def TLineFourPort(rho, gamma):
"""symbolic single-ended four-port transmission line
@param rho string reflection coefficient
@param gamma string gamma
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note use SignalIntegrity.Symbolic.TLine.TLineRho if you want to enter the string
representation of rho based on the characteristic impedance.
@note use SignalIntegrity.Symbolic.TLine.TLineGamma if you want to enter the string
representation of gamma based on a time delay
@note this is the symbolic version of SignalIntegrity.Lib.Devices.TLineFourPort.TLineFourPort except that by
default it's base on rho for symbolic simplicity (Zc complicates the look of things).
"""
Y = 'e^{' + gamma + ' } '
Y2 = 'e^{2 \\cdot ' + gamma + ' } '
D = '2 \\cdot \\left( 1 - ' + Y2 + ' \\cdot \\left[ ' + rho + ' \\right]^2 \\right)'
S1 = lfrac(
' 1 - ' + Y2 + ' \\cdot \\left[ ' + rho + ' \\right]^2 + ' + rho +
' \\cdot \\left( 1 - ' + Y2 + ' \\right) ', ' ' + D + ' ') + ' '
S2 = lfrac(
' \\left( 1 - \\left[ ' + rho + ' \\right]^2 \\right) \\cdot ' + Y +
' ', ' ' + D + ' ') + ' '
S3 = lfrac(
' 1 - ' + Y2 + ' \\cdot \\left[ ' + rho + ' \\right]^2 - ' + rho +
' \\cdot \\left( 1 - ' + Y2 + ' \\right) ', ' ' + D + ' ') + ' '
return [[S1, S2, S3, '-' + S2], [S2, S1, '-' + S2, S3],
[S3, '-' + S2, S1, S2], ['-' + S2, S3, S2, S1]]
def OperationalAmplifier(Zi, Zd, Zo, G):
"""symbolic operationalAmplifier
Operational Amplifier
Port 1 is - input, port 2 is + input and port 3 is output
@param Zi string input impedance between each plus and minus input port
to ground.
@param Zd string input impedance across the plus and minus inputs.
@param Zo string output impedance.
@param G string gain of the op-amp.
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.OperationalAmplifier.OperationalAmplifier
"""
S11 = lfrac(
'\\left(-2\\cdot ' + Zi + ' - ' + Zd + ' \\right)\\cdot Z0^2+ ' + Zi +
' ^2\\cdot ' + Zd + ' ', '\\left(2\\cdot ' + Zi + ' + ' + Zd +
' \\right)\\cdot Z0^2+\\left(2\\cdot ' + Zi + ' \\cdot ' + Zd +
' +2\\cdot ' + Zi + ' ^2\\right)\\cdot Z0+ ' + Zi + ' ^2\\cdot ' + Zd +
' ')
S12 = lfrac(
'2\\cdot ' + Zi + ' ^2\\cdot Z0', '\\left(2\\cdot ' + Zi + ' + ' + Zd +
' \\right)\\cdot Z0^2+\\left(2\\cdot ' + Zi + ' \\cdot ' + Zd +
' +2\\cdot ' + Zi + ' ^2\\right)\\cdot Z0+ ' + Zi + ' ^2\\cdot ' + Zd +
' ')
S32 = lfrac(
'2\\cdot ' + G + ' \\cdot ' + Zi + ' \\cdot ' + Zd + ' \\cdot Z0',
'\\left( ' + Zo + ' +Z0\\right)\\cdot \\left(\\left(2\\cdot ' + Zi +
' + ' + Zd + ' \\right)\\cdot Z0+ ' + Zi + ' \\cdot ' + Zd +
' \\right)')
S33 = lfrac(' ' + Zo + ' -Z0', ' ' + Zo + ' +Z0')
return [[S11, S12, '0'], [S12, S11, '0'], ['-' + S32, S32, S33]]
def RationalString(val):
data = ''
if isinstance(val, int):
data = str(val)
elif isinstance(val, float):
from SignalIntegrity.Lib.Rat import Rat
from numpy import sign
sn = sign(val)
if sn == 0:
return '0'
signStr = ''
if sn == -1:
val = val * -1.
signStr = '-'
(n, d) = Rat(val)
if d == 1:
return signStr + str(n)
elif d < n:
return signStr + str(val)
elif d == n:
return signStr + '1'
elif ((d < 10) and (n < 10)):
return signStr + lfrac(str(n), str(d))
# check for square-root of 2
valsq2 = val * math.sqrt(2.)
