Contents H F T. fur Ingenieurwissenschaften Abteilung Elektrotechnik und Informationstechnik. Fachgebiet Hochfrequenztechnik Prof. Dr. Ing. K.

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1 H F T Fachgebiet Hochfrequenztechnik Prof. Dr. Ing. K. Solbach Fakultat Universitat fur Ingenieurwissenschaften Duisburg Essen Abteilung Elektrotechnik und Informationstechnik Komponenten für die drahtlose Kommunikation Experiment No. 1 Version: April 19, 2012 Last name: First name: Matr.-No.: Tutor: Date: Contents 1 Literature 2 2 Introduction 2 3 Theoretical Background Nonlinear Signal Analysis Analysis Dynamic Range of Nonlinear Two-Ports Dynamic Range Measurements and Evaluation Decibel Scale for Power and Voltage Deriving Taylor-Series Coefficients from Measurement Slope in Logarithmic Plots Homework 18 5 Description of Instruments Spectrum Analyzer Main Setting Parameters R&S FS300 Spectrum Analyzer Signal Combiner Circuits Performing the experiment Check of Test Signals and Spectrum Analyzer Tests of Transistor Amplifier The Third-Order Intermodulation Blocking-Effect Demonstration Mixing demonstration The 1dB-Compression Power Harmonic Generation

2 1 Literature a) D. Pozar: Microwave and RF Design of Wireless Systems New York: John Wiley & Sons, 2001, chapter 3.7 b) Agilent Technologies: Application Note 150, Agilent Spectrum Analysis Basics 2 Introduction Two-port circuits can be subdivided into passive and active circuits. Passive circuits may be linear or nonlinear circuits and active circuits are necessarily nonlinear. Typical examples 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 the active element. As will become clear later, even active circuits may be assumed to be linear 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 at the output-port that is delayed by some delay-time τ and is undistorted otherwise apart from some amplitude scaling: An amplifier should normally scale-up, i.e. increase the amplitude, while passive circuits tend to reduce. Note that a scaling-up may be accomplished also by a simple (passive) transformer, if only output voltage is compared to input voltage; yet this goes along with a 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 v 2 and input voltage u 1 and voltage gain factor g u, yielding u 2 (t) = g u u 1 (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) input signal. 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, leading to frequency-dependent transfer characteristics. The best example for this property is a frequency-filter, which attenuates or passes and shifts phase of certain parts of the spectrum of the input signal. The resultant signal may be restored to the original form, if we apply an inverse filter circuit, which amplifies what has been attenuated 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 sinusoidal signal at the input has to be found at the output as a pure sinusoidal with only amplitude scaling and some phase shift. The term linear also indicates that the superposition theorem applies: If we supply two separate signals to the input of the two-port, we find the output signal to consist of the sum of two output signals, 2

3 where each is the response to one of the input signals. In particular, if we apply two sinusoidal 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 and phase 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 capacitive circuit elements we find additional nonlinear distortion in circuits containing nonlinear circuit elements, e.g. diodes and transistors, which are characterized by curved voltage-to-current characteristics. If we apply a sinusoidal voltage at the input of a nonlinear circuit we find a distorted, non-sinusoidal output signal, which by Fourier-analysis can be decomposed into a sinusoidal component at the original frequency and many harmonics of this fundamental. If we apply simultaneously two sinusoids at the input, we find the distorted output signal composed of fundamentals and harmonics of both signals plus additional sinusoidal signals of combination frequencies of the two input signal frequencies. These so-called products increase in relative magnitude with increasing input signal magnitude and thus limit the usefulness of a two-port for high amplitude operation. In radio frequency communication systems, nonlinear distortion products can create serious interference problems as will be seen later on. Note that the nonlinear distortion effect has long been known from audio amplification where we can hear the non-fidelity in the reproduced acoustic signal from an amplifier/loudspeaker at too high volume setting. 3

4 3 Theoretical Background 3.1 Nonlinear Signal Analysis For the description of characteristics of electronic circuit elements, using the transistor as 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 is acceptable to approximate the curved characteristic of solid state elements by a linear characteristic, e.g. a small-signal conductance due to the first derivative (slope) of the curved diode characteristic at its operating point. More precisely, we need a nonlinear description, which can be given using the Taylor Power Series Approximation, as known from mathematics: We approximate a nonlinear voltage/current characteristic or the voltage-to-voltage transfer characteristic of a two-port containing a non-linear element by a Taylor series expansion which comprises zero-order (bias-) terms, the first order term (linear or smallsignal) plus second and additional higher order terms: y(x) = y 0 + a 1 (x x 0 ) + a 2 (x x 0 ) 2 + a 3 (x x 0 ) 3 + a 4 (x x 0 ) (2) where y(x) is the output quantity (e.g. voltage at the output port of an amplifier), y 0 is the output operating point (e.g. the d.c. bias voltage at the output port of an amplifier), x 0 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 a i are the coefficients of the Taylor series (Fig. 1). y output signal y 0 operating point x 0 x input signal Fig. 1: Transfer Characteristic of an Amplifier As long as the term (x x 0 ) is small, the higher-power terms of the series expansion are even smaller and thus in many cases may be neglected, so that the simple smallsignal approximation is justified. However, if we look closer, we find that the curvature of nonlinear transfer characteristics is the cause of many unpleasant, degrading effects but can also be employed to create the important frequency translation functions in RF systems. In the following, we analyze such behavior by truncating the Taylor series after the third-order term and assume the input signal x to be composed of two sinusoidal timevarying signals of angular frequency ω 1 and ω 2 and amplitudes A 1 and A 2, respectively. It has to be cautioned here that truncation of the power-series will limit the accuracy of the 4

