12
Complex Impedance Synthesis
Substituting the impedance programming resistor, RS, with a
complex programming network provides complex impedance
synthesis.
The reference designators in the programming network
match the evaluation board. The component RS has a
different design equation than the RS used for resistive
impedance synthesis. The design equations for each
component are provided below.
Substituting EQ 17 for VTX with AUX =0 and
IM= -V2W/ZL
gives us EQ 26. Note: AUX input is not used.
Substitute EQ 17 into EQ 21
Substitute EQ 26 into EQ 27
Substitute Equation 19 for RS/8k in Equation 28.
Simplifying
Substitute Equation 30 into Equation 31 and combine terms
where:
VIN = The input voltage at the -IN pinthrough resistor RIN.
AUX = Auxiliary input of SLIC. Not used for AC gains.
VSA = An internal node voltage that is a function of the loop
current and the output of the Sense Amplifier.
IX = Internal current in the SLIC that is the difference between
the input receive current and the feedback current.
IM = The AC metallic current.
RP = A protection resistor (typical 49.9).
RS = An external resistor/network for matching the line
impedance.
VTR = The tip to ring voltage at the output pins of the SLIC.
V2W = The tip to ring voltage including the voltage across the
protection resistors.
ZL = The line impedance.
ZO = The source impedance of the device.
4-Wire to 2-Wire Gain
4-wire to 2-wire gain across the ISL5585 is equal to the V2W
divided by the input voltage VIN, reference Figure 4. The
receive gain is calculated using Equation 32.
Equation 33 expresses the receive gain (VIN to V2W) in
terms of network impedances. From Equation 21, the value
of RS was set to match the line impedance (ZL) to the
ISL5585 plus the protection resistors (Z0 + 2RP). This
results in a 4-wire to 2-wire gain equal to RS/RIN, as shown
in EQ. 33.
2-Wire to 4-Wire Gain
The 2-wire to 4-wire gain is equal to VTX/EG with VIN = 0,
reference Figure 4.
From Equation 30 with VIN = 0
Substituting Equation 35 into Equation 34 and simplify.
FIGURE 5. COMPLEX PROGRAMMING NETWORK
2-WIRE
NETWORK
R1
R2
C2
PROGRAMMING
NETWORK
RSeries
RParallel
CParallel
R
Series
133.3
R
1
2R
P
()
–
()
×
=
(EQ. 22)
R
Parallel
133.3
R
2
×
=
(EQ. 23)
C
Parallel
C
2 133.3
=
(EQ. 24)
I
X
AUX
R
------------- +
V
TX
R
-----------
=
Node Equation
(EQ. 25)
at ISL5585 AUX input, Figure 4
I
X
V
TX
R
-----------
V
IN
R
---------
R
S
R
IN
----------
–
V
2w30
Z
L
------------------
–
R
S
R8k
-----------
==
(EQ. 26)
I
XR - VTR + IXR = 0
Loop Equation
(EQ. 27)
at ISL5585 feed amplifiers and load.
V
TR
2V
IN
R
S
R
IN
----------
–
2V
2w 30
Z
L
----------------------
R
S
8k
--------
+
=
(EQ. 28)
V
TR
2V
IN
R
S
R
IN
----------
–
2V
2w 30
Z
L
----------------------
133.33Z
O
8k
--------------------------
+
=
(EQ. 29)
V
TR
2V
IN
R
S
R
IN
----------
–
V
2w
Z
L
-----------
Z
O
()
+
=
(EQ. 30)
V
2W -IM2RP + VTR = 0
Loop Equation
(EQ. 31)
at Tip/Ring interface
V
2W
Z
L
Z
O
2R
P
++
Z
L
--------------------------------------
2V
IN
R
S
R
IN
----------
=
(EQ. 32)
G
4-2 =
V
2W
V
IN
------------ = 2
R
S
R
IN
----------
Z
L
Z
L
Z
O
+
+ 2
RP
----------------------------------------
2
Z
L
Z
L
Z
L
+
--------------------
R
S
R
IN
----------
==
(EQ. 33)
E
–
G
Z
LIM
2R
PIM
V
TR
–
+
0
=
Loop Equation
(EQ. 34)
V
TR
Z
O V2W
Z
L
--------------------
=
(EQ. 35)
E
G
V
2W
Z
L
2R
P
Z
O
++
Z
L
---------------------------------------
–
=
(EQ. 36)
ISL5585