(n, d) = Rat(valsq2)
if d < n:
return signStr + str(val)
elif ((d < 10) and (n < 10)):
denstr = '\\sqrt{2}'
if d != 1:
denstr = str(d) + '\\cdot' + denstr
return signStr + lfrac(str(n), denstr)
return signStr + str(val)
def TransconductanceAmplifierThreePort(G, Zi, Zo):
"""symbolic three port transconductance amplifier
@param G string transconductance
@param Zi string input impedance
@param Zo string output impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.TransconductanceAmplifier.TransconductanceAmplifierThreePort
"""
D = '3\\cdot Z0^2+ (2\\cdot ' + Zo + '+2\\cdot ' + Zi + '-' + G + '\\cdot ' + Zi + '\\cdot ' + Zo + ' )\\cdot Z0+' + Zo + '\\cdot ' + Zi
S11 = lfrac(
Zo + '\\cdot ' + Zi + '+Z0\\cdot (2\\cdot ' + Zi + '-' + G +
'\\cdot ' + Zi + '\\cdot ' + Zo + ' )-Z0^2', D)
S12 = lfrac('2\\cdot Z0^2', D)
S13 = lfrac('2\\cdot Z0^2+2\\cdot ' + Zo + '\\cdot Z0', D)
S21 = lfrac(
'2\\cdot Z0^2+2\\cdot ' + G + '\\cdot ' + Zi + '\\cdot ' + Zo +
'\\cdot Z0', D)
S22 = lfrac(
Zo + '\\cdot ' + Zi + '+Z0\\cdot (2\\cdot ' + Zo + '-' + G +
'\\cdot ' + Zi + '\\cdot ' + Zo + ' )-Z0^2', D)
S23 = lfrac(
'2\\cdot Z0^2+Z0\\cdot (2\\cdot ' + Zi + '-2\\cdot ' + G + '\\cdot ' +
Zi + '\\cdot ' + Zo + ' )', D)
S31 = lfrac(
'2\\cdot Z0^2+Z0\\cdot (2\\cdot ' + Zo + '-2\\cdot ' + G + '\\cdot ' +
Zi + '\\cdot ' + Zo + ' )', D)
S32 = lfrac('2\\cdot Z0^2+2\\cdot ' + Zi + '\\cdot Z0', D)
S33 = lfrac(
Zo + '\\cdot ' + Zi + '-Z0^2+' + G + '\\cdot ' + Zi + '\\cdot ' + Zo +
'\\cdot Z0', D)
return _LineReplacer([[S11, S12, S13], [S21, S22, S23], [S31, S32, S33]])
def TeeThreePortZ1Z2Z3(Z1, Z2, Z3):
D = '3\\cdot Z0^2+2\\cdot Z0\\cdot\\left(' + Z1 + '+' + Z2 + '+' + Z3 + '\\right)+' + Z1 + '\\cdot ' + Z2 + '+' + Z1 + '\\cdot ' + Z3 + '+' + Z2 + '\\cdot ' + Z3
return [[
lfrac(
Z1 + '\\cdot ' + Z2 + '+' + Z1 + '\\cdot ' + Z3 + '+2\\cdot ' +
Z1 + '\\cdot Z0+' + Z2 + '\\cdot ' + Z3 + '-Z0^2', D),
lfrac('2\\cdot Z0^2+2\\cdot Z0\\cdot ' + Z3, D),
lfrac('2\\cdot Z0^2+2\\cdot ' + Z2 + '\\cdot Z0', D)
],
[
lfrac('2\\cdot Z0^2+2\\cdot Z0\\cdot ' + Z3, D),
lfrac(
Z1 + '\\cdot ' + Z2 + '+' + Z2 + '\\cdot ' + Z3 +
'+2\\cdot ' + Z2 + '\\cdot Z0+' + Z1 + '\\cdot ' + Z3 +
'-Z0^2', D),
lfrac('2\\cdot Z0^2+2\\cdot ' + Z1 + '\\cdot Z0', D)
],
[
lfrac('2\\cdot Z0^2+2\\cdot ' + Z2 + '\\cdot Z0', D),
lfrac('2\\cdot Z0^2+2\\cdot ' + Z1 + '\\cdot Z0', D),
lfrac(
Z1 + '\\cdot ' + Z2 + '+' + Z1 + '\\cdot ' + Z3 + '+' +