5 approximation and that modeling the nonlinear behavior of two-ports by above nonlinear transfer characteristic in principle is only an approximation: This is so, since circuit responses also depend on the slope of time-domain signals (or the frequency), which partly is due to the existence of energy storage elements (and linear distortion); nevertheless, experience shows that good agreement of the theoretical model and measured results is achieved for many practical circuits and in particular, the broadband-type of amplifiers used in this lab Analysis Consider a Taylor series of 3rd order and a two-tone input signal x(t) = A 1 cos(ω 1 t) + A 2 cos(ω 2 t) and bias x 0 = 0. The output signal y(t) as a function of time is y(t) = y 0 + a 1 x(t) + a 2 x 2 (t) + a 3 x 3 (t) = y 0 + a 1 [A 1 cos(ω 1 t) + A 2 cos(ω 2 t)] + a 2 [A 1 cos(ω 1 t) + A 2 cos(ω 2 t)] 2 + a 3 [A 1 cos(ω 1 t) + A 2 cos(ω 2 t)] 3. (3) Using cos 2 α = 1 (1 + cos 2α) 2 cos 3 α = 1 (cos 3α + 3 cos α) 4 cos α cos β = 1 [cos(α β) + cos(α + β)] 2 (a + b) 2 = a 2 + b 2 + 2ab (a + b) 3 = a 3 + b 3 + 3ab 2 + 3a 2 b we can first solve the binomials of second and third powers of our two-tone signals y(t) = y 0 + a 1 A 1 cos ω 1 t + a 1 A 2 cos ω 2 t + a 2 A 2 1 cos 2 ω 1 t + a 2 A 2 2 cos 2 ω 2 t + 2a 2 A 1 A 2 cos ω 1 t cos ω 2 t + a 3 A 3 1 cos 3 ω 1 t + a 3 A 3 2 cos 3 ω 2 t + 3a 3 A 1 A 2 2 cos ω 1 t cos 2 ω 2 t + 3a 3 A 2 1A 2 cos 2 ω 1 t cos ω 2 t (4) and can go on with resolving the resultant terms (products) with respect to frequencies 5

6 y(t) = y 0 + 1a 2 2A a 2 2A 2 2 : direct current (d.c.) + cos ω 1 t [a 1 A 1 + 3a 4 3A a ] 2 3A 1 A 2 2 : fundamental wave (ω 1 ) + cos ω 2 t [a 1 A 2 + 3a 4 3A a ] 2 3A 2 1A 2 : fundamental wave (ω 2 ) + cos 2ω 1 t [ 1 a ] 2 2A cos 2ω2 t [ 1 a ] 2 2A 2 2 : second harmonic (2ω 1, 2ω 2 ) + cos 3ω 1 t [ 1 a ] 4 3A cos 3ω2 t [ 1 a ] 4 3A 3 2 : third harmonic (3ω 1, 3ω 2 ) + cos ( (ω 1 ω 2 )t ) [a 2 A 1 A 2 ] + cos ( (ω 1 + ω 2 )t ) [a 2 A 1 A 2 ] : mixing products (sum- and difference-frequencies) + cos ( (2ω 1 ω 2 )t ) [ 3 a ] ( 4 3A 2 1A 2 + cos (2ω2 ω 1 )t ) [ 3 a 4 3A 1 A cos ( (2ω 1 + ω 2 )t ) [ 3 a ] ( 4 3A 2 1A 2 + cos (2ω2 + ω 1 )t ) [ 3 a ] 4 3A 1 A 2 2 ] : intermodulation in-band : intermodulation out-of-band Fig. 2: Output Spectrum of Second and Third-Order Two-Tone Intermodulation Products, Assuming ω 1 < ω 2 We learn from this table of terms: An input signal x(t) to a transfer characteristic of third order (and with x 0 = 0) creates an output signal y(t) = y 0 + a 1 x(t) + a 2 x 2 (t) + a 3 x 3 (t) (5) which consists of several types of products: a) Direct Current Products y(ω = 0) = y 0 + a 2 (A A 2 2) (6) 2 The d.c. bias point y 0 of a two-port will be offset, increasing with the square of the input signals. This is exploited, e.g. in detector circuits, where a.c. voltage is converted to d.c. voltage b) Fundamental or 1st Harmonic Products ( ) A A 2 2 y(ω 1 ) = A 1 a 1 + 3a 3 4 The factor a 1 is the linear amplification factor of the two-port, but with increasing amplitude of a desired signal A 1, the second term can become large and reduce the amplification (because the third-order term a 3 < 0 as seen later in chapter 3.3.2). This means, the output signal becomes compressed. Such compression also takes place if 6 (7)

