AZ i Z s + Z i. [V s +(V ts + V n + I n Z s )] (1)

Transcription

1 Euivalent Noise Input Voltage Figure shows the amplifier noise model with a Thévenin input source, where V s is the source voltage, Z s = s + jx s isthesourceimpedance,v ts is the thermal noise voltage generated by the source, and V n and I n are the noise sources representing the noise generated by the amplifier. The output voltage is given by Z L V o = AV i = Z o + Z L Z s + Z i Z L Z o + Z L [V s +(V ts + V n + I n Z s )] () where A is the voltage gain and Z i is the input impedance. The euivalent noise input voltage V ni is defined as the voltage in series with V s that generates the same noise voltage at the output as all noise sources in the circuit. It consists of the terms in parenthesis in E. () and is given by V ni = V ts + V n + I n Z s () Note that this is independent of both A and Z i. It is simply the noise voltage across Z i considering Z i to be an open circuit. Figure : v n i n amplifier model with Thevénin source. The mean-suare value of V ni is solved for as follows: vni = (V ts + V n + I n Z s )(Vts + V n + InZ s ) = V ts Vts + V nvn +e V n In Z s + (In Z s )(InZ s ) = 4kT e (Z s ) f + vn + i n e (γzs )+i n Z s (3) where γ = γ r + jγ i is the correlation coefficient between V n and I n anditisassumedthatv ts is independent of both V n and I n. The correlation coefficient is given by v n γ = V nin vn i n (4) Effect of a Series Impedance at the Input Figure shows the input circuit of an amplifier with an impedance Z addedinserieswitha Thévenin source. The noise source V t models the thermal noise generated by Z. As is shown above, the euivalent noise voltage in series with the source can be solved for by first solving for the open-circuit input voltage, i.e. the input voltage considering Z i to be an open circuit. It is given by V i(oc) = V s + V ts + V t + V n + I n (Z s + Z ) = V s + V ni (5)

2 where V ni is the euivalent noise voltage in series with the source. It is given by V ni = V ts + V t + V n + I n (Z s + Z ) = V ts + V ns + I ns Z s (6) where V ns and I ns are the new values of V n and I n onthesourcesideofz. It follows from this euation that V ns = V t + V n + I n Z (7) I ns = I n (8) Note that V ns consists of all terms which are not multiplied by Z s and I ns consists of the coefficient of Z s in the term that is multiplied by Z s. Figure : Amplifier input circuit with Thévenin source and series impedance added at input. It follows that the addition of a series impedance at the input of an amplifier increases the V n noise but does not change the I n noise. If Z is not to increase the noise, Z should be as small as possible. If Z is lossless, it generates no noise itself so that V t =0. The mean-suare values and the correlation coefficient for V ns and I ns are given by vns =4kT e (Z ) f + vn + i n e (γz)+i n Z (9) The mean-suare euivalent noise input voltage is given by v n i ns = i n (0) γ s = γ vn i n + i nz () vns i ns vni = 4kT e (Z s + Z ) f + vn + i n e [γ (Zs + Z)] + i n Z s + Z () Effect of a Shunt Impedance at the Input v n Figure 3 shows the input circuit of an amplifier with an impedance Z addedinparallelwiththe source. The noise source I t models the thermal noise generated by Z. The open-circuit input voltage is given by Z V i(oc) = (V s + V ts ) + V n +(I t + I n ) Z s kz Z s + Z Z = (V s + V ni ) (3) Z s + Z

3 where V ni is the euivalent noise voltage in series with the source. It is given by µ V ni = V ts + V n + Z s +(I t + I n ) Z s Z = V ts + V ns + I ns Z s (4) where V ns and I ns are the new values of V n and I n onthesourcesideofz. It follows from this euation that V ns = V n (5) I ns = I t + V n + I n (6) Z Note that V ns consists of the term which is not multiplied by Z s and I ns consists of the sum of the coefficients of Z s in the terms that are multiplied by Z s. Figure 3: Amplifier input circuit with Thévenin source and shunt impedance added at input. It follows that the addition of a parallel impedance at the input of an amplifier increases the I n noise but does not change the V n noise. If Z is not to increase the noise, Z should be as large as possible. If Z is lossless, it generates no noise itself so that I t =0. The mean-suare values and the correlation coefficient for V ns and I ns are given by v ns = v n (7) µ µ i ns =4kT e f + v n γ Z Z + vn i n e Z γ vn i n + v n Z γ s = vns i ns + i n (8) The mean-suare euivalent noise input voltage is given by Ã! vnis = 4kT e Z s + Z s f + vn Z +Z s Z µ + vn i n e γ + Z s Zs + i Z n Z s (0) Dc bias networks and rf matching networks usually consist of series and parallel elements at the input to an amplifier. One method of analyzing the effect of these elements on the amplifier noise is by transforming the V n and I n sources from the amplifier input back to the source by use of the above relations. This is illustrated in the following example. (9) 3

