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_____ Notes _____ CONTENTS 13.0 Signals and Noise 13.1 Noise Sources: 13.1.1 Combining Noise Voltages 13.1.2 Noise Figure 13.1.2.1 Friiss’ Formula 13.1.2.2 Noise Reduction 13.1.2.3 Noise Temperature 13.2 Review of RLC Networks 13.2.1 Series RLC Circuits 13.2.2 Parallel RLC Circuits 13.2.3 Self Resonance 13.3 Oscillators 13.3.1 Tuned Circuit Oscillators 13.3.2 Crystal Oscillator Assignment Questions For Further Research Wireless Communications Systems 13 - i _____ Notes _____ 13.0 Signals and Noise 13.1 Noise Sources: Man made noise < 500 MHz • Hydro lines • Ignition systems • Fluorescent lights • Electric motors Therefore deep space networks are placed out in the desert. Atmospheric noise - lighting < 20 MHz Solar noise - sun - 11 year sunspot cycle Cosmic noise - 8 MHz to 1.5 GHz Thermal or Johnson noise. Due to free electrons striking vibrating ions. Pn kTf where k 1.38 10 23 J/K This equation applies to copper wire wound resistors, but is close enough to be used for all resistors. Maximum power transfer occurs when the source and load impedance are equal. Resistor Type Carbon Metal film Wire wound Cost Low Mid High Noise High Mid Low White noise - white noise has a constant spectral density over a specified range of frequencies. Johnson noise is an example of white noise. Gaussian noise - Gaussian noise is completely random in nature however, the probability of any particular amplitude value follows the normal distribution curve. Johnson noise is Gaussian in nature. Shot noise - bipolar transistors in 2qI dcf where q electron charge 1.6 10 17 coulombs Excess noise, flicker, 1/f, and pink noise< 1 KHz also Inversely proportional to frequency Directly proportional to temperature and dc current Transit time noise - occurs when the electron transit time across a junction is the same period as the signal. 13.1.1 Combining Noise Voltages The instantaneous value of two noise voltages is simply the sum of their individual values at the same instant. Wireless Communications Systems 13 - 1 Signals and Noise _____ Notes _____ Vtotal inst. V1inst. V2 inst. This result is readily observable on an oscilloscope. However, it is not particularly helpful, since it does not result in a single stable numerical value such as one measured by a voltmeter. If the two voltages are coherent [K = 1], then the total rms voltage value is the sum of the individual rms voltage values. Vtotal rms V1rms V2 rms If the two signals are completely random with respect to each other [K = 0], such as Johnson noise sources, the total power is the sum of all of the individual powers: P Pn1random Pn2 random Total random A Johnson noise of power P = kTB, can be thought of as a noise voltage applied through a resistor. Rs es A example of such a noise source may be a cable or transmission line. The amount of noise power transferred from the source to a load, such as an amplifier input, is a function of the source and load impedances. Rs RL es Thermal Noise Source Amplifier Input If the load impedance is 0 Ω, no power is transferred to it since the voltage is zero. If the load has infinite input impedance, again no power is transferred to it since there is no current. Maximum power transfer occurs when the source and load impedance is equal. 13 - 2 Wireless Communications Systems Signals and Noise _____ Notes _____ PL max es2 4Rs The rms noise voltage at maximum power transfer is: en 4RP 4RkTB R en 4RkTB Thermal Noise Source under matched load conditions Observe what happens if the noise resistance is resolved into two components: en2 4RkTB 4R1 R2 kTB 4R1 kTB 4R1kTB 2 2 en1 en2 From this we observe that random noise resistance can be added directly, but random noise voltages add vectorially: ETotal 2 E2n1 En2 En2 En1 If the noise sources are not quite random, and there is some correlation between them [0 < K < 1], the combined result is not so easy to calculate: PTotal not quiterandom V12 V22 2KV1V2 Ro P1 P2 2K P1 P2 where K correlation 0 K 1 Ro impedance reference level Wireless Communications Systems 13 - 3 Signals and Noise _____ Notes _____ 13.1.2 Noise Figure This parameter is specified in all high performance amplifiers and is measure of how much noise the amplifier itself contributes to the total noise. In a perfect amplifier or system, NF = 0 dB. This discussion does not take into account any noise reduction