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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
]I,V,I,V,I[ 54321
•1I→
onRinV+
−
n1L
2V+
n2Cn3L3I→
+
4Vn4C
n5L5I→
onR outV+
−KVL-KCL-KVL-KCL-KVL (Sequence) * 02)1(1 =−+− VnsLonRIinV
RIV 1'
1 =
][ '121
'1 V
RRVV
sLRV on
nin
−−=
Current Analogs
)1(
0VsCII 2n231 =−−
]VV[sRC
1V '3
'1
n22 −=
0VIsLV 43n32 =−−
]VV[sL
RV 42n3
'3 −=
0IVsCI 54n43 =−−
]VV[sRC
1V '5
'3
n44 −=
0RIIsLV n655n54 =−−
]VR
RV[sL
RV '5
n64
n5
'5 −=
n6
'5
n65out RRVRIV ==
]RRV
RRV[
sLR
RRV
n6out
n64
n5
n6out −=
But
)2(
)3(
)4(
)5(
Active RC Building Blocks
o
inV
r
+−
r
•
+−•
o
o
o
kV
jV
jC
jV
Thus, eq(1) is implemented as follows
o
inV
r
+−
r
•
+−•
o
o
o
2V
'1V
1C
'1V
1R
2R
inR
on
in11
in12
in1in
RLCR
RLCR
RLCR
=
=
=
Cj/j1τ
Cj/j2τ
Cj/j3τ
][ '121
'1 V
RRVV
sLRV on
nin
−−=
o'
1V
r
+−
r
•
+−•
o
o'3V
2C
2V3R
'3R
o
2V
r
+−
r
•
+−•
o
o
4V
3C
'3V4R
'4R
o'3V
r
+−
r
•
+−•
o
o'5V
4C
4V5R
'5R
o
4V
r
+−
r
•
+−•
o
o'5V
5C
'5V6R
'6R
2 3 n 2
2 ' 3 n 2
C R RC C R RC
= =
R / L C R R / L C R
n 3 3 4
n 3 3 ' 4
= =
n 4 4 5
n 4 4 ' 5
RC C R RC C R
= =
n 6 n 5 5 6
n 6 n 5 5 ' 6
R / L C R R / L C R
= =
]VV[sRC
1V '3
'1
n22 −=
]VV[sL
RV 42n3
'3 −=
]VV[sRC
1V '5
'3
n44 −=
]RRV
RRV[
sLR
RRV
n6out
n64
n5
n6out −=
'5V
ino
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
rr
inR 1C
1R
rr
'1V
2R
'3R
2C
r
r
•
3R
'4R
•
3C
r
r'3V
2V
4R
'5R
4C
r
4Vr
5R
'6R
5C
6R
Low-Pass 5th-Order Active –RC Leapfrog Filter
o
inV+
R/Ron
n1SLR
+n2SRC
1+
n3SLR
+n4SRC
1+
n5SLR
'1V '
3V '5V
2V 4V
R/Ron− 1− 1−
o
oV
R/R n6−1−1−
o
inV+
R/Ron
1n1
onR/SL
R/R+
−+
n2SRC1
− +n3SL
R− +
n4SRC1
− +1R/S
R/R
n6n5L
n6+
−
1− 1−
o
oV
1−1−
OTA-C Implementation +
−mjg
+
−mjg
o
o
o
o jV
jCjVjV
'inV
+
−1mg
+
−2mg
o
o
o
o '1V
1C'1V
2V
'inV
+
−6mg
+
−7mg
o
o
o '5V
5C'5V
4V+
−3mg
o
oo 2V
2C
'3V
'1V
+
−4mg
o
oo '
3V
3C4V2V
+
−5mg
o
oo 4V
4C
'5V
'3V
RL
Cg n1
1
1m =
on
n1
2
2mRL
Cg
=
Thus, EQ (1) is implemented as
)2(EQ
n22
3m RCCg
=
)3(EQ
RL
Cg n3
3
4m =
)4(EQ
n44
5m RCCg
=
)5(EQ
on
n5
5
6mRL
Cg
=
on
n5
5
7mRL
Cg
=
][ '121
'1 V
RRVV
sLRV on
nin
−−=
'1V
+
−1mg
o+
−2mg +
−3mg +
−4mg +
−5mg +
−6mg
+
−7mg o
o'5 VV =
5C4C4V
••
•
'3V
3C2C
2V•
1CinV
5th-Order OTA-C Leapfrog Implementation
•
Voltage Scaling of OTA-C All Pole Leapfrog Filters.