Z2 + '\\cdot ' + Z3 + '+2\\cdot Z0\\cdot ' + Z3 + '-Z0^2',
D)
]]
def ShuntZThreePort(Z):
"""symbolic three port shunt impedance
@param Z string impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.ShuntZ.ShuntZThreePort
@remark Ports 1 and 2 are connected together and to port 1 impedance.\n
Port 3 is the other port of the impedance.\n
@todo check the port numbering
"""
D = '2\\cdot ' + Z + '+3\\cdot Z0'
return [[
lfrac('-Z0', D),
lfrac('2\\cdot ' + Z + '+2\\cdot Z0', D),
lfrac('2\\cdot Z0', D)
],
[
lfrac('2\\cdot ' + Z + '+2\\cdot Z0', D),
lfrac('-Z0', D),
lfrac('2\\cdot Z0', D)
],
[
lfrac('2\\cdot Z0', D),
lfrac('2\\cdot Z0', D),
lfrac('2\\cdot ' + Z + '-Z0', D)
]]
def CurrentAmplifierTwoPort(G,Zi,Zo):
"""symbolic two port current amplifier
@param G string current gain
@param Zi string input impedance
@param Zo string output impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.CurrentAmplifier.CurrentAmplifierTwoPort
"""
return [[lfrac(Zi+' - Z0',Zi+' + Z0'),'0'],
[lfrac('2\\cdot '+G+' \\cdot '+Zo+' \\cdot Z0',' ( '+Zi+' +Z0 )\\cdot ( '+Zo+' + Z0 )'),lfrac(Zo+' - Z0',Zo+' + Z0')]]
def TLineRho(Zc, ports=2):
"""string representation of rho from the characteristic impedance
@param Zc string characteristic impedance
@param ports (optional) integer number of ports (defaults to two).
@return string LaTeX representation of rho
@note that rho is different for a four-port single-ended transmission line.
"""
if ports == 2:
return ' ' + lfrac(' ' + Zc + '-Z0 ', ' ' + Zc + ' + Z0 ')
elif ports == 4:
return ' ' + lfrac(' ' + lfrac(' ' + Zc + ' ', '2') + ' - Z0 ',
' ' + lfrac(' ' + Zc + ' ', '2') + ' + Z0 ')
def SeriesZ(Z):
"""symbolic series impedance
@param Z string impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.SeriesZ
"""
return [[
lfrac(Z, Z + '+2\\cdot Z0'),
lfrac('2\\cdot Z0', Z + '+2\\cdot Z0')
], [lfrac('2\\cdot Z0', Z + '+2\\cdot Z0'),
lfrac(Z, Z + '+2\\cdot Z0')]]
def ShuntZTwoPort(Z):
"""symbolic two port shunt impedance
@param Z string impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.ShuntZ.ShuntZTwoPort
@remark Ports 1 and 2 are connected together and to one side of the impedance.\n
The other side of the impedance is tied to ground.
"""
D = '2\\cdot ' + Z + ' +Z0'
return [[lfrac('-Z0', D), lfrac('2\\cdot ' + Z, D)],
[lfrac('2\\cdot ' + Z, D),
lfrac('-Z0', D)]]
def LaTeXTransferMatrix(self):
"""Calculates and stores internally a symbolic representation
of the transfer matrix in LaTeX.
