First-Order Low-Pass Filtered Noise

Transcription

1 Chapter 1 First-Order Low-Pass Filtered Noise Object The object of this experiment is to become familiar with the characteristics of Gaussian noise. A spectrum analyzer, known as a Dynamic Signal Analyzer or DSA, is used to measure the noise bandwidth of a low-pass filter. A low-pass filter with a variable cutoff frequency is used to measure the spectral density of Gaussian noise. Theory The amplitude distribution of Gaussian noise is described by the Gaussian or normal probability density function. The standard deviation of a Gaussian noise voltage is the root-meansquare or rms value of the voltage. When Gaussian noise is band-limited with a filter, its Gaussian characteristics are not changed. However, its rms voltage is decreased. White Gaussian noise has a spectral density that is independent of frequency. Thus the mean-square voltage in a frequency band f is directly proportional to f. When white Gaussian noise is applied to a filter, the mean-square noise voltage at the output of the filter is the spectral density at the input multiplied by the noise bandwidth B n of the filter. For either a single-pole low-pass filter or a two-pole band-pass filter, the noise bandwidth is given by B n = π 2 B 3 where B 3 is the filter 3dB bandwidth. For the single-pole low-pass filter, B 3 = f 0,where f 0 is the pole frequency. For the second-order band-pass filter, B 3 = f 0 /Q, wheref 0 is the center or resonance frequency and Q is the quality factor. When using a voltmeter to measure the rms value of a voltage, the maximum peak voltage that can be applied to the meter without overloading the internal circuits divided by the full scale reading is called the crest factor of the meter. The peak to rms ratio for a sine wave is 2. Thus a meter that is designed to read correctly with a sine wave input must have a crest factor that is greater than 2. A commonly accepted crest factor for Gaussian noise is 4 times the rms value. Therefore, to accurately measure a Gaussian noise voltage, the rms voltage read by the meter should be less than the crest factor of the meter multiplied by the full scale voltage divided by 4. 1

2 The bandwidth of a measuring instrument plays an important role in any noise measurement. If the instrument bandwidth is less than the bandwidth of the signal being measured, then the measurement will be incorrect. Therefore, accurate measurements of noise require that the bandwidth of the measuring instrument be greater than that of the noise signal, preferably by a factor of 10. Often a filter is used to limit the bandwidth of noise to a bandwidth that is less than that of the instrument. The type of meter used to measure noise voltages is an important consideration. A true rms responding meter is the best for such measurements. However, these meters are not universal and many are average responding meters. An average responding meter is normally calibrated to read the correct rms value of the input voltage only for a sine wave with a dc level of zero. If Gaussian white noise is measured on such a meter, the reading must be multiplied by a factor of 1.13 to obtain the correct rms value. With white noise at its input, the rms noise voltage V rms at the output of a filter with a noise bandwidth B n is given by V rms = p S v B n where S v is the voltage spectral density of the white noise source. It follows that if a plot is made of log(v rms ) versus log(b n ) theresultshouldbeastraightlinewithaslopeof1/2 and a y-intercept of log( S v ). This provides an experimental technique of determine the voltage spectral density of a noise source. Because the theoretical prediction is a straight line, well known linear regression techniques may be used to obtain an accurate estimate of S v. Laboratory Procedure First Order Low-Pass Filter Design Figure 1.1 shows two cascaded single-pole low-pass filters with op-amp buffers. It is given that R 1 =10kΩ, R 2 =1kΩ, R 3 =100Ω, andc 1 =100pF.. The input filter has a 3dB frequency of 1.6MHzand prevents possible rf interference with the circuit. This frequency is significantly higher than the 3dBfrequency of the following filter which sets the bandwidth of the circuit. R 3 is included in the circuit to prevent possible oscillations due to capacitive loading on the second op amp. Figure 1.1: Low-pass filter circuit. For the second filter, the available capacitor values for C are 100 pf, 1000 pf, 0.01 µf, and 0.1 µf. For the resistors, standard 1% values are available. Determine suitable values of R and C to obtain 3dB cutoff frequencies in the second filter of 100 Hz, 300 Hz, 1kHz, 3kHz, 10 khz, 30 khz,and100 khz. The 3dBfrequency is given by f 3 =1/ (2πRC). Exact 2