7 we have a second strong signal present (note amplitude A 2 in the second term). Special Case: Single-Tone Operation (A 2 = 0) y saturation region compression linear range 0 A 1 Fig. 3: Compression in Single-Tone Operation For a single-tone operation, we find that the compression effect increases as the third power of the input signal amplitude and relies on the third-order term of the Taylor series. y fund. (t) = [ a 1 A 1 + 3a ] 4 3A 3 1 cos ω 1 t (8) }{{} ŷ(ω 1 ) With increasing amplitude A 1 we first see a compression (reduced gain) as a 3 < 0. At even higher levels, complete saturation with constant output signal level takes place in real systems! However, this is outside our present mathematical model, which is only able to describe a limited amount of compression due to the third-order term (Fig. 3). Special Case: Two-Tone Operation A 2 A 1 RF signal interfering signal with AM 1 2 RF filter width Fig. 4: Radio Reception The second term in y(ω 1 ) contains the effect of a second signal on the first signal by the contribution from A 2 2: y CM (t) = 3 2 a 3 A 1 A 2 2 cos ω 1 t (9) If the second signal is a carrier with constant amplitude, the effect is to increase compression, i.e. to reduce the output from the first signal; we speak of blocking of the first 7

8 signal by the second one (Fig. 4). If the second signal is an amplitude modulated carrier (e.g. in radio broadcast reception), we have A 2 = Â2 cos(ω NF t) (10) and the blocking effect varies in the same way as the modulation of the second signal; in radio broadcast reception we can hear the program of a second radio station overlaid to the 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 or Two-Tone Operation The curvature of nonlinear active elements transfer characteristics is often used in order to produce signals of higher frequency by multiplication of the fundamental frequency by an integer number. Such multiplier circuits need to employ frequency filters to filter-out the desired harmonic signal from the output signal. Second harmonics increase as the square of the input signal amplitudes and rely on the quadratic term in the Taylor series: y n = a 2 2 A2 1 cos(2ω 1 t) y n = a 2 2 A2 2 cos(2ω 2 t) n = 2 (11) Third harmonics increase as the third power of the input signals and rely on the thirdorder term of the Taylor series: y n = a 3 4 A3 1 cos(3ω 1 t) y n = a 3 4 A3 2 cos(3ω 2 t) d) Mixing Products n = 3 (12) The second-order term of the Taylor series is responsible for the creation of signals at the non-harmonic difference- and sum-frequencies of the two tones. This is exploited in RF-systems for the translation of signals from one frequency to another frequency and is called mixing. E.g. in radio receivers, we use to translate the received broadcast signal at ω 1 to a lower frequency in our receiver (the Intermediate Frequency or IF) by mixing with a sinusoidal signal at a frequency ω 2 which is offset by the intermediate frequency ω 1 ω 2 =IF: 8

9 Difference frequency: y IF (t) = a 2 A 1 A 2 cos ( (ω 1 ω 2 )t ) In typical transmitter systems, we translate the low-frequency information signal at ω 1 up into a higher radio frequency (RF) by mixing with a sinusoidal signal of high frequency ω 2 and extracting the sum frequency for further amplification and transmission: Sum frequency: y SF (t) = a 2 A 1 A 2 cos ( (ω 1 + ω 2 )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 as high as possible level of the mixing product due to the factor A 2 in both mixing product terms above. We can also be sure that the frequency-translated signal will be a true replica of the original signal A 1 cos(ω 1 t) with any phase- and amplitudemodulation preserved, since A 1 appears only in its first power ( linear ) and the local oscillator signal is described by a constant amplitude A 2 and constant angular frequency ω 2. In a practical mixing circuit, we again need to employ a filter circuit in order to select the desired 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 the RF-signal fed to the gate, the LO-signal fed to the source and the IF-signal tapped from the drain terminals (Fig. 5). The nonlinear element in the circuit is the approx. quadratic relation between drain current and gate-source voltage u GS. In a typical application for VHF broadcast reception, we would receive a station at f RF = 100MHz, operate the local oscillator at f LO = 110.7MHz and produce an intermediate frequency signal at f IF = 10.7MHz, which is amplified and demodulated in a following stage. Intermediate frequency signal u GS f RF f LO Local Oscillator f IF Amplitude IF RF-band local oscillator! 2! 1! 1! 2 2! 1 2! 2! 1! 2 Fig. 5: Mixer Circuit With a Field-Effect Transistor (FET) 9

