Rules for Sketching a Root Locus
1. Determine the number of branches based on the pole-zero
excess, n-m. Each branch will start at a pole when 𝐾 = 0 and
end at a zero as 𝐾 → ∞.
2. Draw the loci that are on the real axis, right to left, starting at
the first pole or zero and stopping at the next, and so forth.
3. Draw the asymptotes from their “center of gravity” on the
real axis based on the pole-zero excess of the system.
4. Calculate the departure and arrival angles for any complex
poles and zeros.
5. If a locus crosses the imaginary axis, calculate 𝐾 and 𝜔𝑑 by
solving the characteristic equation with 𝑠 = 𝑗𝜔.
6. Calculate any break-out and break-in points that result on the
real axis by using the principle of orthogonality. (Can be
estimated instead)
6/10/2024
Lecture 8 – Root Locus Examples
𝛼=
σ 𝑝𝑖 − σ 𝑧𝑖
𝑛−𝑚
𝑛 − 𝑚 asymptotes
𝜙𝑑𝑒𝑝 = ෍ 𝜓𝑖 − ෍ 𝜙𝑖 − 180°
𝜓𝑎𝑟𝑟 = ෍ 𝜙𝑖 − ෍ 𝜓𝑖 + 180°
𝑎
𝑑𝑏
𝑑𝑎
−𝑏
=0
𝑑𝑠
𝑑𝑠
1
Rules for Drawing Bode Diagrams
1. Put the transfer function in the natural frequency
format so you can determine the steady-state gain, K,
Kv , Ka , etc.
2. Draw the low frequency magnitude asymptote based
on -20 db/decade times the System Type so that it
intersects the steady-state gain at 𝜔 = 1 rad/s.
3. The high frequency magnitude asymptote will be
-20 db/decade times the pole-zero excess.
4. The low frequency phase asymptote will be at -90°
times the System Type.
5. The high frequency phase asymptote will be at -90°
times the pole-zero excess (does not apply to nonminimum phase systems).
6/10/2024
Lecture 16 – Bode Diagrams
2
Time Delay
1
𝐺 𝑠 =
𝑠 𝑠+2
Phase
Margin
6/10/2024
𝑒 −𝑠
𝐺𝑑 𝑠 =
𝑠 𝑠+2
1. We can see in this example how much
phase is lost in frequency range of the 0 db
crossing when a delay of 1 second is
introduced to this transfer function.
2. This system still has 45° of phase margin,
but if the delay is increased to 2.3 seconds,
the system will become unstable.
3. The max time delay a system can accept
before going unstable is:
𝜋
𝑃𝑀
180
𝑇𝑑 =
𝜔
Lecture 17 – Time Delay
3