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1. Circuit Analysis – II (EE-201)Variable-frequency Networks
Part 2

2. Sinusoidal Frequency Analysis
Bode plots to display frequency response data
VARIABLE-FREQUENCY NETWORK
PERFORMANCE
LEARNING GOALS

3. SINUSOIDAL FREQUENCY ANALYSIS
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4. HISTORY OF THE DECIBEL
Originated as a measure of relative (radio) power
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Using log scales the frequency characteristics of network functions
have simple asymptotic behavior.
The asymptotes can be used as reasonable and efficient approximations

5. Bode Plots
• These are Magnitude and Phase based pair of plots
• The Horizontal Access is always a logarithmic scale frequency (f or ꙍ)
• The vertical is a 20log(k + jꙍ) Magnitude plot in 1st part
• The vertical access is tan-1(jꙍ/k) Phase plot in 2nd part
• Most of the times, we take all k’s common from the tf and thus we use
1+jꙍ/k instead k+jꙍ,
• The common terms are multiplied with Constant gain portion so the overall
effect on the graph is none.
• It is calculated for a transfer function of
H(w) = Gain* Polynomial(Zeros)/ Polynomial( Poles)

6. Zeros, Poles and Gains
• The zeros are on the top of transfer function (tf) equation (numerators) so the
magnitude is positive elevation
• Poles are the denominator of tf equation so are negative depressions on graph
• Two polos or Zeros at the same value makes the elevation or depression doubled
• A pair of a pole and a zero cancels out the effect on graph
• When a pole or zero is already drawn, and a second pole or stars its effect, the
previous depression or elevation is effected and thus a composite plot is formed.
• Constant terms with H(w) such as Gains are straight lines with a fixed magnitude.
• We add up all gains, poles and zeros to get our overall plot

7. Drawing a Zero on Bode plot
• A zero has 20dB/decade elevation starting from its position.
• We always convert the zeros and poles to 1+jꙍ/k or 1+s/k form
• Before the value of s or jꙍ is reached on the x-axis, the 20log(1+jꙍ/k) is negligible because ꙍ is
small hence jꙍ/k is more small. Also 20log(1+0) is 0.
• If a zero is at origin, jꙍ=0, then it is a +20dB/decade increasing line, crossing ꙍ axis at 0
• It is an approximation and real plots may be more smooth due to continuous values.

8. Drawing a Pole on Bode plot
• A Pole has -20dB/decade depression starting from its position.
• We always convert the poles to 1+jꙍ/k or 1+s/k form
• Before the value of s or jꙍ is reached on the graph, the -20log(1+jꙍ/k) is negligible because ꙍ is
small jꙍ/k is more small. Also -20log(1+0) is 0.
• If a pole is at origin, jꙍ=0, then it is a -20dB/decade decreasing line, crossing ꙍ axis at 1
• Approximation encounters difference between plots

9. Drawing Phase in bode plot
• We calculate Phase by using tan-1(imaginary/real) method for individual terms in the tf.
• Phase of a constant term is 0 since the imaginary term in such case is zero.
• (jꙍ+0), jw=0 has a 90o phase, always, throughout the graph
• For zero, it is positive
• For pole it is negative
• jꙍ=k has a phase change of +90o or -90o between k and 10 x k
• If we consider a zero such as (1+jꙍ/k), the 0 to 90 degree phase inversion will start from k and will complete at 10k
• If we consider a pole such as (1+jꙍ/m), the 0 to -90 degree phase inversion will start from k and will complete at 10k
• The slope is 45o/decade

14. H(s) =

15. H(s) =

16. Book and Notes available at www.eedmd.weebly.com/ca2