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VARIABLE-FREQUENCY NETWORK PERFORMANCE Variable-Frequency Response Analysis Network performance as function of frequency. Transfer function Sinusoidal Frequency Analysis Bode plots to display frequency response data VARIABLE FREQUENCY-RESPONSE ANALYSIS In AC steady state analysis the frequency is assumed constant (e.g., 60Hz). Here we consider the frequency as a variable and examine how the performance varies with the frequency. Variation in impedance of basic components Resistor Z R R R0 Inductor Z L jL L90 Capacitor Zc 1 1 90 jC C Frequency dependent behavior of series RLC network 2 1 ( j ) 2 LC jRC 1 j RC j ( LC 1) Z eq R jL j C jC jC " Simplifica tion in notation" j s s 2 LC sRC 1 Z eq ( s) sC (RC ) ( LC 1) | Z eq | C 2 2 2 1 LC 1 Z eq tan RC 2 Simplified notation for basic components Z R ( s) R, Z L ( s) sL, ZC 1 sC For all cases seen, and all cases to be studied, the impedance is of the form am s m am 1s m 1 ... a1s a0 Z ( s) bn s n bn1s n1 ... b1s b0 Moreover, if the circuit elements (L,R,C, dependent sources) are real then the expression for any voltage or current will also be a rational function in s 1 sC sL R sRC VS 2 VS R sL 1/ sC s LC sRC 1 s j jRC Vo VS 2 ( j ) LC jRC 1 Vo ( s) R j (15 2.53 103 ) Vo 100 2 3 3 ( j ) (0.1 2.53 10 ) j (15 2.53 10 ) 1 NETWORK FUNCTIONS Some nomenclature When voltages and currents are defined at different terminal pairs we define the ratios as Transfer Functions INPUT Voltage Current Current Voltage OUTPUT TRANSFER FUNCTION SYMBOL Voltage Voltage Gain Gv(s) Voltage Transimpedance Z(s) Current Current Gain Gi(s) Current Transadmittance Y(s) If voltage and current are defined at the same terminals we define Driving Point Impedance/Admittance EXAMPLE To compute the transfer functions one must solve the circuit. Any valid technique is acceptable I 2 ( s) Transadmittance V1 ( s) Transfer admittance V ( s) Gv ( s ) 2 Voltage gain V1 ( s) YT ( s) EXAMPLE VOC ( s) sL V1 ( s) sL R1 We will use Thevenin’s theorem 1 sLR1 1 R1 || sL sC sL R1 sC s 2 LCR1 sL R1 ZTH ( s ) sC ( sL R1 ) ZTH ( s) I 2 ( s) Transadmittance V1 ( s) Transfer admittance V ( s) Gv ( s ) 2 Voltage gain V1 ( s) YT ( s) ZTH (s) VOC (s) sL V1 ( s ) sL R1 VOC ( s) sC ( sL R1 ) I 2 ( s) s 2 LCR1 sL R1 sC ( sL R1 ) R2 ZTH ( s) R2 sC ( sL R1 ) I 2 ( s) R2 V2 ( s ) s 2 LC YT ( s) 2 s ( R1 R2 ) LC s( L R1R2C ) R1 Gv ( s) V2 ( s ) R2 I 2 ( s) R2YT ( s ) V1 ( s ) V1 ( s) POLES AND ZEROS (More nomenclature) am s m am 1s m 1 ... a1s a0 H ( s) bn s n bn1s n1 ... b1s b0 Arbitrary network function Using the roots, every (monic) polynomial can be expressed as a product of first order terms H ( s) K 0 ( s z1 )( s z2 )...( s zm ) ( s p1 )( s p2 )...