techniques such as filtering or dynamic emphasis. Sin Sout G Nin Nout Signal to noise ratio: It is either unitless or specified in dB. The S/N ratio may be specified anywhere within a system. S signal power PS N noise power PN S P 10 log S N dB PN Noise Ratio: is a unitless quantity NR S N S N in out Noise Figure: NF 10 log NR 10 log S N S N in 10log S N in 10 log S N out out S S in out N in dB Nout dB 13.1.2.1 Friiss’ Formula It is informative to examine a cascade of amplifiers to see how noise builds up in a large system. NR Therefore: Sin Nin S N in out Sout Nout Nin Sout G Gain can be defined as: 13 - 4 Sout Sin Wireless Communications Systems Signals and Noise _____ Notes _____ Sin G Therefore the output signal power is: Sout = G Sin Sout GSin and the noise ratio can be rewritten as: NR Sin N N out out Nin GSin GNin The output noise power can now be written: Nout N R GNin From this we observe that the input noise is increased by the noise ratio and amplifier gain as it passes through the amplifier. A noiseless amplifier would have a noise ratio of 1 or noise figure of 0 dB. Consequently, the input noise would only be amplified by the gain. The minimum noise that can enter any system is the Johnson Noise: Nin minimum kTB The minimum noise that can appear at the output of any amplifier is: Nout minimum NR GkTB The output noise of a perfect amplifier would be: Noutperfect GkTB The difference between these two values is the noised added by the amplifier itself: Noutadded Nout minimum Nout perfect N R GkTB GkTB N R 1GkTB This is the added noise, as it appears at the output. The total noise coming out of the amplifier is then given by: N Total GN out perfect Nout added G kTB input noise N R 1GkTB additional noise due to the amp If a second amplifier were added in series, the total output noise would consist the first stage noise amplified by the second stage gain, plus the additional noise of the second amplifier: NTotal G1G2kTB NR1 1G1G2 kTB NR2 1G2kTB Wireless Communications Systems 13 - 5 Signals and Noise _____ Notes _____ If we divide both sides of this expression by the common term: G1G2kTB we obtain: G1 G2 kTB NR1 1G1 G2 kTB N R2 1G2 kTB NTotal G1G2 kTB G1G2 kTB NR Recall: Nout NTotal GNin G1G2 kTB NR overall NR1 Then: N R2 1 G1 This process can be repeated for n stages, and the resulting equation is known as Friiss’ Formula: NR overall NR1 It is more commonly written as: F F1 N R2 1 N R3 1 G1 G1 G2 F2 1 F3 1 G1 G1G2 From an examination of this equation, it is readily apparent that the overall system noise figure is largely determined by the noise figure of the first stage in a cascade. EXAMPLE An amplifier has a noise figure of 8 dB and a power gain of 10 dB. It is followed by a mixer with a noise figure of 25 dB and a gain of -20 dB. The overall noise figure can be determined as follows: Amp G1 = 10 dB F1 = 8 dB X Mixer G2 = -20 dB F2 = 25 dB Convert all numbers to ratios: 13 - 6 G1 = 10 dB 10 G2 = -20 dB 0.01 F1 = 8 dB 6.309 F2 = 25 dB 316.22 Wireless Communications Systems Signals and Noise _____ Notes _____ Apply Friis’ formula: F F1 F2 1 316.22 1 6.31 37.832 G1 10 or in dBs 10log37.832 15.78 dB Note that the overall noise figure is greater than that of the amplifier, but less than that of the mixer. 13.1.2.2 Noise Reduction Inductive and capacitive reactances do not generate thermal noise because they cannot dissipate power. However, they can affect the noise spectrum. The noise spectrum density is defined as: S kT but if it passes through a reactance: 2 S H kT 13.1.2.3 Noise Temperature In microwave applications, it is difficult to speak in terms of currents and voltages since the signals are more aptly described by field equations. Therefore, temperature is used to characterize noise. The total noise temperature is equal to the sum of all the individual noise temperatures. 