10 e) Intermodulation Products The intermodulation products y IM (t) = 3 4 a 3A 2 1A 2 cos ( (2ω 1 ω 2 )t ) y IM (t) = 3 4 a 3A 1 A 2 2 cos ( (2ω 2 ω 1 )t ) (13) are created at non-harmonic frequencies (2ω 1 ω 2 ) and (2ω 2 ω 1 ) which may be close to both of our two tones ω 1 and ω 2, if ω 1 and ω 2 are close. Therefore, in many cases it is hardly possible to filter out these products and we have to live with them although they may cause interference to our system. The amplitudes of intermodulation products may be very small, if both A 1 and A 2 are small, but, as both grow, the products A 2 1A 2 or A 1 A 2 2 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 of the Taylor series and since their amplitudes are determined by a third power of signal amplitudes. Example: Communications Transmitter Amplitude carrier side band Amplitude additional sidebandes 1 2 (a) (b) Fig. 6: Modulated RF-Signal. (a) Single-Sideband, (b) Distorted Modulation and Broadened Spectrum A carrier signal ω 1 and a modulation sideband ω 2 are amplified by a transmitter power amplifier before radiation through the antenna. Due to intermodulation, the spectrum of the transmit signal after the power amplification is broadened (additional, false modulation sidebands). This creates interference in the neighboring channels, which normally are 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 signals and thus may result in serious interference. On the other hand, such interference signals increase at a steeper slope than the original signals: In measurements it is usual to apply the two tones at the same level A 1 = A 2 = A, so that y IM (t) = 3 4 a 3A 3 cos ( (2ω 1 ω 2 )t ). (14) We recognize from this, that the intermodulation product increases with the third power of the input signal amplitude. On a logarithmic scale, the slope of the transfer 10

11 characteristic (y IM versus A) is thus three, while the fundamental output signal increases only proportionally to the input signal (y fund. versus A), i.e. by the first power; thus, the slope for the fundamental signal component is one. When we plot the transfer characteristic, we usually plot output power P out in decibels (dbm) versus input power P in in decibels (dbm) of the various signals, which does not change the above mentioned slopes. Note: The quasi-unit of dbm indicates that before taking the logarithm we normalize the signal powers P to the power of P 0 = 1mW! Dynamic Range Gain IP 3 Intercept Point 1dB P out (dbm) slope=1 D f slope=3 D N 0 P in (dbm) Fig. 7: Transfer Characteristic for Fundamental and Third-Order Signals of a Single Test Signal We recognize (Fig. 7) the straight line under 45 (slope = 1) as the transfer characteristic of the fundamental signals which relates the output signal power to the input signal power 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 curve demands: There is a certain point, where the output power P out stays below the linear power by exactly 1dB. This output power P 1dB is often mentioned in data sheets of amplifiers as the 1dB-compression power and marks the upper limit of useful output power available from the amplifier for many practical applications, or we call it the upper limit of the dynamic range. 11

12 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 on the bandwidth B of measurement and can be calculated by: N 0 [dbm] = 10dB log (G F kt 0 B) ( [ ]) 1 = 10dB log G F Ws B s ( [ ]) 1 = 10dB log G F mws B 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, T 0 the reference temperature of 290 K and B is the system bandwidth (see experiment no. 2 Noise Characterization of Two-Port Networks ). In practical RF systems applications signals carry a certain modulation bandwidth to contain the transmitted information (so this fixes the minimum system bandwidth) and usually require a certain signal-to-noise ratio (S/N), e.g. a factor of 10 or 10dB, in order that 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 noise floor 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 solely defined by the compression effect, but in many applications it is rather defined by the creation of undesired intermodulation signals: We recognize a certain input signal level where the intermodulation transfer characteristic (slope = 3) crosses the noise floor. This marks the level of signals that we can process without noticing the existence of intermodulation 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 range of signal power to this point is called the Spurious-Free Dynamic Range (SF DR = D f ). 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] = P 1dB [dbm] N 0 [dbm]. (15) Spurious Free Dynamic Range D f : Dynamic range without intermodulation products above the noise floor: D f [db] = 2 3 (IP 3[dBm] N 0 [dbm]). (16) Intercept Point of Third-Order IP 3 : Power level marking the cross-over of the linear and the third-order transfer characteristics, where the intermodulation product levels would be equal to the fundamental signal levels. As can be seen, this does not happen in reality because also the intermodulation products compress at high signal levels. The Intercept Point of Third-Order, sometimes called Third Order Intercept (TOI) is the most common single value describing 12