3 values are not necessary. Values within 5% of the specified ones are acceptable. Measure the exact values of the resistors and capacitors using the LCR meter and use these values to calculate the theoretical 3dB cutoff frequencies. First-Order Low-Pass Filter Bandwidth Measurement With reference to Figs. 1.2 and 1.3, assemble the filter circuit on a solderless breadboard. Use a 100 Ω resistor and 100 µf capacitor to decouple each power supply rail. Note that the 100 µf capacitors are electrolytic types which must be inserted with the correct polarity. Use the function generator and the oscilloscopes and/or DSA to measure the upper 3dB frequency for each filter. An input voltage of approximately 0.1Vpeak should be used. Keep the resistors because each of these circuits will be reassembled. Record the measured data. Noise Source Measurement Observe the noise signal with both the oscilloscope and the DSA to observe the nature of the noise. Set the DSA to average the input signal 50 times. Program the DSA to measure the voltage over the frequency band khz. Determine the spot noise spectral density ( V/ Hz) ofthenoisesourceatafrequencyof10 khz. This is given by noise voltage measured at 10 khz divided by the square root of the measurement bandwidth of the signal analyzer. Obtain a screen dump of the display. Filtered Noise With reference to Figs. 1.2 and 1.3, assemble the 100 Hz filter elements on the solderless breadboard. Use the DMM to measure the dc voltage at the input and output of the circuit to assure that the dc voltage is in the millivolt range. If it is not, the input and output must be ac coupled for the next step in the procedure. Apply the noise signal from the DSA to the filter input. Connect oscilloscope to the output of the circuit. Observe the nature of the waveform on the oscilloscope and compare it to what is observed at the source output of the DSA. Spectrum of Output Noise With the noise signal from the source output of the DSA driving the circuit as in the previous procedure step, connect the circuit output to the input of the DSA. Program the analyzer to measure the total noise output and record the value. Also measure the output voltage with the HP3400 true rms voltmeter. Repeat for the other filters. AC Line Corruption UsetheDSAtoexamineanyaclinevoltage corruption at the output of the 100 Hz filter. Set the cursor to 60 Hz and the frequency span to 1kHz and obtain a plot of the display. 3

4 Laboratory Report Noise Bandwidths Calculate the noise bandwidth of each of the filters using the measured half-power cutoff frequencies. Plot the measured output noise voltage as a function of the noise bandwidth of the filter. Use log-log scales for these plots. Use a linear regression analysis to obtain a value for the spectral density of the noise source. Compare this with the value measured directly at the output of the noise source. Conclusions Explain any discrepancy between the values obtained using the different approaches for determining the voltage spectral density of the noise source. Figure 1.2: Pin Outs for 741 Type IC Figure 1.3: Solderless Breadboard Layout 4

5 Chapter 2 Band-Pass Noise Measurements Object The objectives of this experiment are to use the Dynamic Signal Analyzer or DSA to measure the spectral density of a noise signal, to design a second-order band-pass filter, to use the swept sine wave feature to measure its frequency response, and to measure its noise bandwidth with both the rms voltmeter and the DSA. Laboratory Procedure Filter Design Figure 2.1 shows the circuit diagram of a second-order infinite-gain multi-feedback band-pass filter preceded by a unity-gain buffer having an attenuator and low-pass filter at its input. The transfer function for the input buffer circuit is given by V 0 i V i = R B 1 R A + R B 1+(R A kr B ) C A s (2.1) This is a first-order low-pass filter function with a gain constant less than unity. The purpose of the voltage divider is to obtain a gain of unity from the overall circuit at the resonance frequency of the band-pass filter. The purpose of C A is to suppress possible rf interference at the input. The transfer function of the band-pass filter is given by V o V 0 i (1/Q)(s/ω 0 ) = K (s/ω 0 ) 2 +(1/Q)(s/ω 0 )+1 (2.2) where ω 0 is the resonance frequency, K is the gain at resonance, and Q is the quality factor. These are given by R 3 C 1 K = (2.3) R 1 (C 1 + C 2 ) ω 0 = 1 p (R1 kr 2 ) R 3 C 1 C 2 (2.4) 1

6 Figure 2.1: Band-pass filter with input buffer. s 1 R 3 C 1 C 2 Q = (2.5) C 1 + C 2 R 1 kr 2 If we let C 1 = C 2 = C, thevaluesofr 1, R 2,andR 3 for a specified K, ω 0,andQ are given by R 1 = Q R 2 = KR 1 R Kω 0 C 2Q 2 3 =2KR 1 (2.6) K The 3dB bandwidth ω 3 of the band-pass filter is related to the resonance frequency ω 0 and the quality factor Q by ω 3 = ω 0 (2.7) Q Let the lower and upper 3dB cutoff frequencies, respectively, be denoted by ω a and ω b. These frequencies are related to the bandwidth and the resonance frequency by Specifications r 1 ω a = ω 0 4Q Q ω 3 = ω b ω a ω 0 = ω a ω b (2.8) r 1 ω b = ω 0 4Q Q The band-pass filter is to be designed for the following specifications: (2.9) Q =10 f 0 = ω 0 =316Hz 1 K 10 (2.10) 2π Note that f 0 is the geometric mean of 100 Hz and 1kHz, so that it lies half way between these frequencies on a log scale. The following is a possible design procedure: Assume a value for C and K. A recommended range for C is 100 pf to 0.1 µf. Calculate R 1, R 2,andR 3. To minimize any problems with the filter realization, it is recommended that these lie in the range from 1kΩ to 100 kω. However, values in the range from 100 Ω to 1MΩ should give acceptable results. 2

