BODE PLOT - S-PLANE REVIEW - WPI

BODE PLOT - S-PLANE REVIEW

NONINVERTING DC GAIN INVERTING DC GAIN

TERM IN H(s) +K "K

S-PLANE jω jω

! σ σ

!

BODE PLOT: (NOT VISIBLE IN S-PLANE) (NOT VISIBLE IN S-PLANE)
LET s = jω
MAGNITUDE +K = K "K = K

! |H| |H|
! [dB] ! [dB]
PHASE
20 log(K ) 20 log(K )

H ω ω
[deg] !
H
0° ω [deg]

ω

K = tan"1(0 /K) = 0 -180°

"K = tan"1(0 /" K) = "180

LHP ZERO ZERO AT ORIGIN (DERIVATIVE) RHP ZERO

TERM IN H(s) 1+ s" ZL s" ZO 1" s# ZR

jω jω jω

S-PLANE "1 σ σσ
# ZL
! ! !1

" ZR

BODE PLOT: 1+ j"# ZL = 12 + ("# ZL )2 0 + j"# ZO = "# ZO 1" j#$ ZR = 12 + (#$ ZR )2

LET s = jω ! |H| +20dB/dec |H| +20dB/dec |H| ! +20dB/dec
[dB] [dB]
[dB] ! !
! ω ω ω

MAGNITUDE

1 1 1
" ZL " ZR
H H " ZO H
[deg] [deg] [deg]

PHASE +90° ω +90° ω ! ω

+45°! ! -45°
-90°

1+ j"# ZL = tan$1("# ZL ) 0 + j"# ZL = tan$1%'& "# ZL ( = +90o 1" j#$ ZR = tan"1("#$ ZR )
0 *)

TERM IN H(s) LHP POLE POLE AT ORIGIN (INTEGRATOR) RHP POLE
S-PLANE
1 1 1
1 + s" PL s" PO 1 " s# PR

jω jω jω

σ σ σ

! "1 ! !1

# PL " PR

1= 1 1 =1 1= 1
0 + j"# PO "# PO
BODE PLOT: 1+ j"# PL 12 + ("# PL )2 1" j#$ PR 12 + (#$ PR )2

LET s = jω ! |H| |H| |H| !
[dB] [dB] [dB]
-20dB/dec -20dB/dec -20dB/dec
ω! ω! ω
MAGNITUDE !

1 11
" PL
H H " PO H " PR
[deg] [deg] [deg]

PHASE ! ω +90°

-45° ! ω +45°! ω
-90°
-90°

1/(1+ j"# PL ) = $tan$1("# PL ) 1 (0 + j"# PO ) = $tan$1("# PO 0) = $90o 1/(1" j#$ PR ) = tan"1(#$ PR )

GENERAL TRANSFER FUNCTION

ZERO AT LHP RHP
ORIGIN ZERO ZERO

H(s) = ±K s" ZO L(1+ s" ZL )L(1# s" ZR )
s" PO L(1+ s" PL )L(1# s" PR )

POLE AT LHP RHP

ORIGIN POLE POLE

(UNSTABLE MODE!)

!