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Transcript
Noise Figure Calculation Exercise Using Microwave CAD Software 1.0 Derivation of Noise Figure Expression for Computer Simulation A simple noise analysis of the resistive network is carried out. Linear 2-port network Port 1 R1 Port 2 R2 RS RL Vout Vin Figure 1 – A two-port resistive network. The output voltage at Port 2 is given by: R L // R2 Vout = Vin = A ⋅ Vin R L // R2 + R1 + RS (1) In equation (1) A is the effective voltage gain. Deactivating the source voltage Vin and considering the thermal noise from the resistors, the equivalent circuit of Figure 2 is obtained. Port 1 R1 Port 2 R2 RS 4kTR2 Vn1 RL Vout Vn2 4kT (RS + R1 ) Figure 2 – Equivalent circuit with thermal noise source, signal source deactivated. F. Kung 1 June 2004 The noise sources Vn1 and Vn2 are white noise, and the Power Spectral Density (PSD) Sn1 and Sn2 which are assumed constant with respect to frequency. S n1 ( f ) = 4kT (RS + R1 ) (2a) S n 2 ( f ) = 4kTR2 (2b) Assuming a bandwidth of ∆f. Using the fact that Vn1 = ∫ S n1 ( f )df 2 and Vn 2 = ∫ S n 2 ( f )df , the contribution of the noise sources individually at the output node is 2 given as follows: R2 // RL R2 // RL + R1 + RS 2 Vout ( n1) = Vout ( n1) = 2 Vout (n 2 ) = Vout (n 2 ) = 4kT (RS + R1 )∆f (R1 + RS )// RL (R1 + RS ) // RL + R2 4kTR2 ∆f (3a) (3b) The random process due to Vn1 and Vn2 can be combined using Superposition Theorem for linear circuit. Since noise sources Vn1 and Vn2 are uncorrelated, the total PSD at port 2 is (See [1], Section11.4): S ( f ) = S n1 ( f ) + S n 2 ( f ) (4a) Or 2 2 Vout (noise ) = Vout ( n1) + Vout ( n 2 ) This implies: Vout (noise ) = Vout (noise ) 2 = [V R2 // RL = R2 // RL + R1 + RS 2 out ( n1) (4b) 2 + Vout ( n 2 ) 2 ] 1 2 (R1 + RS ) // RL 4kT (RS + R1 )∆f + (R1 + RS ) // RL + R2 2 1 2 (5) 4kTR2 ∆f 2 Not that in computing the total noise voltage, the contribution from the termination at port 2, e.g. RL is not considered. The signal-to-noise ratio at the input, SNRin is given by: Vin2 SNRin = (6) 4kTRS The signal-to-noise ratio at the output, SNRout is given by: ( A ⋅ Vin )2 SNRout = 2 Vout ( noise ) F. Kung 2 (7) June 2004 Thus the noise factor of the network, from (6) and (7) is: 2 Vout (noise ) SNRin = 2 SNRout A ⋅ 4kTRs The noise figure (F) in dB is then given by SNRin F = 10 ⋅ log 10 SNRout NF = (8) (9) Equations (8) and (9) are usually employed by computer simulation software to determine the noise figure F of a linear circuit. 2.0 Numerical Example Let RS = RL = 50, R1 = 50 and R2 = 100. T = 25o C or 298K, k = 1.380×10-23 JK-1. Bandwidth ∆f = 1Hz (spot noise calculation). Then: Vout ( n1) = 0.25 ⋅ 1.28256 × 10 −9 = 320.64 pV Vout ( n 2 ) = 0.25 ⋅ 1.28256 × 10 −9 = 320.64 pV A = 0.25 2 2 × (320.64 pV ) NF = ≅ 4.00 2 0.25 2 ⋅ (906.91 pV ) F = 10 log10 (4.00 ) = 6.02 3.0 ADS Simulation of Small-signal Noise In ADS the noise simulation is typically performed during AC or S-parameter simulation. Therefore it will be convenient if the noise factor NF can be expressed in terms of the Sparameters. Port 1 I1 R1 V2 Linear 2-port network s11 s 21 Vs1 s12 s 22 Port 2 I2 V2 R2 Vs2 Figure 3 – Two-port network. F. Kung 3 June 2004 Consider a two-port network as shown in Figure 3, the normalized incident and reflected voltage waves can be defined according to Chapter 1 of [3]: ai = bi = Vi + Ri I i 2 Ri Vi − Ri I i 2 Ri Vsi = (10a) 2 Ri 2Vi − Vsi = (10b) 2 Ri Where i = 1 or 2. When measuring parameter sij, the source Vsi would be deactivated (e.g. shorted). For s21: b s 21 = 2 a1 a = 0 2 Generator Vs2 = 0. Hence using the definition (10a) and (10b): 2V2 V R1 2 R2 s 21 = V = 2 2 s1 V R 2 s1 2 R1 s 21 2 V =4 2 V1 2 RS 2 RS =4A RL RL (11) Since R1 = RS, R2 = RL, V1=Vin and V2 = Vout. Now consider equation (8) again, 2 NF = Vout (noise ) Vn ( 2− port ) + 4kTRS A 2 2 A ⋅ 4kTRs = 2 RL (4 A R )⋅ kT 2 S RL Here the total mean-square noise voltage at the output is divided into contribution from the source resistance RS and the noise sources within the two-port network. Vn(2-port) is the noise voltage at the output due to noise sources within the two-port network. Now using (11) this can be written as: 2 NF = Vn + kT s 21 RL kT s 21 2 (12) 2 Equation (12) is the expression used by ADS software for noise figure computation when noise calculation is enabled during S-parameter simulation. It can be easily extended to multi-port network as illustrated in the online documentation in ADS. F. Kung 4 June 2004 4.0 ADS Simulation Example Here the numerical example of Section 3.0 is repeated in ADS (version 2003C is used). The schematic and data display are shown in Figure 4 and Figure 5 respectively. The S-parameter is performed at a single frequency of 1.0GHz. Figure 4 – The schematic and “Noise” tab setting in the S-parameter control. Figure 5 – The data display. Here we are only interested in the noise figure with output being taken at port 2. So we select nf(2) to be shown in the list. Note that the software also calculates the minimum noise figure NFmin and the corresponding optimum source reflection coefficient Sopt. See [4] on how to find Sopt and NFmin. In Figure 5, the contribution of the individual noise F. Kung 5 June 2004 source at the output port is also displayed. We are only interested in the listing under port2.NC.name and port2.NC.vnc for output taken at port 2. Compare the values with those obtained in Section 3.0. References 1. B. P. Lathi, “Modern digital and analog communication systems”, 3rd edition 1998, Oxford University Press. 2. B. Razavi, “RF microelectronics”, 1998, Prentice-Hall Inc. 3. G. D. Vendelin, A. M. Pavio, U. L. Rohde, “Microwave circuit design using linear and nonlinear techniques”, 1990, John Wiley & Sons. 4. R. Ludwig, P. Bretchko, “RF circuit design – theory and applications”, 2000, Prentice-Hall Inc. F. Kung 6 June 2004