7 ByjudiciouschoiceofC and K, it should be possible to obtain acceptable values for the resistors. The filter elements should be calculated so that the required numerical values can be obtained within an error of ±5%. Because precise capacitor values are not available in the laboratory stock, use the RCL meter to measure the value of all capacitors used in the filter. Note that two reasonably matched capacitors are required. After the gain constant K has been chosen, choose R A and R B such that the gain of the input voltage divider is 1/K. This gives an overall gain of unity at the resonance frequency of 316 Hz. An error of ±5% is acceptable. Choose C A so that the 3dB cutoff frequency of the rf filter is in the range from 1MHz to 2MHz. Filter Assembly Assemble the filter circuit on the solderless breadboard. Use a RC decoupling network on each power supply rail consisting of a 100 Ω resistor and a 100 µf capacitor. Frequency Response Oscilloscope With the oscilloscope connected to the circuit output, drive its input from the function generator and determine if it exhibits the correct frequency response. Determine the frequency f 0 at which the gain is a maximum, the gain at this frequency, and the two frequencies f a and f b at which the gain is down by a factor of 1/ 2. Calculate the quality factor from Q = f 0 (2.11) f b f a Use the oscilloscope to observe the sine wave and square wave responses at several frequencies, e.g. at f 0,atf 0 /k, andatkf 0,wherek is suitably chosen to obtain the most interesting waveforms for the square wave input case. Note any ringing on the waveform. DSA Swept Sine Wave Measurement Measure the dc voltage at the input and output nodes of the circuit with the DMM. It should be in the millivolts range. If it isn t something is seriously wrong with the circuit. Correct the problem and proceed. Set the inputs for Channels 1 & 2 on the HP35665A DSA for ac coupling. To do this, press Input, select the menus for Channels 1 & 2 setup, and select ac coupling. Connect Channel 1 to the circuit input (V i ) and Channel 2 to the output (V o ). Connect the Source out to the input node (V i ). Press the Source button and set the level to 0.1V. Press Inst Mode and select swept sine wave. Press Freq and set the start frequency to 100 Hz, the stop frequency to 1kHz,and the Steps to log. 3

8 Press Trace Coord and switch the frequency axis to log. Press Disp Format and select Bode. Set the marker to 316 Hz. Printthedisplay. Press Disp Format and select Single. Print the display. Use the frequency response data that you obtain to determine f 0 and Q. Compare the results with those obtained using the function generator and the oscilloscope. Disconnect the circuit from the DSA. Noise Analysis NoiseSourceSetup The filter circuit is not connected to the DSA for this part. Press Inst Mode on the DSA and select FFT. Set the Source output to produce a random noise voltage having a rms value of approximately 0.1 V. Turn the source on. Measure the source output voltage with the HP 3468A DMM, and the DSA programmed to measure the voltage over the extended audio range (0 100 khz). For this step, connect the Source output to the Channel 1 input of the DSA. Noise Spectral Density Measurement The filter circuit is not connected to the DSA for this part. Press Avg on the DSA, set the number of averages to 50, and turn averaging on. Set the marker or cursor on the DSA to 316 Hz (or whatever the actual center frequency of the filter is) and record the voltage level and the band over which it is measured (known as the measurement bandwidth). To determine it press Disp Format and select Measurement State. Obtain a plot of the noise spectrum. Use the data to calculate the noise voltage at 316 Hz in V/ Hz and the spectral density in V 2 / Hz. Spectra of Filtered Noise The filter circuit is connected to the DSA for this part. Apply the random noise signal at the Source output of the DSA to the input of the filter. Connect the filter output to the Channel 1 input of the DSA. 4

