This is
the user sandbox of Zen-in . A user sandbox is a subpage of the user's
user page . It serves as a testing spot and page development space for the user and is
not an encyclopedia article .
Create or edit your own sandbox here . Other sandboxes: Main sandbox | Template sandbox
Submit your draft for review!
Negative impedance converter Transfer function derivation
edit
First, assigning Va = voltage at + terminal = Vs
and assigning Vb = voltage at - terminal
and assigning Vo = voltage at output
I
s
+
V
o
−
V
s
R
3
=
0
,
{\displaystyle I_{s}+{\frac {V_{o}-V_{s}}{R_{3}}}=0,}
V
a
=
V
o
+
I
s
R
3
,
{\displaystyle V_{a}=V_{o}+{I_{s}}{R_{3}},}
V
a
{\displaystyle V_{a}}
V
o
{\displaystyle V_{o}}
I
s
{\displaystyle I_{s}}
V
o
=
V
s
−
I
s
R
3
,
{\displaystyle V_{o}=V_{s}-{I_{s}}{R_{3}},}
V
o
{\displaystyle V_{o}}
V
s
{\displaystyle V_{s}}
I
s
{\displaystyle I_{s}}
V
b
R
1
+
V
b
−
V
o
R
2
=
0
,
{\displaystyle {\frac {V_{b}}{R_{1}}}+{\frac {V_{b}-V_{o}}{R_{2}}}=0,}
V
b
=
V
o
R
1
R
1
+
R
2
,
{\displaystyle V_{b}={\frac {{V_{o}}{R_{1}}}{R_{1}+R_{2}}},}
V
o
=
(
V
a
−
V
b
)
α
,
{\displaystyle V_{o}=(V_{a}-V_{b})\alpha ,}
α
{\displaystyle \alpha }
(
V
o
+
I
s
R
3
−
V
o
R
1
R
1
+
R
2
)
α
=
V
o
,
{\displaystyle (V_{o}+{I_{s}}{R_{3}}-{\frac {{V_{o}}{R_{1}}}{R_{1}+R_{2}}})\alpha =V_{o},}
V
a
{\displaystyle V_{a}}
V
b
{\displaystyle V_{b}}
α
V
o
(
1
−
R
1
R
1
+
R
2
)
+
α
I
s
R
3
=
V
o
,
{\displaystyle \alpha {V_{o}}(1-{\frac {R_{1}}{R_{1}+R_{2}}})+\alpha {I_{s}}{R_{3}}=V_{o},}
V
o
(
α
(
1
−
R
1
R
1
+
R
2
)
−
1
)
=
−
α
I
s
R
3
,
{\displaystyle {V_{o}}(\alpha (1-{\frac {R_{1}}{R_{1}+R_{2}}})-1)=-\alpha {I_{s}}{R_{3}},}
V
o
(
1
−
R
1
R
1
+
R
2
)
=
−
I
s
R
3
,
{\displaystyle {V_{o}}(1-{\frac {R_{1}}{R_{1}+R_{2}}})=-{I_{s}}{R_{3}},}
α
{\displaystyle \alpha }
V
o
=
V
s
−
I
s
R
3
,
{\displaystyle V_{o}=V_{s}-{I_{s}}{R_{3}},}
V
s
(
1
−
R
1
R
1
+
R
2
)
−
I
s
R
3
(
1
−
R
1
R
1
+
R
2
)
+
I
s
R
3
=
0
,
{\displaystyle V_{s}(1-{\frac {R_{1}}{R_{1}+R_{2}}})-{I_{s}}{R_{3}}(1-{\frac {R_{1}}{R_{1}+R_{2}}})+{I_{s}}{R_{3}}=0,}
V
o
{\displaystyle V_{o}}
V
s
(
R
2
R
1
+
R
2
)
=
−
I
s
R
3
(
R
1
R
1
+
R
2
)
)
,
{\displaystyle V_{s}({\frac {R_{2}}{R_{1}+R_{2}}})=-{I_{s}}{R_{3}}({\frac {R_{1}}{R_{1}+R_{2}}})),}
V
s
I
s
=
−
R
3
R
1
R
2
{\displaystyle {\frac {V_{s}}{I_{s}}}=-{\frac {{R_{3}}{R_{1}}}{R_{2}}}}
Kirchoff's current law
rearranging to express 
in terms of 
and 
rearranging to express 
and 
Kirchoff's current law
rearranging
where 
= the opamp open loop gain
substituting for 
rearranging
rearranging
cancelling out the 
substituting for 