+ −1mg
o
aV
+ −2mg
•o
cV• o
dV
+ −3mg
o
bV•
2C
xV
1C 3C
2dc2mx sC/)VV(gV −=
+ −3mg
o
aV
+ −k/gm2
o
cV• o
dV
+ −3mkg
o
bV•
2C
k/Vx
•
1C 3C
+ −1mkg
•
+ −3mg
Changing without changing k
VV xx → dcba Vand,V,V,V
+ −1mg
o
inV
+ −3mg o
oV
+ −4mg
+ −2mg
1C
2C
3C
4C+ −
6mg
+ −5mg
+−7mg5C
o
+inV
+ +
+ +
o
RRon
RRon−
'1V oV
RRon−n5SL
R
n3SLR
n1SLR
n2SRC1−
n4SRC1−
System Level Representation: 5th-Order LP Leapfrog Filter
SCALING MAXIMUM DYNAMIC RANGE
4max4
maxo
2max2
maxon6n54
1max1
maxo3
max3
maxon6n5o4
in
kVV
kVV
;R
RR
Ls)s(TV
kVV
;kVV
;R
RR
LsVV
1V
=
=
+=
==
+=
=
o
+inV
+ +
+ +
o
4321on kkkkR
R
1k
RRon−
'1V '
3V '5V oV
4V2V
2k
3k
4k1k1
2k1
3k1
4k1
RRon−n5SL
R
n3SLR
n1SLR
n2SRC1−
n4SRC1−
Current Level Scaling Technique Case 1
oV
+
−mg o
C R
oIinIo
R represents input resistance of a current-mirror
If needs to be constant, then R must be decreased by k. oV
→→→→
k/RR&k/CC2optionk/RR&kgg1option
summaryIn mm
Case 2
oV
+
−mg o
1C 2C
oIinIo
represents input impedance of next block 2C
sC/gI mo −=
sCRgRIV m
oo −==
→→
−=k/CC2option
kgg1optionsCgkkI mmm
o
RkIV oo =
1
mo sC
gI −=
2oo sC
1IV =
LC Ladder Filters with Finite Zeros
'1iY −
'1iY +
1iZ − 1iZ +oo o2i − i
'1iI −↑
3iI −→
2iY − iY 2iY +
2i + 3iI +→+
2iV −
−
+
iV−
+
2iV +
−
1iI −→
1iI +
1iZ − 1iZ +oo o2i − i3iI −
→
2iY − iY 2iY +
3iI +→+
2iV −
−
+
iV−
+
2iV +
−
1iI +→1iI −→
+i2i,i VK −
−
+2ii,2i VK ++
−+
2ii,2i VK −−−
+i2i,i VK +
−
− − −
i1i,2iY −− 1i,i,1iY +− 2i,1iY ++
Dependent Sources to Replace the Bridging Admittances
In a similar for KCL for node I+2 and solving for results 2iV +
2iatKCL −
]VV[YVYII i2i'
1i2i2i1i3i −+=− −−−−−−or
'1i2i
1i3ii'
1i2i
'1i
2i YYIIV
YYYV
−−
−−
−−
−−
+−
+
+=
1i,2i
1i3ii2i,i2i Y
)II(VKV−−
−−−−
−+=
+−+−−=− +−−+− ii'
1ii2i'
1iii1i1i VV(Y)VV(YVYIIor
'1ii
'1i
1i1i2i'
1ii'
1i
'1i
2i'1ii
'1i
'1i
i YYYIIV
YYYYV
YYYYV
+−
−++
+−
+−
+−
−
++−
+++
+++
=
1i,i,1i
'1i
'1i
2ii,2i2ii,2ii RYVVVKVKV
+−
−+++−−
−++=
'2i,i
'3i
'1ii2i,i2i RY/)VV(VKV +++++ −+=
KCL @ NODE i
LC Ladder Simulation 1. Start from the desired LC LP prototype network satisfying SPECS.
2. Eliminate shunt elements in the series branches or series element in the shunt branches.
3. Normalize and transform the network if the realization is not low-pass. 4. Select the state variables such that
5. Solve for each state variable in terms of the other states
and output variable including the proper scaling.