13 the large-signal handling capabilities of nonlinear components, like amplifiers, and can range around a few dbm (mw) in small-signal receiver stages. Cell phone base station transmitters are very critically limited with respect to the broadening of the transmit spectrum, so that intermodulation generation must be suppressed to a minimum. This requirement can be fulfilled by operating a transmitter of many 100W output power capability (thus very high IP 3 of the order of 1kW) and reducing the drive signal power to such a level that only 10% of the power capability is actually realized as generated output power to the antenna; this mode of operation is called back-off. Since the backoff mode is very expensive in equipment cost and supply power consumption, today we employ linearization schemes that improve the linearity of amplifiers by means of extra equipment, e.g. by pre-distortion or distortion cancellation. 3.3 Measurements and Evaluation Decibel Scale for Power and Voltage Nonlinear two-ports can be modeled by their nonlinear transfer characteristic using the Taylor 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 a logarithmic scale in dbm: A sinusoidal signal is measured as peak-voltage U over a load resistor R to give absorbed power P P = 1 2 U 2 R 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 P 0 = 1 mw and given as dbm by: ( ) ( ) P P P [dbm] = 10dB log = 10dB log 1mW P 0 (17) (18) while the voltage U 0 from P 0 = 1mW of power is U 0 = 2 P 0 R = 2mW R (19) i.e., for the standard system impedance of R = 50Ω we have: U 0 = 0.316V. (20) Instead of giving the power in decibel over P 0 = 1mW, we could also give voltage in decibel over U 0 = 0.316V (50 Ω-impedance system) ( ) ( ) U U U [db] = 20dB log = 20dB log (21) U V which leads us to exactly the same decibel-numbers as for power, when we express power by voltage squared: ( ) ( ) ( ) P U 2 U P [dbm] = 10dB log = 10dB log = 20dB log. (22) 1 mw (0.316 V) V 13

14 3.3.2 Deriving Taylor-Series Coefficients from Measurement a) When we measure the power ratio of a fundamental tone, with levels far below the compression level, we know this as the linear power gain ( ) Pout G[dB] = 10dB log. (23) P in We can determine the ratio of voltages from this by G[dB] = 20dB log U out U in (24) and the Taylor series coefficient a 1 becomes a 1 = U out U in = 10 G[dB]/20 db with [a 1 ] = 1. (25) b) We can evaluate the measured 1 db-compression output power P 1dB by inspection of the full fundamental wave output signal formula based on the Taylor series and using only a single tone x(t) = A cos ωt y fund. (t) = cos(ωt) [a 1 A + 34 ] a 3A 3. (26) The condition for 1dB-compression is that the output fundamental signal amplitude is 1dB smaller than it would be if there were no compression. At 1dB-reduction means a factor of so that 10 1/20 = (or 90%) (27) a 1 A a 3A 3 }{{} linear term plus compression term = a 1 A }{{} only linear term (28) or a 1 ( ) a 3A 2 = 0 (29) or a 3 = (30) a 1 4 A2 This shows, that a 3 will be negative if a 1 is positive! 14

15 With the linear gain G given, we know a 1 and from the output power P 1dB we calculate the uncompressed power P unc. to: P unc. [dbm] = P 1dB [dbm] + 1 db. (31) From P unc. and the linear gain G we conclude on the input signal power P in = P unc. [dbm] G[dB] and on the signal voltage squared A 2 = ( U 0 1) 2 10 P 1dB [dbm]+1db G[dB] 10 db (32) since voltage U squared is proportional to power P. c) We can determine the second-order coefficient a 2 of the voltage Taylor series from a suitable measurement, e.g. the second order harmonic generation from a single-tone excitation. The appropriate expression from our table of products is: [ ] 1 y 2 (t) = cos(2ωt) 2 a 2A 2. (33) For the determination of a 2 we need a measurement of the second harmonic signal voltage amplitude at the output 1 2 a 2A 2 at a given input signal voltage A. If both signal powers are measured in Watts and assuming negligible frequency variation of the two-port (flat frequency response, no filter included) P out = 1 2 ( 1 2 a 2A 2) 2 R and P in = 1 2 A2 R we can solve for a 2 : with R = 50Ω a 2 = 2R Pout R P in with [a 2 ] = 1 V (34) 15

16 d) The intercept point of third order IP 3 can be constructed as the meeting point of the fundamental wave line of slope = 1 and the third-order products line of slope = 3 in a logarithmic plot. We need at least two different voltages for the two input tones to construct each of the two lines; more measurement points can improve the accuracy of the graphical construction. From the intercept point of third order IP 3 we may derive the third-order coefficient a 3 in the following way: The condition for the intercept point of third order IP 3 is that the linear fundamental output signal voltage is of equal magnitude as the output intermodulation signal voltage when we use equal amplitudes two-tones of amplitude A: or a 1 A IP3 = 3 4 a 3A 3 IP 3 a 3 = 4 3 a 1 A 2 IP 3 (35) where A IP3 is the signal input voltage producing the output power of the intercept point of third order IP 3 by amplification due to a 1 P IP3 = 1 2 (a 1A IP3 ) 2. (36) R Note that we use a negative sign in order to compensate that the coefficient a 3 < 0. We now solve for a 3 : a 3 1 a 3 = 2 3 P IP3 R with [a 3 ] = 1 V 2. (37) Note: Should the intercept point of third order IP 3 be given in dbm, we need to convert to power in Watt: P IP3 = 10 IP 3 [dbm] 10 db 10 3 = 10 IP 3 [dbm] 10 db 3 16