@return self
@see Symbolic
"""
self.Check()
self._LaTeXSi()
numMeas = len(self.pMeasurementList)
vemsi = MatrixMultiply(
self.VoltageExtractionMatrix(self.pMeasurementList),
self.SIPrime(True))
oneElementVemsi = False
if len(vemsi) == 1:
if len(vemsi[0]) == 1: oneElementVemsi = True
vemsi = Matrix2LaTeX(vemsi, self._SmallMatrix())
veosi = MatrixMultiply(self.VoltageExtractionMatrix(self.pOutputList),
self.SIPrime(True))
oneElementVeosi = False
if len(veosi) == 1:
if len(veosi[0]) == 1: oneElementVeosi = True
veosi = Matrix2LaTeX(veosi, self._SmallMatrix())
if self.pStimDef is None:
numDeg = len(self.SIPrime(True)[0])
inverse = '^{-1}' if numDeg == numMeas else '^\\dagger'
if oneElementVemsi and oneElementVeosi:
line = lfrac(' ' + veosi + ' ', ' ' + vemsi + ' ') + ' '
elif oneElementVemsi and not oneElementVeosi:
line = veosi + ' ' + lfrac('1', ' ' + vemsi + ' ') + ' '
elif not oneElementVemsi and oneElementVeosi:
line = '\\left( ' + veosi + '\\right)\\cdot ' + vemsi + inverse
elif not oneElementVemsi and not oneElementVeosi:
line = veosi + '\\cdot ' + vemsi + inverse
else:
numDeg = len(self.pStimDef[0])
inverse = '^{-1}' if numDeg == numMeas else '^\\dagger'
D = Matrix2LaTeX(self.pStimDef, self._SmallMatrix())
fveosi = '\\left( ' + veosi + '\\right)' if oneElementVeosi else veosi
fvemsi = '\\left( ' + vemsi + '\\right)' if oneElementVemsi else vemsi
line = '\\left[ '+fveosi+'\\cdot '+D+'\\right]\\cdot'+\
'\\left[ '+fvemsi+'\\cdot '+D+'\\right]'+inverse
if len(self.pMeasurementList) == 1 and len(self.pOutputList) == 1:
H = 'H'
else:
H = '\\mathbf{H}'
return self._AddEq(H + ' = ' + line)
def CurrentControlledVoltageSource(G):
"""symbolic current controlled voltage source
@param G string transresistance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of
SignalIntegrity.Lib.Devices.CurrentControlledVoltageSource.CurrentControlledVoltageSource
"""
return [['0', '1', '0', '0'], ['1', '0', '0', '0'],
[
'-' + lfrac(G + ' ', '2\\cdot Z0'),
lfrac(G + ' ', '2\\cdot Z0'), '0', '1'
],
[
lfrac(G + ' ', '2\\cdot Z0'),
'-' + lfrac(G + ' ', '2\\cdot Z0'), '1', '0'
]]
def IdealTransformer(a=1):
"""symbolic ideal transformer
Ports 1 and 2 are the primary.
Ports 3 and 4 are the secondary.
The dot is on ports 1 and 3.
