UMBC A B M A L T F O U M B C I M Y O R T 1 - PowerPoint PPT Presentation

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots Previously, we plotted the magnitude and phase of a network function, N(j ω) , as: N j ω ( ) vs. ω Arg N j ω ( ( ) ) vs. ω Here, we will use a logarithmic scales for magnitude and for ω (not phase). This allows piecewise linear line segments to be fit to the curves. Consider the following transformation of our network function definition: ) e jArgN j ω ( ) N j ω ( ) N j ω ( ( ln of both sides) = e jArgN j ω ( ) (product becomes sum) ( N j ω ( ) ) N j ω ( ) ( ) ln = ln + ln ( N j ω ( ) ) N j ω ( ) jArgN j ω ( ) ln = ln + So the natural logarithm of the network function expresses the real (mag) and imag (phase) as a sum. L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 1 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots The units of N j ω ( ) neper (mag in dB = 8.6859 X mag in nepers) ln jArgN j ω ( ) radians (arg in degrees = 57.2958 X arg in radians) Amplitude to decibels : A dB A dB A A 0 1.00 30 31.62 3 1.41 40 100.00 6 2.00 60 10 3 10 3.16 80 10 4 15 5.62 100 10 5 20 10.00 120 10 6 Note that - 3 dB corresponds to 1/1.41 = 0.707. Take the reciprocal of A when A dB is negative. L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 2 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots Real, first order poles and zeros : K j ω ( ) z 1 + N j ω ( ) = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - j ω j ω ( ) p 1 + Put it in standard form by dividing out the poles and zeros : ( j ω z 1 ⁄ ) Kz 1 1 + N j ω ( ) = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - p 1 j ω ( ) 1 ( j ω p 1 ⁄ ) + Re-expressing N(j ω ) in polar form: j ω z 1 ⁄ ∠ φ 1 K o 1 + ( ) p 1 ⁄ K o Kz 1 N j ω ( ) = with = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ω ( ∠ 90 ° ) 1 j ω p 1 ⁄ ∠ θ 1 + Rearranging terms: j ω z 1 ⁄ K o 1 + N j ω ( ) ∠ ( φ 1 90 ° θ 1 ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = – – ω 1 j ω p 1 ⁄ + L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 3 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots Separating into magnitude and phase terms: j ω z 1 ⁄ K o 1 + N j ω ( ) = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ω 1 j ω p 1 ⁄ + tan 1 – φ 1 ( ω z 1 ⁄ ) = ArgN j ω ( ) φ 1 90 ° θ 1 = – – with tan 1 – θ 1 ( ω p 1 ⁄ ) = Consider the amplitude: converting to decibels : j ω z 1 ⁄ K o 1 + A dB = 20 log - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ω 1 j ω p 1 ⁄ + ( j ω ) z 1 ⁄ ω j ω p 1 ⁄ A dB K o = 20 log + 20 log 1 + – 20 log – 20 log 1 + Plotting involves plotting each term separately and then combining them graphically. We approximate each term with a straight line. L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 4 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots The term K o is a straight line (not a function of ω ). Note that its value is zero when K o = 1. We use two straight lines to approximate: j ω z 1 ⁄ When ω is small 20 log 1 + This function -> 0 this term is ~1 ω → 0 When ω is large ( ω z 1 ⁄ ) 20 log this term is ~ ω /z 1 ω → ∞ On a log frequency scale, 20log( ω /z 1 ) is a straight line with a slope of 20 dB / decade (a 10-to-1 change in frequency). The line intersects the 0 dB at ω = z 1 . This value of w is called the corner frequency . Similarly, the term -20log ω is a line with slope -20 dB /decade And the term -20log|1+j ω /p 1 |is approximated by two lines. L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 5 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots Straight line approximations for: First order zero First order pole 25 5 p 1 20 0 20log(w/z 1 ) 15 -5 -20log(w/p 1 ) 10 -10 A dB A dB -15 5 0 -20 z 1 -5 -25 1 2 5 10 20 50 100 1 2 5 10 20 50 100 ω (rad/s) ω (rad/s) The function with K o = sqrt(10), z 1 = 0.1 rad/s and p 1 = 5 rad/s. ( j ω ) 0.1 ⁄ ω j ω 5 ⁄ A dB = 20 log 10 + 20 log 1 + – 20 log – 20 log 1 + 50 20log|1+j ω /z 1 | 20log |N(j ω )| 40 30 -20log ω A dB 20 20log K o 10 -20log|1+j ω /p 1 | 0 -10 -20 ω (rad/s) 0.1 1.0 10 100 L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 6 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots j ω R L ⁄ + N j ω ( ) L=100mH C=10mF = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 + 2 ω j ω R L ⁄ R=11 Ω V s V o – + + - - - - - - - - LC 0.11 j ω - N j ω ( ) = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - j ω ω     j ω 110 j ω 110 j N j ω ( ) 1 + - - - - - - 1 + - - - - - - - - -     = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ( j ω ) j ω ( ) 10 100 – 2 + 10 + 100 ω j ω 110 + + 1000 j ω j ω j ω A dB = 10 log 0.11 + 20 log – 20 log 1 + - - - - - - – 20 log 1 + - - - - - - - - - 10 100 40 30 20 10 0 A dB -10 -20 -30 -40 -50 -60 ω (rad/s) 1.0 10 100 1000 L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 7 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots The accuracy of the amplitude plot can be improved by correcting at the cor- ner frequencies (+/- indicates this applies to zeros and poles ): ± ± ≈ ± A dB c j 1 3 dB = 20 log 1 + = 20 log 2 Similar corrections can be made at 1/2 c and 2 c j 1 5 ± ± ≈ ± A dB c 1 dB - - - = 20 log 1 + = 20 log - - - 2 4 Graphically, this amounts to: 25 20 3 dB 15 1 dB 10 1 dB 5 A dB 0 -5 1 dB -10 -15 -20 -25 c/2 c 2c L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 8 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots Straight line approximations can be drawn for phase as well: The phase angle of a constant, K o , is 0. First order zeros and poles at the origin are a constant +/- 90 degrees For first order zeros and poles not at the origin: • For ω less than 10X the corner frequency, its 0 • For ω greater than 10X greater, its +/- 90, for zeros and poles , respectively. 90 45 degrees 60 Actual 30 ArgN 0 -30 -60 -90 p 1 z 1 /10 p 1 /10 z 1 10z 1 10p 1 L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 9 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6

Digital Systems Basic Circuit Theory Review IV CMPE 650 Bode Plots From our previous example: 0.11 j ω N j ω ( ) = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - j ω ω     j 1 + - - - - - - 1 + - - - - - - - - -     10 100 0.11 j ω N j ω ( ) ∠ α 1 β 1 β 2 = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - – – j ω ω     j - - - - - - - - - - - - - - - 1 + 1 +     ArgN j ω ( ) ∠ α 1 β 1 β 2 = – – 10 100 ω ω     tan 1 – tan 1 – α 1 90 ° β 2 β 1 = = - - - - - - = - - - - - - - - -     10 100 ArgN 90 60 30 0 -30 -60 -90 ω (rad/s) 1 5 10 50 100 500 1000 L A N R Y D UMBC A B M A L T F O U M B C I M Y O R T 10 (4/7/03) I E S R C E O V U I N N U T Y 1 6 9 6