13.2 Review of RLC Networks 13.2.1 Series RLC Circuits R Wireless Communications Systems L C 13 - 7 Signals and Noise Voltage _____ Notes _____ fo Frequency Rs L Voltage R fo Frequency C The total impedance of the RLC circuit is given by: Z R jX L jX C R j X L XC where XC 1 C and X L L When the magnitude of the inductance exceeds that of the capacitor, the impedance at a specific frequency may be graphically represented as: +j XL XL - X C Resultant Impedance Z R XC -j X 2 2 Z Z R X tan 1 R modulus phase angle By definition, the series RLC circuit is resonant when = 0. This occurs when X = XL - XC = 0. The magnitude of the output voltage is determined by applying the voltage divider rule to the source and load resistances. The circuit impedance is its minimum value at resonance. The frequency at which resonance occurs can be determined as follows: 13 - 8 Wireless Communications Systems Signals and Noise _____ Notes _____ X L X C 0 or X L XC o 1 LC or fo or o L 1 oC 1 2 LC The quality factor or Q is a measure of the amount of energy stored in a reactor. It can be defined as the ratio of voltage across the inductor to voltage across the resistor. Q Since at resonance o L VL X L o L VR R R 1 1 , then: Q oC o RC Typical values of Q in RLC circuits are: 10 < Q < 300. BANDWIDTH A sketch of the magnitude of the impedance as a function of frequency resembles: Z 2R R frequency f1 fo f2 The impedance is sometimes expressed in terms of Q and a new parameter we shall call y. 1 1 L Z R j L R 1 j R 1 C R RC o R 1 j Q R1 jyQ o let = y o L o 1 j RC o R Z R 1 y 2Q 2 Wireless Communications Systems 13 - 9 Signals and Noise _____ Notes _____ At resonance, the current flowing in the circuit is determined by R. The 3 dB point is defined when the current drops by 3 dB from its resonant value, at which point the modulus increases by 2 . The magnitude of the impedance is therefore given by: Z 2 R The 3 dB bandwidth can be determined by evaluating the impedance function at the 3 dB points: 2 2 R 1 y Q R 2 y 1 Q at the 3dB points At the upper cutoff frequency we obtain: y f2 fo 1 fo f 2 Q 2 f f 2 f2 o o fo 2Q 2Q At the lower cutoff frequency we obtain: y fo f1 1 f1 f o Q 2 f f 2 f1 o o fo 2Q 2Q The 3 dB bandwidth is the difference between these two frequencies: B3 dB f2 f1 fo Q 13.2.2 Parallel RLC Circuits R C L 13 - 10 Wireless Communications Systems Signals and Noise _____ Notes _____ Parallel circuits are easier to analyze in terms of admittance instead of impedance. Y 1 R jL jC 2 jC R jL R 2 L2 R L j C 2 2 2 2 2 R L R L 2 At resonance, the admittance is real and the imaginary components cancel. The resonant frequency can be determined as: oC oL R 2o L2 o 1 R2 2 LC L 2 It is interesting to note that in a parallel resonant circuit, the resonant frequency is influenced by R, but in series resonant circuit, it is not. Usually 1 R2 2 , and therefore: o LC L 1 . LC Since the admittance at resonance is purely conductive: Yo R R 2o L2 2 1 L RC , therefore Yo . Since Yo , the Zo C L . L impedance at resonance is given by: Zo Q2 R . RC Recall that R2 2o L2 f f is similar to that of Z vs for the series circuit. It fo fo fo therefore follows that B3 dB . Q The graph of Y vs The impedance of a parallel resonant circuit is high and is often referred to as the dynamic impedance where: RD 1 L Q Yo RC o C 13.2.3 Self Resonance A real coil has winding resistance and capacitance in addition to its inductance. As a result, it acts like a parallel tuned circuit. In some applications, it may be necessary to consider these secondary affects. Wireless Communications Systems 13 - 11 Signals and Noise _____ Notes _____ 13.3 Oscillators Many amplifiers and oscillators can be represented by: Vin + G Vout - H The transfer function of this network can easily be determined as: Vo G closed loop gain Vin 1 GH G open loop gain GH loop gain This expression is sometimes written: A Ao 1 Ao Ao open loop gain feedback factor As an example of this, we may consider an opamp. 10 K 1K Vout + Vin If the open loop gain for this amplifier is: Ao = 104, the closed loop gain is given by: ACL 10 4 1 104 10.9879 1K 1K 10K The closed loop gain can be very closely approximated by: ACL 1 13 - 12 10K 11 1K Wireless Communications Systems Signals and Noise _____ Notes _____ Note that the feedback factor is 1/11 of the output signal. With negative feedback the closed loop gain is less than of open loop gain, and the circuit is stable. With positive feedback the closed loop gain is greater than the open loop gain and the circuit is unstable. For continuous oscillations: loop must be GH 1 and the total phase change around the o n 360 . If GH 1 , oscillations build up until something limits the process. Creating an output with no input means that the closed loop gain is equal to infinity. This occurs when 1 + GH = 0. The loop gain at this point is GH = -1. This condition is known as the Barkhausen criteria for oscillation. 