a is the turns ratio specified as (secondary/primary) windings
@param a integer, float or string turns ratio\n
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.IdealTransformer
"""
if isinstance(a, str):
try:
a = int(a)
except ValueError:
try:
a = float(a)
except ValueError:
pass
if isinstance(a, int) or isinstance(a, float):
if a == 1:
a = '1'
asq = '1'
denom = '2'
else:
asq = str(a * a)
denom = str(a * a + 1)
a = str(a)
elif isinstance(a, str):
asq = a + '^2'
denom = asq + ' + 1'
one = ' ' + lfrac(' 1 ', denom) + ' '
a2 = ' ' + lfrac(' ' + asq + ' ', ' ' + denom + ' ') + ' '
a1 = ' ' + lfrac(' ' + a + ' ', ' ' + denom + ' ') + ' '
na = ' -' + lfrac(' ' + a + ' ', ' ' + denom + ' ') + ' '
return [[one, a2, a1, na], [a2, one, na, a1], [a1, na, a2, one],
[na, a1, one, a2]]
def TransresistanceAmplifierFourPort(G,Zi,Zo):
"""symbolic four port transresistance amplifier
@param G string transresistance
@param Zi string input impedance
@param Zo string output impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.TransresistanceAmplifier.TransresistanceAmplifierFourPort
"""
D11=lfrac(Zi,Zi+'+2\\cdot Z0')
D12=lfrac('2\\cdot Z0',Zi+'+2\\cdot Z0')
D31=lfrac('2\\cdot '+G+'\\cdot Z0',' ('+Zo+'+2\\cdot Z0 )\\cdot ('+Zi+'+2\\cdot Z0 )')
D33=lfrac(Zo,Zo+'+2\\cdot Z0')
D34=lfrac('2\\cdot Z0',Zo+'+2\\cdot Z0')
return [[D11,D12,'0','0'],
[D12,D11,'0','0'] ,
[D31,'-'+D31,D33,D34],
['-'+D31,D31,D34,D33]]
def TeeWithZ2(Z2):
D = '3\\cdot Z0^2+2\\cdot Z0\\cdot ' + Z2
return [[
lfrac('-Z0^2', D),
lfrac('2\\cdot Z0^2', D),
lfrac('2\\cdot Z0^2+2\\cdot Z0\\cdot ' + Z2, D)
],
[
lfrac('2\\cdot Z0^2', D),
lfrac('-Z0^2+2\\cdot Z0\\cdot ' + Z2, D),
lfrac('2\\cdot Z0^2', D)
],
[
lfrac('2\\cdot Z0^2+2\\cdot Z0\\cdot ' + Z2, D),
lfrac('2\\cdot Z0^2', D),
lfrac('-Z0^2', D)
]]
def TeeThreePortSafe(Zt):
D = '3\\cdot \\left(' + Zt + '+Z0\\right)'
return [[
lfrac('3\\cdot ' + Zt + '-Z0', D),
lfrac('2\\cdot Z0', D),
lfrac('2\\cdot Z0', D)
],
[
lfrac('2\\cdot Z0', D),
lfrac('3\\cdot ' + Zt + '-Z0', D),
lfrac('2\\cdot Z0', D)
],
[
lfrac('2\\cdot Z0', D),
lfrac('2\\cdot Z0', D),
lfrac('3\\cdot ' + Zt + '-Z0', D)
]]
def ShuntZOnePort(Z):
"""symbolic four port shunt impedance
This is simply a termination impedance to ground.
@param Z string impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.TerminationZ.TerminationZ
"""
return [[lfrac(' ' + Z + ' -Z0', ' ' + Z + ' +Z0')]]
def Tee(P=3):
"""symbolic Tee
@param P (optional) integer number of ports (defaults to three)\n
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic equivalent of SignalIntegrity.Lib.Devices.Tee.Tee
"""
D = str(P)
DiagEle = '-' + lfrac(str(-(2 - P)), D)
OffDiagEle = lfrac('2', D)
M = []
for r in range(P):
row = []
for c in range(P):
if r == c:
ele = DiagEle
else:
ele = OffDiagEle
row.append(ele)
M.append(row)
return M
def VoltageAmplifierThreePort(G,Zi,Zo):
"""symbolic three port voltage amplifier
@param G string voltage gain
