHz/dBm

Rel.Power/dB

010200

10

20

Attenuator Setting in dB

Rel.P(ower)(M

ixingProduct)in

dB

Fig. 23: Plot of Relative Output Power Rel.Power Versus AttenuatorSettings a

34


(c) Draw best fit straight line (linear regression) through your measurement pointsand determine its slope - compare to the theoretical prediction:

SlopeMixing product = dB(output)/dB(input)

- Do you see a linear or cubic dependence?

35


6.2.4 The 1dB-Compression Power

We remove the second signal generator and the signal combiner circuit to feed the DUTwith just one signal (single-tone) of variable power, Fig. 24.

Fig. 24: Set-Up for the Measurement of the Single-Tone Compression Power

Tune the Spectrum Analyzer to frequency f = 868MHz, set the reference level to 10dBm:

• AMPT→ REF LEVEL→ 10dBm → ENTER

and activate the marker

• MKR→ MARKER1→ PEAK

to measure the DUT output power Pout and set the variable attenuator to a = 20dB inorder to create a very low level of input power Pin to the DUT where we may assumelinear operation!

(a) Measure the signal output power Pout at the given attenuator settings a, also checkthe input power level Pin at the given attenuator settings by removing the DUTconnecting directly to the RF input of the Spectrum Analyzer. Plot the pairs ofinput- and output power levels in Fig. 25!

a/dB 20 15 10 5 4 2 0

Pin/dBm

Pout/dBm

36


−80 −70 −60 −50 −40 −30 −20 −10 0−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Pin (dBm)

Pout(dBm)

Fig. 25: Graphs of Output Power Pout Versus Input Power Pin for Single-ToneDrive

(b) Draw a best fit curve through the measured points (straight line of slope 1dB/1dBwith curved upper end).

(c) Compare to an extended straight line of slope 1dB(output)/1dB(input) for an ideallinear amplifier and find the output level Pout,1dB where the measured power is 1dBbelow the ideal linear output level.

Pin,1dB = dBm

Pout,1dB = dBm

37


6.2.5 Harmonic Generation

We use the same measurement set-up as before, but now we investigate the output signalat the third harmonic frequency f = 3 x 868MHz = 2604MHz. Theory predicts that thisis a third-order product depending on the third power, i.e. the slope of power increaseshould be 3dB(output)/1dB(input). We test this by measuring the 3rd harmonic outputpower Pout,3rd with the input power Pin varied by the attenuator a.

(a) Measure the signal input power level Pin at the given attenuator settings a.

• Set the Spectrum Analyzer to f = 868MHzFREQ/SPAN→ CENTER→ 868MHz

• Set the reference power levelAMPT→ REF LEVEL→ 0dBm

• Activate the marker to the signal peakMKR→ MARKER1→ PEAK

(b) Measure the signal output level Pout,3rd at the given attenuator settings a and plotthe measured pairs of power levels in Fig. 25.

• Now set the Spectrum Analyzer to f = 2604MHzFREQ/SPAN→ CENTER→ 2604MHz

• Lower the reference power level to display the 3rd harmonic signalAMPT→ REF LEVEL→ -30dBm

• Activate the marker to the signal peakMKR→ MARKER1→ PEAK

a/dB 20 15 10 8 6 4 2 0

Pin/dBm

Pout,3rd/dBm

(c) Draw the best fit line through the points of higher levels (a = 0 − 8dB) and asecond line through the points of lower levels (a = 10− 20dB).

Slope3rdharm. = dB(output)/dB(input)

(d) Determine from the slope whether we see a linear, quadratic or a cubic dependence!

38

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