13.3.1 Tuned Circuit Oscillators Note: The following analysis only works for high output impedance amplifiers such as transistors. In opamp circuits, the low output impedance effectively shorts out Z1. Transitor Amplifier Ro Vout Vin Zin AVin Vin I1 Z2 Z1 I2 Z3 Writing KVL in the output loop we obtain: AVin I1Ro Z1 I2Z1 Writing KVL in the feedback loop we obtain: 0 I1Z1 I2Z1 Z2 Z3 From the input loop we obtain: I2 Vin Z2 The loop gain can be determined as: Wireless Communications Systems 13 - 13 Signals and Noise _____ Notes _____ Vin Z1 Z2 A 2 Vin Z1 Z1 Z2 Z3 Ro Z2 In order for oscillations to occur, the loop gain must be greater than 1, therefor an inversion is necessary in the circuit to change the sign of the gain. Such an inversion does take place between the base and collector of a transistor. Recall that at resonance, a series RLC circuit has low impedance. Therefor Z1 Z2 Z3 0 and the loop gain reduces to: Vin Z A 2 Vin Z1 Oscillator Type Hartley Colpitts Clapp Z1 L C C Z2 L C C Z3 C L L+C 13.3.1.1 Hartley Oscillator +V R1 RFC C2 C4 R2 R3 n2 Z2 C3 n1 L Z1 C1 Z3 From our previous analysis, the loop gain at resonance is given by: A A Z1 L A 1 Z2 L2 In order to guarantee oscillations, when the power is first turned on, the loop gain must be greater than 1. 13 - 14 Wireless Communications Systems Signals and Noise _____ Notes _____ EXAMPLE A worst-case design: the typical gain of a transistor is 35, but at the worst case operating temperature, it falls 30%. Determine the value of to ensure oscillations with a 2:1 safety margin. Solution The lowest gain expected is 35 x.7 = 24.5 For a 2:1 safety margin, assume the gain is 24.5/2 = 12 Since A = 1 and A = 12, then = 1/12 Z2 L2 Z1 L1 L2 12L1 The operating frequency is determined by: fo 1 2 LC 2 1 L1 L2 C1 13.3.1.2 Colpitts Oscillator This is the same as the Hartley oscillator, but with the inductors and capacitors reversed. C2 C1 L The feedback factor is given by: X C 2 C1 X C1 C 2 The oscillation frequency is given by: fo 1 2 LC 1 C C 2 L 1 2 C1 C2 13.3.1.3 Clapp Oscillator This more stable variation of the Colpitts oscillator has a reduced tuning range. Wireless Communications Systems 13 - 15 Signals and Noise _____ Notes _____ +V RFC C1 C2 L1 C3 The capacitors C1 and C2 must be much larger than the transistor junction capacitance. The oscillation frequency is given by: fo 1 2 L1C3 13.3.2 Crystal Oscillator Crystal Oscillators by MXCOM The equivalent RLC circuit of a crystal is: Rs Ls Cp Cs Applying an electric potential across the crystal causes it to deform. But since the crystal is a piezo-electric device, deforming it causes a slight electrical potential to develop across it. As a result, the crystal can act as an electromechanical resonator if the right excitation is applied. The oscillation frequency is determined by the size and shape of the crystal. Crystals can be used to make very stable oscillators ranging from 15 KHz to 100 MHz. 13 - 16 Wireless Communications Systems Signals and Noise _____ Notes _____ XL fs fp frequency XC Placing an inductor in series with the crystal lowers the series resonant frequency, and placing an inductor in parallel to the crystal increases the parallel resonant frequency. 13.3.2.1 Crystal Filter Output Input Attenuation C Large C Small C frequency Wireless Communications Systems 13 - 17 Signals and Noise _____ Notes _____ Assignment Questions Quick Quiz 1. Friiss’ formula shows that noise figure is proportional to bandwidth. [True, False] 2. Johnson noise is also called [pink, white, blue] noise. 3. Admittance at resonance is purely conductive. [True, False] Analytical Questions 1. Determine the total noise power if two thermal noise sources of 12 µW and 16 µW are connected in series. 2. A two stage RF amplifier has a 3 dB bandwidth of 150 KHz determined by an LC circuit at the input and operates at 27o C. First stage: PG = 8 dB NF = 2.4 dB Second stage: PG = 40 dB NF = 6.5 dB Output Load = 300 Ω In testing the system, a 100 KΩ resistor is applied at the input. Determine: a) Input noise voltage b) Input noise power c) Output noise voltage d) Output noise power e) 3. System noise figure Prove that under matched load conditions, the rms Johnson noise voltage is: en 4RkTB Composition Questions 13 - 18 Wireless Communications Systems Signals and Noise _____ Notes _____ For Further Research http://www.mth.msu.edu/~maccluer/Lna/noisetemp.html Wireless Communications Systems 13 - 19