9 Program the signal analyzer to measure the spectrum of the filter output signal. Because the spectral density of the input noise signal is flat (white) over the filter bandwidth, the DSA display should resemble the shape of the frequency response of the filter. Program the analyzer to measure the total rms noise voltage over the frequency interval from 0 to 100 khz. Obtain a plot of the spectrum. Also measure the noise output with the the HP 3468A DMM and compare the measurements with the signal analyzer value. Use the results of the measurements to calculate the noise bandwidth B n of the bandpass filter. Compare this with the theoretical value given by B n = πf 0 /2Q. Laboratory Report 1. Turn in all plots properly labeled, i.e. title, axes, scales, and data identified. 2. Perform all calculations asked for in the procedure section. 3. Derive the transfer function of the active filter used in the experiment. Derive the equations for its two 3dB frequencies and use these equations to verify that f 0 = fa f b and Q = f 0 / (f b f a ). From the transfer function, derive the noise equivalent bandwidth of the filter. 4. Compared the measured and/or calculated values obtained with the different procedures. 5. Perform SPICE simulations to obtain the transfer function (magnitude and phase versus frequency) of the filter. First, assume that the op amp can be modeled by a voltage-controlled voltage source having a gain of Second, perform the simulation using an appropriate ac macromodel for the particular op amps used. Compare these to each other and to the experimental Bode plot. 6. Discuss anything interesting, unusual, or new that occurred in the experiment. SPICE Op-Amp Macromodel Circuits Quasi-Ideal Macromodel Macromodel circuits are used to represent the op amp in SPICE simulations. The simplest macromodel is a voltage-controlled voltage source having a voltage gain equal to the dc gain of the op amp. In SPICE, a voltage-controlled voltage source is modeled by an E amplifier. The SPICE subcircuit code for such a macromodel with a gain A = is given below. This is referred to as the quasi-ideal macromodel subcircuit in the following. The non-inverting input is node 1, the inverting input is node 2, and the output is node 3. This subcircuit model is useful in SPICE simulations of most op-amp circuits where the op amp is considered to be ideal in the design of the circuit. 5

10 *QUASI-IDEAL OP-AMP SUBCIRCUIT.SUBCKT OPAMP EOUT30122E5.ENDS OPAMP The node numbers in the op-amp subcircuit code are dummy node numbers. The actual node numbers in a circuit may are used in the subcircuit call. For example, the code line X1049OPAMP refers to op amp X1 with non-inverting input node 0, invertinginputnode4, and output node 9. Small-Signal AC Macromodel A more elaborate macromodel is required if the ac frequency response behavior of the op amp is to be modeled. Such a model is given in Fig This circuit models the small-signal input resistance, output impedance, and voltage-gain transfer function of the op amp. Only asinglepoleismodeledinthevoltage-gaintransferfunction. TheSPICEsubcircuitcodes for the 741 and the LF351 op amps are given below. These are referred to as the small-signal AC macromodel subcircuits. The non-inverting input is node 1, the inverting input is node 2, and the output is node 3. The ac macromodel is useful in SPICE simulations of most op-amp circuits where the small-signal frequency response of the op amp is to be modeled. Figure 2.2: *741 OP-AMP SUBCIRCUIT *LF351 OP-AMP SUBCIRCUIT.SUBCKTOA SUBCKTOA RIN122E6 RIN122E12 GM E-4 GM E-4 R1401E5 R1401E5 CC4520E-12 CC4515E-12 GM GM RO RO13550 RO RO25025.ENDS OA741.ENDS OA351 The resistor R O shown connected in series with the output of the op amps in this experiment is included for practical reasons, viz. to reduce the possibility that the circuit will 6

11 oscillate when a capacitive load such as an oscilloscope probe is connected to it. It should not be included in the theoretical derivations, calculations, or simulations. For these analyses the output of the circuit should be taken directly at the output node of the of the appropriate op amp. 7

12 Chapter 3 Op-Amp Noise Measurements Object The object of this experiment is to measure the equivalent input noise voltage and noise current of a number of commonly used integrated circuit op amps. These are the 741, OP-27, TL071, and LF351. Laboratory Procedure Gain Stage Design Because the noise output voltage of the op amps is very low, it will be necessary to amplify it before it can be analyzed on the Dynamic Signal Analyzer. The object of this step is to assemble a 40 db gain stage for this purpose using one of the LF351 op amps. The stage is to be operated in the non-inverting configuration with a series feedback resistor of 10 kω and a shunt feedback resistor of 100 Ω. A 330 µf capacitor should be used in series with the shunt resistor to provide 100% feedback at dc so that the dc offset of the op amp being measured will not be amplified by the full 40u db gain of this stage. Assemble the 40u db gain stage near one end of the solderless breadboard. The reason for putting this stage at one end of the board is to minimize the possibility of oscillations due to coupling from input to output of the final circuit. A power supply decoupling network is to be used on the op amp rails on the breadboard; this consists of a 100 µf capacitor from the rail to ground and a 100 Ω resistor in series with the output of the power supply and the power supply rail. After the gain stage is assembled, measure its midband gain and its lower and upper half-power cutoff frequencies. The test signal amplitude should be low enough so that slew-rate limiting does not occur during the measurements. After making the preceding measurements, ground the input to the op amp and use the dynamic signal analyzer to measure the spectrum of its output noise from 0Hz to 1kHz and from 0Hz to 100 khz. Obtain plots of these. There will probably be some 60 Hz and integer multiplies on the 0 to 1kHz plot; this will not show up in the other plot because the resolution on the 0 to 100 khz is not sufficient to show it. In all further measurements, the 40 db gain stage will be used to amplify the output noise of another op amp. It should be observed during these measurements that the noise 1