@param Zi string input impedance
@param Zo string output impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.VoltageAmplifier.VoltageAmplifierThreePort
"""
D='-'+Zo+'\\cdot '+Zi+'-2\\cdot '+Zo+'\\cdot Z0-2\\cdot '+Zi+'\\cdot Z0-3\\cdot Z0^2+'+G+'\\cdot '+Zi+'\\cdot Z0'
S11=lfrac('-'+Zo+'\\cdot '+Zi+'-2\\cdot '+Zi+'\\cdot Z0+Z0^2+'+G+'\\cdot '+Zi+'\\cdot Z0',D)
S12=lfrac('-2\\cdot Z0^2',D)
S13=lfrac('-2\\cdot Z0\\cdot ('+Zo+' +Z0 )',D)
S21=lfrac('-2\\cdot Z0 \\cdot ('+G+'\\cdot '+Zi+' +Z0 )',D)
S22=lfrac('Z0^2-2\\cdot '+Zo+'\\cdot Z0+'+G+'\\cdot '+Zi+'\\cdot Z0-'+Zo+'\\cdot '+Zi,D)
S23=lfrac('2\\cdot Z0\\cdot ('+G+'\\cdot '+Zi+'-'+Zi+'-Z0 )',D)
S31=lfrac('2\\cdot Z0\\cdot (-Z0+'+G+'\\cdot '+Zi+'-'+Zo+' )',D)
S32=lfrac('-2\\cdot Z0\\cdot ('+Zi+'+Z0 )',D)
S33=lfrac('-'+Zo+'\\cdot '+Zi+'+Z0^2-'+G+'\\cdot '+Zi+'\\cdot Z0',D)
return [[S11,S12,S13],
[S21,S22,S23],
[S31,S32,S33]]
def ShuntZFourPort(Z):
"""symbolic four port shunt impedance
@param Z string impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.ShuntZ.ShuntZFourPort
@remark Ports 1 and 3 are connected to port 1 of the device D provided.\n
Ports 2 and 4 are connected to port 2 of the device D provided.\n
@todo check the port numbering
"""
D = '2\\cdot (' + Z + '+Z0 )'
return [[
lfrac('-Z0', D),
lfrac('Z0', D),
lfrac('2\\cdot ' + Z + '+Z0', D),
lfrac('Z0', D)
],
[
lfrac('Z0', D),
lfrac('-Z0', D),
lfrac('Z0', D),
lfrac('2\\cdot ' + Z + '+Z0', D)
],
[
lfrac('2\\cdot ' + Z + '+Z0', D),
lfrac('Z0', D),
lfrac('-Z0', D),
lfrac('Z0', D)
],
[
lfrac('Z0', D),
lfrac('2\\cdot ' + Z + '+Z0', D),
lfrac('Z0', D),
lfrac('-Z0', D)
]]
def TransconductanceAmplifierFourPort(G, Zi, Zo):
"""symbolic four port transconductance amplifier
@param G string transconductance
@param Zi string input impedance
@param Zo string output impedance
@return list of list of string s-parameter matrix
containing LaTeX or ASCII strings for each element.
@note strings can be any valid LaTeX
@note this is the symbolic version of SignalIntegrity.Lib.Devices.TransconductanceAmplifier.TransconductanceAmplifierFourPort
"""
return [
[
lfrac(Zi, Zi + '+2\\cdot Z0'),
lfrac('2\\cdot Z0', Zi + '+2\\cdot Z0'), '0', '0'
],
[
lfrac('2\\cdot Z0', Zi + '+2\\cdot Z0'),
lfrac(Zi, Zi + '+2\\cdot Z0'), '0', '0'
],
[
lfrac('2\\cdot ' + Zi + '\\cdot ' + Zo + '\\cdot Z0\\cdot ' + G,
' (' + Zi + '+2\\cdot Z0 )\\cdot (' + Zo + '+2\\cdot Z0 )'),
'-' +
lfrac('2\\cdot ' + Zi + '\\cdot ' + Zo + '\\cdot Z0\\cdot ' + G,
' (' + Zi + '+2\\cdot Z0 )\\cdot (' + Zo + '+2\\cdot Z0 )'),
lfrac(Zo, Zo + '+2\\cdot Z0'),
lfrac('2\\cdot Z0', Zo + '+2\\cdot Z0')
],
[
'-' +
lfrac('2\\cdot ' + Zi + '\\cdot ' + Zo + '\\cdot Z0\\cdot ' + G,
' (' + Zi + '+2\\cdot Z0 )\\cdot (' + Zo + '+2\\cdot Z0 )'),
lfrac('2\\cdot ' + Zi + '\\cdot ' + Zo + '\\cdot Z0\\cdot ' + G,
' (' + Zi + '+2\\cdot Z0 )\\cdot (' + Zo + '+2\\cdot Z0 )'),
lfrac('2\\cdot Z0', Zo + '+2\\cdot Z0'),
lfrac(Zo, Zo + '+2\\cdot Z0')
]
]