4 Example Figure 4shows the input circuit of an amplifier. It is given that s =75Ω, = kω, C =0nF, =00Ω, vn =nv, i n =.5 pa, γ =0.+j0.. Thenoisespecifications are for a freuency f =00kHz and a bandwidth f =Hz. Calculate vni in series with the source by transforming V n and I n back to the source with Es. (9) through () and (7) through (9). Solution. To the left of " i 4kT f na = γ a = Figure 4: Amplifier input circuit. γ vna = vn =nv µ / + v n γ + vn i n e + i n# =4pA v n i n + v n i na v na = j The capacitor impedance is Z C =/jπfc = j59 Ω. TotheleftofC v / nb = vna + vna i na e (γ a ZC)+i n Z C =4.3 nv i nb = i na =4pA γ b = γ a vna i na + i naz C = j0.885 vnb i nb To the left of vnc = vnb =4.3 nv Ã! i vnb nc = + 4kT f / + i nb =6.3 pa γ b vnb i nb + v n γ c = = 0.53 j0.807 vnc i nc 4

5 The euivalent noise voltage in series with the source is v ni = µ 4kT s f + vnc + vnc i nc e (γ c s )+i nc s = 5.69 nv The following example illustrates the calculation of vni for the circuit of Fig. 4 by first calculating the open-circuit input voltage due to all sources in the circuit, factoring out the coefficient of V s, and assigning all remaining noise terms to V ni. Example For the circuit of Example, calculate vni by calculating the open-circuit input voltage, factoring out the coefficient of V s, and assigning all remaining noise terms to V ni. Solution. The euivalent source impedance seen by the amplifier is given by µ Z e = k jωc + k s =68.6 j9.4 The thermal noise voltage generated by s,,and has a mean-suare value of 4kT e (Z e ). Denote the phasor value of this voltage by V t(e). The open-circuit input voltage is given by / V i(oc) = V s + s + k s +/jωc + V t(e) + V n + I n Z e = (0.9 + j0.73) V s + V t(e) + V n + I n Z e µ = (0.9 + j0.73) V s + V t(e) + V n + I n Z e j0.73 It follows that the euivalent noise input voltage is given by vni = 4kT e (Z e)+vn + vn i n e γze + i n Z e j0.73 = 5.69 nv This is the same as that found in Example. Note that the method used in this example is more straightforward because it is necessary to transform only one source through the network. That source is the signal source V s. Noise Factor and Noise Figure The noise factor F of an amplifier is defined as the ratio of its actual SN and the SN if the amplifier is noiseless, where the temperature is taken to be the standard temperature T 0.Whenit is expressed in db, it is called noise figure and is given by NF = 0 log (F ). Consider the amplifier model in Fig.. If the amplifier is noiseless, the signal-to-noise ratio given by SN = vs /v ts,where vs is the mean-suare source voltage and vts is the mean-suare thermal noise voltage generated by the source impedance. When the amplifier noise is included, the signal-to-noise ratio is given by SN = vs/v ni.thusthenoisefactorisgivenby ³ F = vs /v ts ³ vs/v ni = v ni v ts =+ v n + v n i n e (γz s )+i n Z s 4kT 0 s f / () 5

6 It follows from this expression that a noiseless amplifier has the noise factor F =. A useful relation which follows from the definition of F is This relation is used below in the method for measuring F. v ni = F v ts = F 4kT 0 e (Z s ) () Example 3 Calculate F and NF for the amplifier in Example for which s =75Ω and 5.69 nv. Assume f =Hz and T = T 0 = 90 K. v ni = Solution. The mean-suare thermal noise voltage of the source is v ts =4kT s = V. Thus the noise factor and noise figure are F =. 0 8 =7.0 NF =0log7.0 =4.3 db When f =Hz,asinthisexample,F is called the spot-noise factor and NF is called the spot-noise figure. Measuring the Noise Factor This method is the most general one because it does not reuire knowledge of either the amplifier gain or its noise bandwidth. Consider the noise model of an amplifiergiveninfig.5.considerthe source to be a white noise source having the spectral density S v (f) =V s Vs / f = vs/ f. The total noise voltage at the output can be written Z i V o = A (V s + V ts + V n ) + I n (Z s kz i ) Z s + Z i = (V s + V ts + V n + I n Z s ) (3) Z s + Z i The mean-suare value is given by v o = = = Z s + Z i S v (f) B n +4kT 0 e (Z s ) B n + vn + i n e (γz s )+i n Z s v n h i Z s + Z i S v (f) B n + vni Z s + Z i [S v (f) B n + F 4kT 0 e (Z s ) B n ] where B n is the amplifiernoisebandwidthande. ()hasbeenusedtorelatevni to the thermal noise voltage of the source. Let vo be the value of v o with the noise source at the input set to zero, i.e. S v (f) =0.Now, let S v (f) be increased until the rms output voltage increases by a factor r, i.e. vo = r vo. It follows by taking the ratio of the two mean-suare voltages that r =+ S v (f) B n S v (f) =+ F 4kT e (Z s ) B n F 4kT 0 e (Z s ) (4) 6

7 Figure 5: Amplifier driven by a white noise source. Note that the noise bandwidth B n cancels. The above euation can be solved for F to obtain F = S v (f) (r ) 4kT 0 e (Z s ) (5) In making measurements, a commonly used value for r is r =. In this case, the output noise voltage increases by 3 db when the source is activated. Note that the expression for F is independent of B n, A, andz i. Other methods of measuring F reuire knowledge of these parameters. 7