13 of the cascade combination of the two op amps will be much greater than that generated by the 40 db gain stage by itself. For this reason, it can be assumed that the 40 db gain stage has negligible noise during these measurements. Device Under Test On the other end of the breadboard, configure the test op amp as a 40 db non-inverting amplifier stage using a 1 kω series feedback resistor and a 10 Ω shunt feedback resistor. Because these values are so small, the noise generated in the feedback resistors can be neglected in the calculation. Connect the output of this op amp to the input of the 40 db gain stage assembled in the preceding step. With the non-inverting input of the test op amp grounded (R S =0), use the Dynamic Signal Analyzer to measure the spectrum of the output noise from 0 to 1kHz and from 0 to 100 khz. Placethecursorormarkerataspotonthe spectrum which appears to be reasonably flat. Obtain plots of these. Use the same marker frequency for all of the measurements. Repeat the measurements of the preceding paragraph for source resistor values of R S = 1kΩ, 10 kω, and100 kω. Recordthemarkervalueataportionofthespectrumwhichis reasonably flat. Oscillations may occur due to capacitive coupling from the output of the second 40 db gain stage to the non-inverting input of the first stage with the higher resistor values. This can be minimized by carefully routing all wires and leads to keep the capacitive coupling between output and input at a minimum. Even through oscillation problems may not be experienced, some high frequency gain peaking in the 0 to 100 khz spectrum may be observed. This is an indicator of positive feedback. Attempt to repeat the preceding measurements for a source resistance of R S =1MΩ. If the circuit oscillates, reduce R S to the largest value between 100 kω and 1MΩ for which oscillations do not occur and perform the measurements. Repeat the previous step for the two remaining test op amps. Repeat all measurements using the shielded boxes and batteries as the dc power supplies. Laboratory Report 1. From the measured data, calculate v ni for each value of R S at the marker frequency. On separate graphs, plot v ni versus R S for each of the op amps that were tested. 2. From the data obtained with R S =0,calculatev n at the marker frequency for each op amp. From the data obtained with the highest value of R S,calculatei n at the marker frequency for each op amp. In the latter calculation, the value obtained previously for v n with R S =0to obtain i n. Assume that R i = foralltheopampsinthiscalculation. Compare the experimental results for v n and i n with textbook and manufacturers published data. 3. Address the question as to whether or not the largest value of R S was sufficient to place v ni on the asymptote of the v ni versus R S curve. Explain and justify the answer with numerical calculations. 4. What would be an advantage and a disadvantage of operating the 40 db gain stage in the inverting configuration rather than the non-inverting configuration? 2

14 5. Derive a least squares curve fitting routine to obtain least squares estimated of v n and i n from the experimental data of v ni versus R S. Use the theoretical equation for v ni as a function of R S and v n and i n and determine the appropriate LSQ algorithm. See Appendix A following this experiment. 6. Discuss any unusual and interesting results obtained. 7. Due Date. The written laboratory report is due one week from the date the experiment was completed unless otherwise stated. This is a two week experiment. During the first week all measurements are made using laboratory bench power supplies for the op amps and an unshielded breadboard. During the second week, the measurements are repeated using battery power supplies and a shielded box. Appendix A LSQ Curve Fitting Introduction The major goal of the third experiment is to obtain v n and i n for each of the op-amps tested at some particular frequency. For each op-amp K data points were taken of (R, v o )wherer is the input resistance and v o istheoutputnoisevoltagemeasuredatafrequencyf where the measurement bandwidth is B. The theoretical expression is given by vo 2 = A 2 vb 4kTR + vn 2 + i 2 nr 2 (A3.1) where A v is the gain of the measurement circuit which is approximately 80 db, oralinear gain of 10, 000. As this equation states, v n can be obtained if R is set to zero and i n can be obtained if R is made sufficiently large. There are two practical problems with this approach: (1) R cannotbemadeexactlyzeroand(2)forlargevaluesofr many of the circuits will either oscillate or saturate which makes the data useless. (The resistor R = R s + R 1 kr 2 where R s is the resistor connected to the noninverting input and R 1 and R 2 are the two resistors in the feedback network for the input op-amp.) Values can be obtained for v n and i n by performing a least squares curve fit ofthe experimental data (R, v o ) to the theoretical equation (Eq. A3.1). This is done by selecting v n and i n so that the curve passes between the data points so that the squared error is minimized. The equation to be fit is a quadratic polynomial in R. Least Squares Curve Fitting for Quadratic Equations The mathematical problem is to fit K data points (x i,y i ) to a quadratic equation ŷ = ax 2 + bx + c (A3.2) Thiscanbedonebyformingthesquarederror = KX (ŷ y i ) 2 (A3.3) i=1 3