17 3.3.3 Slope in Logarithmic Plots When we plot the output power P out in dbm versus the input power P in in dbm we apply the logarithm to the square of the voltages at output and input: ( ) U P out [dbm] = 20dB log out, (38) V ( ) U P in [dbm] = 20dB log in. (39) V For the linear term in the Taylor series we find U out = a 1 U in and ( ) U P out [dbm] = 20dB log a 1 in = C dB log V ( U in V For the quadratic term in the Taylor series we find U out = a 2 U in 2 and [ ( ) ] 2 U P out [dbm] = 20dB log a V in V ( U = C dB log in V and for the cubic term in an analogue way P out [dbm] = C dB log Introducing (39) into (40) - (42) gives ( U in V P out,linear [dbm] = C P in [dbm], P out,quadr. [dbm] = C P in [dbm], P out,cubic [dbm] = C P in [dbm] where C 1, C 2, C 3 are constants. Plotting these relations as P out [dbm] vs. P in [dbm] shows: ). (40) ( ) 2 U = C dB log in V ), (41) ). (42) a) The output power P out in dbm is a linear function of the input power P in in dbm with 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

18 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 f 1 = 11MHz and frequency f 2 = 12MHz excites a nonlinear 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 f 0 = 11.5MHz? 4.3 A short-wave receiver is troubled by nonlinear distortion at the presence of very strong local transmitters. Instead of just receiving the really existing radio station signals, our receiver produces additional signals, which interfere with real signals from 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 levels and thereby reduce nonlinear distortion. Our signal strength-meter indicates the following changes in the level of three different signals when we switch-in a 10dB attenuation: Signal Signal level 0 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 signals, either a third-order intermodulation product or a second-order mixing product. (b) What is the improvement in intermodulation suppression (ratio of fundamental and intermodulation product in db) that we achieve per db of attenuation? 18

19 4.4 A broadband transistor amplifier is excited by a single-tone generator. We measure a linear gain of G = 20dB for the fundamental wave and a second harmonic product output power of P out = 1mW at an input power of P in = 1mW when we use generator and load impedance of R = 50Ω. (a) Determine the Taylor series coefficients a 1 and a 2. (b) What will be the second harmonic output power P out at an input power of P in = 2mW? 4.5 A transistor amplifier is characterized by a system bandwidth of B = 1MHz, a noise figure of F = 10dB and by the following two-tone measurements (power levels per tone or product). P in P out fundamentals intermodulation -10 dbm 10 dbm -35 dbm 0 dbm 20 dbm -5 dbm (a) Determine the intercept point of third-order output level IP 3 in dbm. (b) Determine the spurious free dynamic range SF DR in db. 19

20 5 Description of Instruments 5.1 Spectrum Analyzer In our experiment we use a spectrum analyzer for the measurement of single-tone and two-tone signal levels at the input and output of our nonlinear two-ports. The principle of operating of a spectrum analyzer is explained based on Fig. 8. Attenuator RF-Filter Mixer IF- Amp. Detector IF-Filter f Signal In Low-pass Filter Sweep Generator Local Oscillator Display Fig. 8: Principle Block Diagram of Spectrum Analyzer A signal from the input terminal has to pass a bandpass filter before being mixed down to an intermediate frequency f IF and being amplified, detected and displayed. The RF-filter and the local oscillator are tuned simultaneously in such a way that the signal can mixed to a fixed frequency f IF. Before detecting of the frequency translated signal by a detector circuit, the IF-signal has to pass through a filter bank of selectable bandwidth B; the selected bandwidth defines the system bandwidth with respect to the noise floor. Since in many situations the signal frequency is unknown in the beginning, the RF-filter and LO-frequency can be varied using the sawtooth sweep generator: In sweep mode, the receiver system continuously searches a predetermined frequency band (selectable from the front panel) at a selectable repetition rate. If there is just one signal within the search range we observe a single peak at the display; using a marker we are able to determine the precise frequency of the signal. If there are several signals, there will be several peaks displayed, with the lowest frequency at left and the highest frequency at right; 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. Using a logarithmic scale, we may choose a resolution of 10dB per division on the screen. This means, the highest peak is the strongest signal and precise values can be read-out using the marker. Without any signal at the input we observe the noise floor at the lower end of the display. Note that the noise floor varies with the selected bandwidth B! 20