( s pn ) z1, z2 ,..., zm zeros of the network function p1, p2 ,..., pn poles of the network function The network function is uniquely determined by its poles and zeros and its value at some other value of s (to compute the gain) EXAMPLE zeros : z1 1, poles : p1 2 j 2, p2 2 j 2 H (0) 1 H ( s) K 0 ( s 1) s 1 K0 2 ( s 2 j 2)( s 2 j 2) s 4s 8 1 H (0) K 0 1 8 H ( s) 8 s 1 s 2 4s 8 SINUSOIDAL FREQUENCY ANALYSIS A0e j ( t ) B0 cos( t ) H (s ) A0 H ( j )e j ( t ) B0 | H ( j ) | cos t H ( j ) Circuit represented by network function To study the behavior of a network as a function of the frequency we analyze the network function H ( j ) as a function of . Notation M ( ) | H ( j ) | ( ) H ( j ) H ( j ) M ( )e j ( ) Plots of M ( ), ( ), as function of are generally called magnitude and phase characteristics. 20 log10 (M ( )) BODE PLOTS vs log10 ( ) ( ) HISTORY OF THE DECIBEL Originated as a measure of relative (radio) power P2 |dB (over P1 ) 10 log P2 P1 V2 V22 I 22 PI R P2 |dB (over P1 ) 10 log 2 10 log 2 R V1 I1 2 V |dB 20 log10 | V | By extension I |dB 20 log10 | I | G |dB 20 log10 | G | Using log scales the frequency characteristics of network functions have simple asymptotic behavior. The asymptotes can be used as reasonable and efficient approximations General form of a network function showing basic terms Poles/zeros at the origin Frequency independent K 0 ( j ) N (1 j1 )[1 2 3 ( j3 ) ( j3 )2 ]... H ( j ) (1 ja )[1 2 b ( jb ) ( jb )2 ]... log( AB ) log A log B First order terms N log( ) log N log D D Quadratic terms for complex conjugate poles/zeros | H ( j ) |dB 20 log10 | H ( j ) | 20 log10 K 0 N 20 log10 | j | 20 log10 | 1 j1 | 20 log10 | 1 2 3 ( j3 ) ( j3 ) 2 | ... 20 log10 | 1 ja | 20 log10 | 1 2 b ( jb ) ( jb ) 2 | .. z1z2 z1 z2 H ( j ) 0 N 90 Display each basic term z1 2 z1 z2 1 1 separately and add the 3 3 tan tan ... 1 z2 results to obtain final 1 (3 ) 2 2 bb tan 1 a tan 1 ... 1 (b ) 2 answer Let’s examine each basic term Constant Term the x - axis is log10 this is a straight line Poles/Zeros at the origin ( j ) N | ( j ) N |dB N 20 log10 ( ) ( j ) N N 90 | 1 j |dB 20 log10 1 ( ) 2 1 j Simple pole or zero (1 j ) tan 1 1 | 1 j |dB 0 low frequency asymptote (1 j ) 0 1 | 1 j |dB 20 log10 high frequency asymptote (20dB/dec) The two asymptotes meet when 1(corner/break frequency) (1 j ) 90 Behavior in the neighborhood of the corner corner octave above octave below distance to FrequencyAsymptoteCurve asymptote Argument 3dB 3 45 1 0dB 2 6dB 7db 1 63.4 0 .5 0dB 1dB 1 26.6 Asymptote for phase Low freq. Asym. High freq. asymptote Simple zero Simple pole LEARNING EXAMPLE Draw asymptotes for each term Generate magnitude and phase plots Gv ( j ) 10(0.1 j 1) ( j 1)(0.02 j 1) Breaks/corners : 1,10,50 Draw composites dB 40 20 10 |dB 20dB / dec 0 20dB / dec 20 90 45 / dec 45 / dec 0.1 1 10 100 90 1000 asymptotes DETERMINING THE TRANSFER FUNCTION FROM THE BODE PLOT This is the inverse problem of determining frequency characteristics. We will use only the composite asymptotes plot of the magnitude to postulate a transfer function. The slopes will provide information on the order A. different from 0dB. There is a constant Ko A B C K 0 |dB 20 K 0 D E K 0 |dB 10 20 B. Simple pole at 0.1 ( j / 0.1 1)1 C. Simple zero at 0.5 ( j / 0.5 1) D. Simple pole at 3 ( j / 3 1)1 E. Simple pole at 20 G ( j ) 10( j / 0.5 1) ( j / 0.1 1)( j / 3 1)( j / 20 1) ( j / 20 1)1 If the slope is -40dB we assume double real pole. Unless we are given more data