15 and then setting a = b = c =0 (A3.4) and then solving the three equations that result for the least squared estimates for a, b, and c. Fortunately, the linear term is known in Eq. A3.1 so that least squared estimates are needed for only a and c. Thesolutionisthengivenby where a c 1 a11 a = 12 β1 a 22 β 2 a 21 a 11 = KX a 12 = a 21 = β 1 = x 4 i i=1 KX i=1 x 2 i a 22 = K KX yi x 2 i bx 3 i i=1 (A3.5) (A3.6) (A3.7) (A3.8) (A3.9) b =4kTBG 2 (A3.10) KX β 2 = (y i bx i ) (A3.11) and x i are the K values for R and y i are the K values for v 2 o. Once a and c are known, the least squares estimates for v n and i n are given by i=1 î n = 1 A v r a B A/ Hz (A3.12) ˆv n = 1 A v r c B V/ Hz (A3.13) The goodness of fit may be determined by plotting the data and then plotting Eq. A3.1 using the least squares estimates for v n and i n. 4

16 Chapter 4 Bipolar Junction Transistor (BJT) Noise Measurements Object The objective of this experiment is to measure the mean-square equivalent input noise, v 2 ni, and base spreading resistance, r x, of some NPN Bipolar Junction Transistors (BJTs). Theory Equivalent Input Noise It can be shown that vni 2, the mean-square equivalent input noise measured over a narrow frequency band f centered at frequency f, of a resistively loaded BJT amplifier with zero small-signal impedance from both base to ground and emitter to ground is given by µ vni "4kTr 2 = x + 2q + K µ f IC f β r2 x +2qI rx C β + V # 2 T f (4.1) I C where, r x is the base spreading resistance, β = I C / I B is the small-signal current gain, I C isthedccollectorcurrent,i B isthedcbasecurrent,k = is Boltzmann s constant, T is the Kelvin temperature, q = is the electronic charge, V T = kt/q is the thermal voltage, K f is the flicker noise-coefficient, and f is the frequency at which the mean-square noise voltage is measured. If the noise measurement is made at a frequency f where the flicker noise may be ignored, the expression for the mean-square equivalent input noise becomes " vni 2 = 4kTr x +2q I C β r2 x +2qI C µ rx β + V # 2 T f (4.2) I C which is not a function of f. Thus Eq. (4.1) may be used to calculate the equivalent input noise of a BJT if the collector current and transistor parameters are known. The small-signal current gain β may be readily measured from either the output or transfer characteristics of the transistor. 1

17 But attempting to measure the base spreading resistance, r x, from the dc characteristics is essentially impossible. However, it can be determined from noise measurements. Base Spreading Resistance Figure 4.1: Circuit for measuring base spreading resistance. The base spreading resistance of a BJT is one of the more prickly parameters to accurately measure. It can be measured using the circuit shown in Fig If it is assumed that the op amp is ideal and that the thermal noise in the feedback resistor R F can be ignored, the mean-square output voltage of the op amp is given by µ vno 2 = RF 2 G 2 m 4kTr x + 2qI b + K fi B rx 2 + 2qI C f (4.3) f G 2 m where G m = 1 r x β + V (4.4) T I C If the measurement is made at a high enough frequency so that the flicker noise component can be neglected, the base spreading resistance satisfies the quadratic equation A β 2 2qI C 2AVT β f rx 2 + 4kT f r x + AV T 2 =0 (4.5) βi C IC 2 2