21 One problem connected with the frequency sweeps is that in each sweep a signal is caught only for a short moment, depending on the sweep rate, the RF sweep width and the IF bandwidth. We must carefully select suitable combinations of the adjustments in order to allow the signal to build-up through the narrow-band IF-filter (minimum time roughly τ = 1/B) in order to allow precise measurement results. Modern spectrum analyzers indicate a warning to the user if incorrect settings are chosen or select valid settings automatically Main Setting Parameters Spectrum analyzers usually provide the following elementary setting parameters: Frequency Display Range The 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 Range This 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. The resolution/div defines the level distance between two horizontal grid lines on the CRT display. The attenuation of an input RF attenuator depends on the reference level and will generally be adjusted automatically. Frequency Resolution For analyzers operating on the heterodyne principle, the frequency resolution is set via the bandwidth of the IF filter. The frequency resolution is therefore referred to as 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 from noise, and to smooth the trace that is subsequently displayed so that the display is stabilized. Sweep Time The time required to record the whole frequency spectrum that is of interest is described as sweep time. This value will generally be adjusted automatically, since it is limited by the video bandwidth. 21

22 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 communication 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 automatic settings controlled by the instrument firmware. The instrument front panel is divided into a data field, where parameters can be adjusted by a tuning knob or by keyboard and 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 Cursor Keys until the freq/span key is highlighted and the associated menu is presented. (Function area III at the right side of the display) 22

23 Fig. 10: R&S FS300 Spectrum Analyzer Display Now, select the center function key and enter the frequency by numerical entry and confirming by pressing the blue MHz-unit key (key pad at the right side). Other menus appear after selection using the Cursor key again. The ampt menu is used to set the RF input attenuator and the reference level, the mkr menu for setting the marker to the signal peaks for read-out of the levels and frequencies and the bw/sweep menu for setting the 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 the preset state (Factory) using the sys Key (F9)! For most of our measurements, the Spectrum Analyzer should be set to: FREQ/SPAN CENTER FREQ 868MHz SPAN 5MHz AMPT RF ATTEN MANUAL 20dB Enter REF LEVEL -10dBm Enter MKR MARKER 1 PEAK or NEXT PEAK LEFT or NEXT PEAK RIGHT BW/SWEEP RES BW MANUAL 10kHz Enter 23

24 5.2 Signal Combiner Circuits Signal Generators DUT Combiner Spectrum Analyzer 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 into our nonlinear Device Under Test (DUT). Since we want to investigate the creation of nonlinear products in our DUT, we need pure sinusoidal signals to feed into the DUT. Therefore we have to avoid to terminate the signal generators by unmatched impedances and have to avoid that the signal from our generator feeds into the other generator producing nonlinear products there. Hence, combining the signal of the two generators requires special 3-ports combiner circuits, which match the generator ports and 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 threeport with 6dB of insertion loss and 6dB of isolation of the two generators. Other combiner/splitter circuits have been developed, e.g. the Wheatstone-bridge and the Wilkinson-combiner, which give lower insertion loss and higher isolation but at the price of restricted bandwidth. In our case, the generators need more isolation and therefore we employ the special power combiner circuit of Fig. 12 with block diagram in Fig. 13, which provides two double nonreciprocal isolators connecting to the generators and a 3dB-hybrid directional coupler to combine the two signals; this circuit operates over a frequency range of about f = 800MHz to f = 1200MHz and affects an insertion loss of about 4dB and an isolation of about 50dB to the generator signals. 24

25 Fig. 12: Circuit for the Combination of two Signal Generators (Generators Inputs at the Left, Combined Signal Output at the Right) 50W C G1 3dB 50W 50W 50W G2 50W Output 50W Monitor Fig. 13: Power Combiner - Block Diagram 25

26 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 after the combiner circuit. Our signal generators are the synthesizer circuits of our ME1000 RF Training Kit (both the RX- and the TX-Boxes), Fig. 14, which we activate through the 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 frequency of f = 868MHz and the second one (TX-Box) to a frequency of f = 869MHz. We use two coaxial cables to connect the output ports of both synthesizers to the two input ports of the signal combiner with a 10dB coaxial SMA attenuator inserted to each of the connecting lines. The two signals from the combiner circuit travel through the variable attenuator and enter the RF input of the Spectrum Analyzer. We check the signals with the attenuator 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 precise read out!) at approximately the right frequencies (some inaccuracy is due to the frequency 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 P in at frequency f = 868MHz so that we can refer to this maximum input power in later measurements! P in = dbm 26

27 Fig. 15: Measurement Set-Up to Verify Low Distortions in the Combined Test-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 noise floor 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, we have a clean combined signal with little distortion which is worthy of feeding into our DUT. Otherwise: If the input signal already shows considerable distortion, the harmonics and intermodulation products will be amplified by the DUT and it becomes impossible to separate the contribution of the DUT to the harmonics and products at its output. 27

28 6.2 Tests of Transistor Amplifier The Third-Order Intermodulation We extend our measurement set-up by inserting our Device Under Test (DUT), the Low Noise 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 P in 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 to a = 6dB and read-out from the Spectrum Analyzer display the amplitudes of the two fundamentals at frequency f = 868MHz and frequency f = 869MHz as well as of the two intermodulation products at frequency f = 867MHz and frequency f = 870MHz. Take the average of both pairs of amplitudes than enter in table and plot in Fig

