Bode plot — magnitude and phase vs frequency: Foundations

3.5.41 · D1 · Physics › Guidance, Navigation & Control (GNC) › Bode plot — magnitude and phase vs frequency

Is page par kuch bhi assumed nahi hai. Agar parent note ne koi symbol bina build kiye use kiya, toh hum use yahan build karte hain, us order mein jo har idea ko pehle waale par tikne deta hai.

1. Ek sine wave, aur uske teen knobs

Pehle "kitni badi" ya "kitna late" ki baat karne se pehle, hum yeh agree karte hain ki sine wave kya hai aur hum usmein kya badal sakte hain.

Figure — Bode plot — magnitude and phase vs frequency

Figure dekho. Kaali wave reference hai. Laal wave ki speed same hai lekin shifted hai — woh shift hi phase $ϕ$ hai. Uski height alag hai — woh amplitude $A$ hai. Yeh do farq, amplitude aur phase, wahi do cheezein hain jo ek well-behaved system ek sine ke saath karta hai. Isliye poora topic exactly do graphs hai.

ω

t

t

time hai (seconds).

ω

angular frequency hai (radians per second) — har second mein circle ke kitne radians sweep hote hain.

Amplitude ek word mein

Height.

Phase ek word mein

Timing (kitna left/right shifted hai).

2. Frequency $ω$ aur log axis

$ω$ (Greek letter omega) haari shaking speed hai. Ek control system ko speeds ki parwah hai bahut slow se (fuel sloshing, ~0.01 rad/s) se bahut fast tak (metal bending, ~1000 rad/s) — ek lakh to ek ka range.

Isko kyun chahiye: normal ruler par, 0.01 aur 1000 ek page share nahi kar sakte — ek origin par ek dot hai, doosra agले kamre mein hai. Log axis par, har ×10 step (ek decade) same jagah leta hai, toh saari speeds fit ho jaati hain aur evenly spread ho jaati hain.

Decade vs octave

Decade ×10 hai. Octave ×2 hai. Bode slopes decade per quote hoti hain.

lo g_{10} (100)

2

, kyunki

1 0^{2} = 100

3. Complex numbers: ek arrow jo dono answers store karta hai

Yahan clever bit hai. Hum har frequency par do numbers report karna chahte hain (amplitude change aur phase shift). Ek complex number ek aisa object hai jo exactly do numbers pack karta hai — ek length aur ek angle. Perfect fit.

Figure mein laal arrow complex number hai. Uski length $r$ draw ki gayi hai; rightward axis se uska angle $θ$ mark hai. Dono descriptions same arrow hain.

Topic ko yeh kyun chahiye: frequency response $G (j ω)$ ek arrow hai jiska length amplitude change batata hai aur angle phase shift batata hai. Ek complex number, dono answers — phir hum ise do Bode graphs mein split karte hain.

j

matlab...

"upar point karo" — arrow ke vertical (imaginary) part ka marker.

3 + 4 j

ki magnitude

9 + 16 = 5

4. $arctan$ — angle-finder (yeh tool kyun)

Parent note baar baar $arctan (ω / ω_{c})$ likhta rahta hai. Yeh kahaan se aata hai?

§3 mein arrow phir dekho. Uska tilt angle ek right triangle ke andar baitha hai: horizontal side $a$ (angle ke adjacent) aur vertical side $b$ (angle ke opposite). Us angle ka tangent define hota hai

$tan θ = \frac{opposite}{adjacent} = \frac{b}{a} .$

Yeh ratio arrow ki steepness hai — ek steep arrow (bada $b$ , chhota $a$ ) ka tangent bada hoga. Toh ratio $b / a$ secretly angle encode karta hai.

Yahi tool kyun aur koi nahi: phase pane ke liye humare paas arrow ke side lengths hain aur angle chahiye. Tangent angle → ratio map karta hai; humare paas ratio hai aur angle chahiye, toh tangent ka inverse chahiye. Woh exactly $arctan$ hai.

arctan (1)

4 5^{\circ}

— woh angle jiska opposite uske adjacent ke barabar ho (45° tilt).

arctan

undo karta hai...

tangent ko.

5. $e^{j θ}$ — ek spinning unit arrow

Parent $e^{j ω t}$ aur $e^{j ϕ}$ likhta hai. Yeh kya hai?

