Study of Rauch Low-Pass Filters using Pascal’s TriangleMinhTri Tran*, Anna Kuwana, Haruo Kobayashi
Division of Electronics and Informatics, Gunma University Email: [email protected]
1. Research Objective
[1] M. Tran, A. Kuwana, H. Kobayashi, "Ringing Test for Tow-Thomas Low-Pass Filters", Int. Conf. on Promising Electronic Technologies (ICPET 2020), Jerusalem and Gaza City, Palestine, Dec. 2020.
[2] M. Tran, A. Kuwana, H. Kobayashi, "Study of Behaviors of Electronic Amplifiers using Nichols Chart", IEEE the 3rd Int. Conf. on Electronics and Communication Engineering (ICECE 2020), Xi'an, China, Dec. 2020.
[3] M. Tran, A. Kuwana, H. Kobayashi, "Ringing Test for 2nd-order Sallen-Key Low-Pass Filters", IEEE 2nd Int. Conf. on Circuits and Systems (ICCS 2020), Chengdu, China, 10-13, Dec. 2020.
[4] M. Tran, A. Kuwana, H.Kobayashi "Ringing Test for Negative Feedback Amplifiers" 11th IEEE Annual Information Technology, Electronics and Mobile Communication Conf. (IEMCON 2020), Canada, Nov. 2020.
References
2. Research Background
3. Stability test for Rauch LPF
5. ConclusionRinging test for Rauch low-pass filters using Pascal’s triangle Observation of coefficients and phase margin can help us determine the operating regions of high-order systems. Theoretical concepts of stability test are verified by SPICE
simulations and practical measurements.Future work: Stability test for parasitic components in printed circuit boards, physical layout layers, transmission lines…
[P7] 15:15 ～ 16:30, Feb. 2, 2021ICEIC 2021Jan. 31st(Sun) - Feb. 3rd(Wed), 2021 / Jeju Shinhwa World, Republic of Korea
4. Experimental Results
(Technology limitations)
Over-damping:Phase margin is 87 degrees.Critical damping:Phase margin is 80 degrees.Under-damping: Phase margin is 50 degrees.
20 1 ; L a j ja3 2 3
2 3 1 2 2 3 21 1
10 0; ; ;
R R RR R C C RbR
a R CaR
R1= R2 = 1 kΩ, R3 = 10 kΩ, C1 = 6 nF,C2 = 200 pF, at f0 = 20 kHz.• Over-damping (C1 = 6 nF),• Critical damping (C1 = 2 nF), and• Under-damping (C1 = 0.1 nF).
Fully differential Rauch LPF
Nyquist plot of loop gain
Re
Im
0
-1
Nichols plot of loop gain Nichols chart in Network Analyzer?
Bode plot of transfer function Transient response
91o
103.5o130o
Nichols plot of self-loop functionOver-damping:Phase margin is 89 degrees.Critical damping:Phase margin is 76.5 degrees.Under-damping: Phase margin is 50 degrees.
21dB
16dB18dB
Simulated results
Measured results
Bode plot of transfer function Transient response
Nichols plot of self-loop function
(Very complicate)(Unclear operating region)
Overshoot
Undershoot
10 1 11 ... 1
n n nnj a j a j a j
n = 1 1 1n = 2 1 2 1n = 3 1 3 3 1n = 4 1 4 6 4 1
Characteristics of Pascal’s triangle
Transfer function
( )A : Numerator function
Self-loop function
Input Output( )H
( )inV ( )outV
oMagnitude-frequency plotoAngular-frequency plot
oPolar chart Nyquist chart
oMagnitude-angular diagram Nichols diagram
Bode plots
1()
)(( )
AH
L
Linear system ( ), ( ) in o u tV V : periodic signalswith angular frequency variable
Bode plot of transfer function
Nichols plot of self-loop function
1 (( )(
()
))
( )
out
in
AL
VV
H• Critical damping:
21 1 1 11
1( ) ( 1 62
)
H Bjj
dH
3 32 0.316( 11( ) 6)1
123
dBHjj
H
222 11( ( 1)
2) 6
21
dBH
jjH
• Under-damping:
• Over-damping:
1 : 1 : 1
1 : 2 : 1
1 : 3 : 198o
103.7o
128o
1dB
-10dB
-6dB
PM 52o
PM 82o
PM 76.3o
PM 50oPM
89o
PM 76.5o
7dB
-5dB0dB
93o 100o 130o
PM 50o
PM 87o
PM 80o
Transfer function
Self-loop function
2
0
0 1
;1
Hja
ba j
Single ended Rauch LPF
Ringing in electronic systems
Solving the stability test problems:• Overshoot, undershoot, ringing • Loop gain, Nyquist diagram, Nichols chart
o Easiest selection of circuit componentso Simplest design in fully differential forms and complex topologies
Merits of Rauch low-pass filters
o Nichols chart of self-loop function A useful tool for stability test of feedback networks o Use of Pascal’s triangle Fast ringing test for high-order systems
Innovation of this work
LAB tool
LAB tool
Unused tool
Stability test
Simple tool
Simple
Simple
Simple
