LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form of this technique is called “Leapfrog Technique” • Fundamental Building Blocks are - Integrators - Second-order Realizations • Filters considered - LP - BP - HP - BE - Zeros ω j ECEN 622 (ESS) TAMU-AMSC
39
Embed
LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
LEAP FROG TECHNIQUE
Operational Simulation of LC Ladder Filters
• RLC prototype low sensitivity • One form of this technique is called “Leapfrog Technique” • Fundamental Building Blocks are - Integrators - Second-order Realizations • Filters considered - LP - BP - HP - BE - Zeros ωj ECEN 622 (ESS)
TAMU-AMSC
Ladder Networks
Elements connected in series and in parallel
Zeros are easily recognized: Zseries=infinity, or Zshunt=zero
Problem: How to design active ladder filters? How to design inductors?
Ladder Filters
434
3423
2312
1211
Z)0i(vY)vv(i
Z)ii(vY)vv(i
−=−=−=−=
i1 i3 v2 v4
v1
gnd
Y1 Y3
Z2 Z4
By means of a simple example it is illustrated how to implement an active filter based on the equation of a passive RLC prototype.
434
3423
2312
1211
Z)0i(vY)vv(i
Z)ii(vY)vv(i
−=−=−=−=
- +
R
Zx
v1
-v2
RZvvv xx /)( 21 −−=
Next we match the equations coefficients with the implementations.
Vx’=i1 Rdummy For R dummy=1 one can make v’x=i1
Two key transformations: 1. If needed converter current equations to voltage equations by multiplying by a dummy (artificial) resistor value of 1. 2. Express the relation of current and voltage of equations always as integrator. The integrator is the basic building block.
General Principles 3i − 1i − 1i + 3i +
2i −i
2i +
3iI −
3iV =
2iV − −
2iI −
+
1iI −
1iV −−
iIiV−
+1iI + + 1iV +
−
2iV ++
−
2iI +
3iI +
3iV ++ −
Interior Portion of a General LC Ladder Network • Interior components are reactive elements only. • • Branch voltages are voltage and current • How to select the proper “STATE” variable ? a) b) c)
)finite(0Rand0Ror0R Lss ≠=≠
o o+ −cv
Ci
o o+ −v
LLi
o o+ −v
Li C
∫=⇒= dt)t(iC1vi
C1
dtdv
cc
∫=⇒= dt)t(vL1iv
L1
dtdi
LL
∫ ∫∫ +−
=⇒+= dt)t(vL1dt)t(i
LC1i
Ci
dt)t(idL
dtdv 2
2
2
Systematic approach by writing voltage (KVL) and current (KCL) equations
3i − 1i − 1i + 3i +
2i −i
2i +
3iI −
3iV =
2iV − −
2iI −
+
1iI −
1iV −−
iIiV−
+1iI + + 1iV +
−
2iV ++
−
2iI +
3iI +
3iV ++ −
] I I [ Y
1 V 1 3 i 2 i
2 i i− − −
− − =
]VV[Z
1I i2i1i
1i −= −−
−
]II[Y1V 1i1i
ii +− −=
]VV[Z
1I 2ii1i
1i ++
+ −=
]II[Y
1V 3i1i2i
2i +++
+ −=
- Immittance Functions - Voltage Transfer Functions, convert I to V functions
RIV.,e.i k'k =
Terminations of LC Ladder Filters Source a) b)
1Z
2V 2Y+
−
3I
−+
inV
oR
1I
]VVR
RV[ZRRIV 2
'1
oin
11
'1 −−==
]VV[RY
1V '3
'1
22 −=
1Y 1V+
−−
+inV
oR2I
2Z
3V+
−
−−=⇒−
−= '
21ino1
12o
1in11 V)VV(
RR
RY1VI
RVVVY
]VV[ZRV 31
2
'2 −=
Load termination
2nV −
+
−
+
outn VV =nR
1nI −1nZ −+
−
a) or b)
'1n
nnout V
RRVV −==
]Z/)VV[(RV 1nout2nnout −− −=
)VV(R
RZ
RV out2nn
1nout −= −
−
1n
n
2nn
out
ZR1
VR
R
V
−
−
+=
Lossy Integrator
1nV −
+
outn VV =nR2nI −
2nZ −
−1nY −
]VRRV[
RY1V out
n
'2n
1nout −= −
−
The approach to map a passive RLC prototype is to pick the state-variables which can be expressed as integrators, since integrators are the basic building block. By applying KCL, KVL, KCL, …, as many times as the order of the filter, one can write the state-equations that can be implemented by active filters. an example is shown below:
2IoV
LRinV 2C1C 1V+
−−+
SR
A Typical Passive RLC Filter
( )
−−= − '
21inS1
1 VVVRR
SRC1V ( )a15
( )b15
( )16
( )17
,RIV 2'2 = R is an arbitrary value
KCL
where
KVL
KCL
( )o1'2 VV
SLRV −=
−= o
'2
2o V
RLRV
SRC1V
SRR
1S
SRR
−
∑
∑
∑
1SRC1
2S
3S 4S
5S
2SRC1 LR
R−
1−
1−1
1'2V
SLR
Signal Flow Graph of RLC Prototype of Fig. Shown in previous page.
Low-Pass Ladder Filters (Zeros at Infinity) (All Poles)
Example: A Fifth-Order LP Filter • 5 State Variables • 5 State Equations