6. Synthesize each equation with its corresponding building block.
)s,x(fdtdx
µ=
Example: Low-Pass With Finite Zeros ωj
n2L
2I→
oonR
n2L
n2C n4C
4I→
o
n5Cn1C n3C onR outV
−
+
3V
−
+
1V
−
+
iV
−
+
oonR
n2L n4L
4I→
o
n45Cn12Cn234C
onR
outV
−
+
3V
−
+
1V
−
+inV
−+
2I→
+
− 331VK
+
−553VK
+
− 113VK
+
−335VKn
n
n
n
n
n
n
n
nnn
nnnn
nnn
CCK
CCK
CCK
CCK
CCCCCCC
CCC
45
435
234
213
234
453
12
231
5445
432234
2112
=
=
=
=
+=++=
+=
n45
n435
n234
n213
n234
n453
n12
n231
n5n4n45
n4n3n2n234
n2n1n12
CCK
CCK
CCK
CCK
CCCCCCC
CCC
=
=
=
=
+=++=
+=
)1(VKRV
RV
RV
SC1V 331
'2
on
1
on
in
n121 +
−−=
r3V •CK31
ro
+
− 4R
+
−o
1C
1V2R
o1V3R
o'2V
inV
]VV[SL
RV 31n2
'2 +=
]VV[SL
RV 31n2
'2 −= )2(
)2( '
r ro
+
− 4R
+
−o
2C
'2V
3V
ro1V
•
)3(VKVK]VV[SRC
1V 131553'4
'2
n2343 ++−=
131553'2
'4
n2343 VKVK]VV[
SRC1V −−−= )3( '
r
5V 353CK
ro
+
−
5R
+
−o
3C
3V
6R̂o'
2V
'4V
o
1V 331CKo
]VV[SL
RV out3n4
'4 −−= )4(
o
6R
+
−o
4C
'4VoutV
3V o
7R
R/LCRR/LCR
n447
n446==
)5(VKRV
RV
SC1VV 335
n6
out'4
n456out −
−==
r ro
+
−
8R
+
−o
5C
outV
9RooutV
'4V3V 535CK
o
inV o
+−
+−
+−
+−
+−
+−
+− +
−
r r
1C
1R
r r
1V
3R
4R2C
6R̂•
r
6R•
3C
r'2V
8R5C
CK31
•
'2V
8R
3V
7R
r
9R
+−
1V
•
inVo
353CK 331CK
r
r
Active RC Fifth-Order Elliptic Filter
OTA-C Elliptic Implementation
)1(VKRV
RV
RV
SC1V 331
'2
on
1
on
in
n121 +
−−=
+− 1mg
+
−1mg
o
o
o
o
1V1C'
2V1VinV
3V o
+− 31g
+
−rg o
=
=
31r
31
onn121r
1m31
Kgg
RCCggg
131
onn121m
r3131
CK
RCg
gKg
=
=
]VV[SL
RV 31n2
'2 += )2(
+− 2mg
+
−2mg
o
o
o
2C
3V
1V'2V
RL
Cg n2
2
2m =
131553'2
'4
n2343 VKVK]VV[
sRC1V −−−= )3(
+− 3mg
+
−5mg
o
o
o
3V3C
'2V
5V
4V
1V
o
+− 4mg
+
−rg o
]VV[sL
RV out3n4
'4 −−= )4(
+
−6mg
+
−6mg
o
o
o
4CoutV
3V'4V
R/LCg
n44
6m =
)5(VKRV
RV
sC1VV 335
on
out'4
n455out −
−==
+− 7mg
+
−mxg
o
o
outV5C
3V
'4V
outV
o
+− 35g
+
−rg o
35r
35
n455
7m
r
35
Kgg
RCC
ggg
=
=
−+o
−+ −
+
−+
−+
•
−+
−+
•
−+
−+
•
−+
−+
−+
−+
•
−+
o
in V1 C
1 V
2 C
'2 V
3 C3 V
4 C
'4 V
5 V5 C
out V
Fifth-Order OTA-C Elliptic Filter
i1 -vi
R1
C1 L2/R2
R
- v2
i2
C4 - R
- vout
R1
R R5
R
R1 vi
C3
L2
R5 C4 C1
V2 vout
i1 i2
i3
-1
-1
i2
C3
C3
i3
i3 C3 C3 -1
Simulating Floating Capacitors (Elliptic Filters)
3sC)vout(3sC)2v(3i −=
Finite Zeros: An Alternative Approach ωj
n4Lo
onR
2I
4I
o
n5Cn1C n3C onR oV
−
+
3V
−
+
1V
−
+
inV−+
n2L•••
•n2C n4C
5V
−
+
)1()VV(sCRVI
RVVsC 31n2
on
12
on
in1n1 −−−−=
RVI;RIV
'2
22'2 ==
)VV(sCRV
RV
RVVsC 31n2
on
1'2
on
in1n1 −−−−= )1( '
31
'2
n2 VVRVsL −=
)VV(sC)VV(sCRV
RVVsC 53n431n2
'4
'2
3n3 −−−+−=
53'4
n4 VVVR
Ls −=
)VV(sCRV
RVVsC 53n4
n6
5'4
5n5 −+−=
)2(
)3(
)4(
)5(
Component Calculations for an OTA-C implementation onn1
1m
1 RCgC
= n34m
3 RCgC
=
onRR =R
LgC n4
5m
7 =
R1g 6m =
on2m R
1g =
RL
gC n2
3m
6 =n6
7m R1g =
Note that if then the OTA # 7 can be eliminated and the output of OTA # 6 can be simply connected to its negative terminal.
RR n6 =