18 where A = v2 no 2qI RF 2 C f (4.6) Thus Eq (4.5) may be solved to determine r x using the measured value of vno. 2 Only the positive solution for r x should be used since the negative value has no physical meaning. The capacitor C 1 is a coupling capacitor which prevents dc current from the transistor from flowingintotheresistorr F whileforcingtheentiresignalcomponentofthecollector current to flow though this feedback resistor. The op amp inverting terminal is at a virtual ground which means that the signal component of the collector voltage is zero which eliminates the Early effect. The capacitor C 2 is a bypass capacitor which places the emitter at signal ground. Both of these capacitors are chosen to be large so that the low frequency noise spectrum is not altered. This means that these capacitors are electrolytic and the polarity is shown. Laboratory Procedure Base Spreading Resistance Assemble the circuit shown in Fig. 4.1 on a solderless breadboard using a 2N4401 NPN BJT. Use an OP27 as the op amp. Use V + =+15Vand V = 15 V (these may be reduced to 9 V if the experimenters choose to assemble the circuits in the shielded boxes). Pick C 1 =10µF and C 2 =100µF. The power supply decoupling network consisting of 100 Ω resistors and 100 µf capacitors should be used. Insert a 100 Ω resistor between the output node of the circuit and the lead to the oscilloscope or signal analyzer. Insert a 100 pf capacitor between the base and emitter terminals to eliminate possible rf electromagnetic interference. Bias the circuit so that the collector current is 1mA. The collector current is given by I C = V V BE R E (4.7) where V BE may be assumed to be 0.65 V. Eq (4.7) may be used to determine R E.(Itshould be borne in mind that V is a negative voltage so V is a positive voltage.) Select R C = R E /2. This places the collector-emitter bias voltage at approximately one half the positive power supply voltage. The choice of this bias is somewhat arbitrary. The selection of the feedback resistor R F is somewhat arbitrary. The larger R F is the larger the output noise will be. But the larger R F, the larger the thermal noise produced by this resistor will be. A value of R F =100kΩ should suffice. Measure the collector current by using the DMM (Digital Multimeter) to measure the dc voltage across R C and then use Ohm s law to determine the current. Measure the dc voltage at each terminal of the transistor. Use the Dynamic Signal Analyzer to measure the mean-square output noise voltage v 2 no at a frequency that is large enough so that the flicker noise may be neglected and at a low enough frequency so that the op amp and transistor combination have not begun rolling off the frequency response, i.e. make the measurement at a frequency where the output voltage is white or flat as a function of frequency. 3

19 Repeat the measurement for bias currents of 3mA and 5mA for the 2N4401 NPN BJT. Repeat the measurement for the 2N3904 NPN BJT. NPN Equivalent Input Noise The mean-square input noise vni 2 is related to the mean-square output noise by v 2 ni = v2 no G 2 mr 2 F (4.8) This expression is to be compared with Eq. (4.1) once the base spreading resistance and small-signal current gain are determined. Transistor Parameters Use a transistor curve tracer to measure the small-signal current gain, β, of the 2N4401 and 2N3904 NPN BJTs for collector currents of 1mA, 3mA,and5mA that were used above. Resistance Measurement Use the DMM (Digital Multimeter) or the LCR meter to measure the value of each resistor that was used. Measurement Bandwidth Record the measurement bandwidth that was used by the HP35665A Dynamic Signal Analyzer. Press Disp Format and then Measurement State. Laboratory Report Bias Tabulate the quiescent bias voltages and currents for each transistor for each of the three values of collector current for which data was taken. Base Spreading Resistance From the data obtained calculate the r x of each transistor at the three collector bias currents that were used. What is the average value of r x for each transistor type at the three collector bias currents? Also tabulate the values of r x for each of the transistors. Equivalent Input Noise Use the values of that r x were obtained to calculate v 2 ni using Eq. (4.1). Compare these results to those obtained from Eq. (4.8) and the measured values of v 2 no. Explain any significant differences between these results. 4

20 Chapter 5 JFET Noise Object The objects of this experiment are to measure the spectral density of the noise current output of a JFET, to compare the measured density to the theoretical density, and to determine the lower corner frequency below which excess noise generated by generation-recombination centers in the intrinsic region of the JFET dominates. Theory Figure 5.1: JFET noise equivalent circuit. ThenoiseequivalentcircuitofaJFETisgiveninFig. 5.1wherei 2 td is the mean-square thermal drain noise current and i 2 fd is the excess or mean-square flicker noise current. These are given by µ i 2 2gm td =4kT f (5.1) 3 i 2 fd = K fi D f (5.2) f 1

21 where g m is the small-signal JFET transconductance and K f is the flicker noise coefficient. The small-signal transconductance of the JFET is related to its pinchoff voltage V P and its drain-to-source saturation current I DSS by g m = 2 p ID I DSS (5.3) V P where I D is the quiescent drain current. This current is related to the quiescent gate-to-source voltage by I D = I DSS 1 V 2 GS (5.4) V P where it is assumed that the JFET is operated in its saturation region and that the drain current is not a function of the drain-to-source voltage in this region. It should be borne in mind that both the gate-to-source voltage and the pinch off voltage are negative for an N channel JFET. This makes the small signal transconductance g m positive. The relationship between the drain current and gate-to-source voltage may also be expressed using as alternative set of parameters known as the threshold voltage V TO and transconductance parameter β for the JFET. These are the JFET parameters used with SPICE. (One should not confuse this beta with the one used as the current gain for a BJT.) These parameters are related to the drain-to-source saturation current and pinch off voltage by V TO = V P (5.5) and β = I DSS VP 2 (5.6) which means that I D = β (V GS V TO ) 2 (5.7) and g m =2 p βi D (5.8) The flicker noise coefficient is given by K f = 16kT r β 3 f L (5.9) I D which means that the mean-squared flicker noise current can also be expressed as i 2 fd =4kT µ 2gm 3 fl f f (5.10) This illustrates that the corner frequency f L is the frequency at which the mean squared thermal and flicker currents are equal in magnitude. Thus the corner frequency may be experimentally determined by measuring the frequency at which the total noise current increases by 3dB over the value in the region at which thermal noise dominates where the response is flat or white. 2