Figure mein laal dot unit circle par ride karta hai; uski height $sin θ$ hai aur floor par uska shadow $cos θ$ hai. Toh $e^{j ω t}$ ek hand hai jo speed $ω$ par spin kar raha hai — aur uska horizontal shadow exactly $cos ω t$ hai, haari sine wave. Isliye topic $e^{j ω t}$ use karta hai: yeh disguise mein sine wave hai, aur complex exponentials ko raw sines se kahin zyada aasaani se system ke through push kar sakte hain.

e^{j θ}

ki length...

exactly

1

hai (yeh unit circle par rehta hai).

e^{j 0}

cos 0 + j sin 0 = 1

(arrow daayein point karta hai).

6. Transfer function $G (s)$ aur substitution $s = j ω$

K

(ek constant)

ek plain number jo poori cheez multiply karta hai — ek flat gain, koi frequency dependence nahi.

ω_{c}

(corner frequency)

woh special speed jahan pole ya zero "kick in" karta hai — uska factor

1 + j ω / ω_{c}

ke real aur imaginary parts equal hote hain.

n

(integrator count)

1/ s

ke kitne factors hain — har ek pure integrator hai.

7. Decibels: gain ko friendly number mein badalna

20 kyun aur 10 nahi: Bode magnitude ek amplitude ratio hai, lekin power amplitude ke square ke saath badhti hai. Kyunki $lo g (A^{2}) = 2 lo g A$ , power-decibel $10 lo g_{10}$ amplitude ke liye $20 lo g_{10}$ ban jaata hai. (Yeh parent ke main "mistakes to avoid" mein se ek hai.)

Decibels kyun hain: log-frequency axis ke saath mila kar, yeh har simple factor ke contribution ko ek straight line banate hain, aur products aankhon se add karne wali lines ka sum ban jaata hai.

0

dB matlab gain = ?

1

(unchanged amplitude), kyunki

20 lo g_{10} 1 = 0

20 kyun 10 nahi

magnitude amplitude hai; power ∝ amplitude², aur

lo g (A^{2}) = 2 lo g A

8. Yeh sab topic ko kaise feed karta hai

Is map mein har arrow ek "yeh pehle chahiye" hai. Koi bhi path trace karo aur do Bode graphs tak pahuncho.

Equipment checklist

Har ek tab reveal karo jab tum ise zor se jawaab de sako.

Main sine wave ke teen knobs bata sakta hoon

Amplitude

A

(height), angular frequency

ω

(speed), phase

ϕ

(timing shift).

Main jaanta hoon

lo g_{10} (1000)

kya hai aur kyun

3

, kyunki

1 0^{3} = 1000

; log ×10 ko +1 mein badalta hai.

Main complex number ko length + angle mein convert kar sakta hoon

∣ z ∣ = a^{2} + b^{2}

(length),

∠ z = arctan (b / a)

(angle).

Main jaanta hoon phase ke liye

arctan

kyun use karte hain

Humare paas arrow ka side ratio hai aur uska tilt angle chahiye; arctan tangent ko undo karke angle deta hai.

Main jaanta hoon plain

arctan

pole/zero factors ke liye safe kyun hai

Woh factors

1 + j ω / ω_{c}

ka real part

+ 1

hota hai, isliye arrow right half-plane mein rehta hai jahan arctan correct hai.

Main

e^{j θ}

ko picture ki tarah padh sakta hoon

Length-1 arrow

θ

par tilted; ise speed

ω

par spin karne se

e^{j ω t}

milta hai, jiska shadow

cos ω t

hai.

Main jaanta hoon

s = j ω

kyun set karte hain

Speed

ω

ki pure sine ke liye system ka response sample karne ke liye;

G (j ω)

ek arrow ban jaata hai jiska length gain hai aur angle phase hai.

Main jaanta hoon decibels 20 kyun use karte hain, 10 nahi

Bode magnitude ek amplitude ratio hai aur power ∝ amplitude², isliye

10 lo g_{10} (A^{2}) = 20 lo g_{10} A

Main jaanta hoon frequency axis logarithmic kyun hai

Control systems kai decades span karte hain; log spacing sab fit kar leta hai aur simple factors ko straight lines banata hai.

Jab upar ki har line kuch nayi nahi reveal karti, tum parent note ke liye ready ho.