29 (c) Increase the attenuation a of the variable attenuator in 10dB-steps from a = 10dB to a = 30dB and read-out from the Spectrum Analyzer display the amplitudes of the two fundamentals at frequency f = 868MHz and frequency f = 869MHz. Take the average of both pairs of amplitudes, enter in table and plot in Fig. 19 as points P out versus P in for the fundamental signals. a/db P in /dbm P out /dbm P IM /dbm From the ratio of the output power P out and the input power P in determine the amplifier 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: Slope fundamental = db(output)/db(input) Slope IMproduct = db(output)/db(input) (e) Extend the fundamental line and the 3 rd order line and determine the intercept point IP 3 : IP 3 (output) = dbm (f) Enter the noise floor as displayed by the Spectrum Analyzer and indicate the Spurious Free Dynamic Range D f in the graph and give the numerical value: Noise floor = dbm D f = db 29

30 Pout (dbm) P in (dbm) Fig. 19: Plot of Output Power P out Versus Input Power P in for Fundamental and IM-Products 30

31 6.2.2 Blocking-Effect Demonstration We now insert an additional amplifier, the Power Amplifier of the TX-Box, to boost the signal 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 and Fig. 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 this signal, but not the second signal. The second synthesizer is shifted to frequency f = 870MHz (TX-Box). The Spectrum Analyzer now displays signals at different levels. Set the Analyzer to a higher reference level: AMPT REF LEVEL 0dBm and set the marker to the f = 870MHz-signal MKR MARKER1 PEAK NEXT PEAK RIGHT. 31

32 We assume that the second signal at frequency f = 870MHz is the signal that we want to receive in a receiver. The first signal is an interfering signal which can block the reception of our wanted signal: (a) Start with the measurement of the signal strength of the wanted signal when the interfering signal is highly attenuated by setting the attenuator to a = 30dB. This is approximately the same as if no attenuated signal were present. Now increase the interferer strength by reducing the attenuation a until the wanted signal strength reduces by 1dB - the interferer signal strength in this situation is the 1dB-blocking level P 1dB block at the amplifier input: a 1dB block = db P 1dB block = dbm (b) The blocking turns into cross-modulation when the interfering signal is amplitude modulated. You can demonstrate this by switching in and out the attenuator 10dB-step. Note: With a 10dB amplitude modulation depth in a AM broadcast system we would hear the second station! 32

33 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 interested in the mixing effect between the two signals which creates mixing products at frequency f = 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 the other 868MHz-signal can be varied in amplitude. The theoretical mixing product amplitude depends on the product of both amplitudes; the theoretical dependence can be checked by variation of the first signal amplitude by stepping-down the attenuation from a = 20dB to a = 0dB with observing the mixing power P 2MHz : Mixing product at 20dB attenuation P 2MHz,20dB = dbm (a) Normalize the mixing signal amplitude P 2MHz to the amplitude at the a = 20dB attenuator setting and plot this point as 0dB rel. power Rel.P ower over a = 20dB attenuator setting. Rel.P ower = P 2MHz /P 2MHz,20dB Rel.P ower/db = P 2MHz /dbm P 2MHz,20dB /dbm 33

34 (b) Measure the mixing signal amplitudes P 2MHz at attenuator increments of a = 5dB and plot the points in Fig.23. a/db P 2MHz /dbm Rel.P ower/db 20 Rel. P(ower) (Mixing Product) in db Attenuator Setting in db 0 Fig. 23: Plot of Relative Output Power Rel.P ower Versus Attenuator Settings a 34

35 (c) Draw best fit straight line (linear regression) through your measurement points and determine its slope - compare to the theoretical prediction: Slope Mixing product = db(output)/db(input) - Do you see a linear or cubic dependence? 35

36 6.2.4 The 1dB-Compression Power We remove the second signal generator and the signal combiner circuit to feed the DUT with 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 P out and set the variable attenuator to a = 20dB in order to create a very low level of input power P in to the DUT where we may assume linear operation! (a) Measure the signal output power P out at the given attenuator settings a, also check the input power level P in at the given attenuator settings by removing the DUT connecting directly to the RF input of the Spectrum Analyzer. Plot the pairs of input- and output power levels in Fig. 25! a/db P in /dbm P out /dbm 36

37 Pout (dbm) P in (dbm) Fig. 25: Graphs of Output Power P out Versus Input Power P in for Single-Tone Drive (b) Draw a best fit curve through the measured points (straight line of slope 1dB/1dB with curved upper end). (c) Compare to an extended straight line of slope 1dB(output)/1dB(input) for an ideal linear amplifier and find the output level P out,1db where the measured power is 1dB below the ideal linear output level. P in,1db = dbm P out,1db = dbm 37