22 Laboratory Procedure Curve Tracer Measurements The power supply voltages for this circuit are V + and V, the positive and negative power supply voltages respectively. For symmetry s sake these will be selected as V + = V. Either ±15 V may be chosen if the circuit is assembled using the laboratory bench dc power supply or ±9V if batteries are employed. Use the transistor curve tracer to measure the I D of the JFET at a gate-to-source voltage of 0V and a drain-to-source voltage of V + /2. If the JFET is pinched off, this drain current will be I DSS. Next, use the input generator on the curve tracer to reverse bias the gateto-source junction until the drain current has been reduced to I DSS /2. Recordthevalueof the gate-to-source voltage (remember that both the gate-to-source voltage and the pinchoff voltage are negative for an N channel JFET). From the data obtained, calculate the pinch off voltage of the JFET and its transconductance at a quiescent drain current of I DSS /2. Figure 5.2: JFET noise measurement. Biasing The circuit shown in Fig. 5.2 is the test circuit for measuring the noise produced by the JFET. When this circuit is assembled on the solderless breadboard, the power supply rails should be decoupled with a 100 Ω resistor and a 100 µf capacitor. If available, use batteries as the power supply and use the shielded boxes to enclose the breadboard. The JFET is to be operated at a drain-to-source voltage of V + /2 and a drain current of I DSS /2 as determined in the previous step. For this current, the correct value of R S is given by R S = V V GS I D (5.11) 3

23 where V GS is the gate-to-source voltage determined in the previous step at which I D = I DSS /2. (Remember that both V is negative and V GS is also negative for an N channel JFET.) To bias the drain-to-source voltage at V + /2, the correct value of R D is given by R D = V + /2+V GS I D (5.12) With the calculated values of R S and R D, assemble the circuit on the breadboard and verify that the bias current and voltages are correct, i.e. that they are within 5% of the design values. The choice of R F is somewhat arbitrary. The larger R F is the larger the voltage at the output will be. Initially, select R F = 100 kω. Frequency Response Experimentally determine the frequency response of the JFET and the bifet-op-amp combination. Disconnect the short from the gate to ground, place a resistor (11 kω) from the gate to ground and place a large coupling capacitor (e.g. 10 µf) between the gate and the function generator output. The theoretical value of the voltage gain is A v = V o V g = g m R F (5.13) where V g is the voltage at the output of the function generator. Assemble the circuit again as shown in Fig Namely, remove the function generator, coupling capacitor, and resistors from the gate to the positive and negative power supplies. Connect a short from the gate to ground. Background Noise, JFET Noise, and Corner Frequency After it has been verified that the circuit is functioning properly, remove the JFET from the circuit and connect the Dynamic Signal Analyzer to the output of the op amp. Using a total analysis band of 100 khz, measure the spectra of the background noise generated by R D and the op-amp circuit. After doing this, reconnect the JFET to the circuit and remeasure the spectra of the noise. (For accuracy in the calculations, the noise with the JFET should be several db higher than the noise without the JFET. If it isn t a larger value of R F may be used.) From the spectra observed, decide on an optimum frequency to measure the noise so that the midband white noise generated by the JFET is not corrupted with the excess or flicker noise. Plot the spectra obtained. From the display, determine the corner frequency, f L, which will be the frequency at which the output voltage of the op amp increases by 3dB from the value in the flat region. Laboratory Report Formula Verification Verify the formulas for g m, R S,andR D that were given in the theory and procedures sections. 4

24 Comparison of Experimental and Theoretical Spectra Compare the measured noise spectral density for the white noise region of the spectrum to that predicted by the theoretical formula for i 2 td. What is the lower corner frequency below which the excess noise dominates? A major source of error in this calculation will be the excess noise in the measurement system. If this dominates, the excess noise generated by the JFET may be difficult to determine. Computation of v 2 n Reflect i 2 td and i2 fd noise sources to the gate of the JFET and compute the v2 n for this transistor. Use the experimental data from 5 of the procedure. (Divide the voltage at the output of the op amp by g m R F.) Due Date The written laboratory report is due one week from the date the experiment was performed unless otherwise stated. 5