N
N-1
V1
OUT
= GG
0
X [V
L
+
5V
INO
/5
J-1
] = GG
0
X [V
L
+
V
INO
/5
J-1
]
J-2
J-1
N
And V1
OUT
= GG
0
X [V
L
+
V
IN0
/5
J-1
– V
IN0
/5
N-1
];
J-1
FOR V
IN
= 25V
IN0
,
N
V2
OUT
= GG
0
X [2V
L
+
V
IN0
/5
J-1
– (V
IN0
/5
N-1
+ V
IN0
/5
N-2
)]
J-1
So for V
IN
= 5
M
V
IN0
,
N
V
MOUT
= GG
0
X [MV
L
+
V
IN0
/5
J-1
– (V
IN0
/5
N-1
+ V
IN0
/5
N-2
… + V
IN0
/5
N-M
)]
J-1
For M < N-2, the subtracted portion is < V
in0
/25, and we can see that the difference
between V
Mout
and V
M-1out
is approximately a constant, which is exactly what is needed for
a logarithmic output.
For M = N-1 and M = N the approximation starts to fail, and corrective measures such as
"linear extension" are needed. In the case of DLVAs, such measures are needed in any
case, because of detector roll-off.
It is easy to see that the series is only valid for V
in0
> V
L
/5; otherwise the difference
between the first two terms would be too small, because the lead amp, A1, would not yet
limit. This consideration basically determines the start of logging. The actual shape of the
transfer curve is approximated in Fig. A2.
Fig. A2
-A2-
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