Bode Plot Gain Margin Phase Margin | Crossover Frequency

Let G(s)H(s) be the open loop transfer function (OLTF) for which a bode plot is drawn. Then

Gain cross-over frequency  ( W_gc ) 

Gain cross-over frequency is the frequency at which OLTF magnitude is equal to 1 or 0dB.

i.e     |G(jw)H(jw)|=1  or  0dB          , where  W=W_gc

Phase cross-over frequency  ( W_pc ) 

Phase cross-over frequency is the frequency at which phase of OLTF  is equal to -180⁰.

i.e     ∠G(jw)H(jw) = -180⁰              , where  W=W_pc

Gain Margin (GM) 

Gain Margin is the factor by which the gain factor K can be multiplied before the closed loop system reaches to the verge of instability.

It is calculated as 

GM = -20 log X        (in dB)

where,   X=|G(jw)H(jw)|                , at  W=W_pc

Phase Margin (PM)  

Phase Margin is defined as the allowable phase lag before the closed loop system reaches to the verge of instability.

It is calculated as 

PM = 180+ Ф         

where,   Ф = ∠G(jw)H(jw)             , at  W=W_gc

Stability Rules of Bode Plot

(1) If  GM and PM both are positive, or ,  W_gc < W_pc , the closed loop sytem is stable.

(2) If  GM and PM both are negative, or ,  W_gc > W_pc , the closed loop sytem is unstable.

(3) If  GM and PM both are zero, or ,  W_gc = W_pc , the closed loop sytem is marginally stable.

Gain Margin Phase Margin Solved Example

Q. Construct bode plot for the open loop transfer function

and find 

(a) Gain cross-over frequency

(b) Phase cross-over frequency

(c) Gain Margin

(d) Phase Margin

(e) Closed loop stability of the system

Sol :- The bode plot comes out to be as shown below

(a) |G(jw)H(jw)|=1                                              at  W=2.5 rad/sec

therefore, gain cross-over frequency = 2.5 rad/sec

(b) ∠G(jw)H(jw) = -180⁰                   at  W=5 rad/sec

therefore, phase cross-over frequency = 5 rad/sec

(c) GM = -20 log |G(jw)H(jw)|           at  W=W_pc=5 rad/sec 

We get   GM=11.21 dB

(d) PM = 180+ Ф        

where,   Ф = ∠G(jw)H(jw)                 at  W=W_gc=2.5 rad/sec

we get   PM= 180⁰-153⁰ = 27⁰

(e) Since both GM and PM are positive , therefore, closed loop system is stable.

Related :- 
     Bode Plot step-by-step
     What is Root locus | solved examples
     Routh Stability Criterion