Меню Главная Случайная статья Настройки
Четырёхполюсник — электрическая цепь, разновидность многополюсника , имеющая четыре точки подключения[ 1] Содержание Общие сведения U 1 , I 1 ) и выходной (U 2 , I 2 ) цепях. В этой системе параметров линейный четырёхполюсник описывается системой из двух линейных уравнений, причём два из четырёх параметров состояния являются исходными, а два других — определяемыми. Для нелинейных четырёхполюсников зависимость может носить более сложный характер. Например, выходные параметры через входные можно выразить системой
{
U
2
=
b
11
U
1
+
b
12
I
1
I
2
=
b
21
U
1
+
b
22
I
1
;
(
U
2
I
2
)
=
(
b
11
b
12
b
21
b
22
)
(
U
1
I
1
)
{\displaystyle {\begin{cases}U_{2}=b_{11}U_{1}+b_{12}I_{1}\\I_{2}=b_{21}U_{1}+b_{22}I_{1}\\\end{cases}};~~~{\begin{pmatrix}U_{2}\\I_{2}\end{pmatrix}}={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}{\begin{pmatrix}U_{1}\\I_{1}\end{pmatrix}}}
Системы параметров Линейный четырёхполюсник, не содержащий независимых источников (напряжения и/или тока ), описывается четырьмя параметрами — два напряжения и два тока. Любые две величины из четырёх можно определить через оставшиеся две. Поскольку число сочетаний 2 из 4 — 6, используется одна из шести систем записи формальных параметров четырёхполюсника[ 2] A-форма U1 =AU2 +BI2 ; I1 =CU2 +DI2 , где A, B, C, D — A-параметры, обобщенные[ 3] [ 4]
Y-форма I1 =Y11 U1 +Y12 U2 ; I2 =Y21 U1 +Y22 U2 , где Y11 , Y12 , Y21 , Y22 — Y-параметры, или параметры проводимостей[ 5]
Z-форма U1 =Z11 I1 +Z12 I2 ; U2 =Z21 I1 +Z22 I2 , где Z11 , Z12 , Z21 , Z22 — Z-параметры, или параметры сопротивлений[ 5]
H-форма U1 =h11 I1 +h12 U2 ; I2 =h21 I1 +h22 U2 , где h11 , h12 , h21 , h22 — h-параметры, которые применяются при рассмотрении схем с транзисторами[ 3]
G-форма I1 =G11 U1 +G12 I2 ; U2 =G21 U1 +G22 I2 , где G — это параметр который используется при рассмотрении ламповых схем.
B-форма U2 =B11 U1 +B12 I1 ; I2 =B21 U1 +B22 I1 генераторы тока и управляемые генераторы напряжения , которые управляются входными сигналами четырёхполюсника. Поэтому в качестве четырёхполюсников рассматриваются, как правило, эквивалентные схемы по переменному току.
Системы уравнений, эквивалентные схемы, измерение параметров
Тип
Система уравнений
Эквивалентная схема
Измерение параметров
G
{\displaystyle G}
(
I
1
U
2
)
=
(
g
11
g
12
g
21
g
22
)
(
U
1
I
2
)
{\displaystyle {\begin{pmatrix}I_{1}\\U_{2}\end{pmatrix}}={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}{\begin{pmatrix}U_{1}\\I_{2}\end{pmatrix}}}
g
11
=
I
1
U
1
|
I
2
=
0
g
12
=
I
1
I
2
|
U
1
=
0
{\displaystyle g_{11}=\left.{\frac {I_{1}}{U_{1}}}\right|_{I_{2}=0}\quad g_{12}=\left.{\frac {I_{1}}{I_{2}}}\right|_{U_{1}=0}}
g
21
=
U
2
U
1
|
I
2
=
0
g
22
=
U
2
I
2
|
U
1
=
0
{\displaystyle g_{21}=\left.{\frac {U_{2}}{U_{1}}}\right|_{I_{2}=0}\quad g_{22}=\left.{\frac {U_{2}}{I_{2}}}\right|_{U_{1}=0}}
H
{\displaystyle H}
(
U
1
I
2
)
=
(
h
11
h
12
h
21
h
22
)
(
I
1
U
2
)
{\displaystyle {\begin{pmatrix}U_{1}\\I_{2}\end{pmatrix}}={\begin{pmatrix}h_{11}&h_{12}\\h_{21}&h_{22}\end{pmatrix}}{\begin{pmatrix}I_{1}\\U_{2}\end{pmatrix}}}
h
11
=
U
1
I
1
|
U
2
=
0
h
12
=
U
1
U
2
|
I
1
=
0
{\displaystyle h_{11}=\left.{\frac {U_{1}}{I_{1}}}\right|_{U_{2}=0}\quad h_{12}=\left.{\frac {U_{1}}{U_{2}}}\right|_{I_{1}=0}}
h
21
=
I
2
I
1
|
U
2
=
0
h
22
=
I
2
U
2
|
I
1
=
0
{\displaystyle h_{21}=\left.{\frac {I_{2}}{I_{1}}}\right|_{U_{2}=0}\quad h_{22}=\left.{\frac {I_{2}}{U_{2}}}\right|_{I_{1}=0}}
Y
{\displaystyle Y}
(
I
1
I
2
)
=
(
y
11
y
12
y
21
y
22
)
(
U
1
U
2
)
{\displaystyle {\begin{pmatrix}I_{1}\\I_{2}\end{pmatrix}}={\begin{pmatrix}y_{11}&y_{12}\\y_{21}&y_{22}\end{pmatrix}}{\begin{pmatrix}U_{1}\\U_{2}\end{pmatrix}}}
y
11
=
I
1
U
1
|
U
2
=
0
y
12
=
I
1
U
2
|
U
1
=
0
{\displaystyle y_{11}=\left.{\frac {I_{1}}{U_{1}}}\right|_{U_{2}=0}\quad y_{12}=\left.{\frac {I_{1}}{U_{2}}}\right|_{U_{1}=0}}
y
21
=
I
2
U
1
|
U
2
=
0
y
22
=
I
2
U
2
|
U
1
=
0
{\displaystyle y_{21}=\left.{\frac {I_{2}}{U_{1}}}\right|_{U_{2}=0}\quad y_{22}=\left.{\frac {I_{2}}{U_{2}}}\right|_{U_{1}=0}}
Z
{\displaystyle Z}
(
U
1
U
2
)
=
(
z
11
z
12
z
21
z
22
)
(
I
1
I
2
)
{\displaystyle {\begin{pmatrix}U_{1}\\U_{2}\end{pmatrix}}={\begin{pmatrix}z_{11}&z_{12}\\z_{21}&z_{22}\end{pmatrix}}{\begin{pmatrix}I_{1}\\I_{2}\end{pmatrix}}}
z
11
=
U
1
I
1
|
I
2
=
0
z
12
=
U
1
I
2
|
I
1
=
0
{\displaystyle z_{11}=\left.{\frac {U_{1}}{I_{1}}}\right|_{I_{2}=0}\quad z_{12}=\left.{\frac {U_{1}}{I_{2}}}\right|_{I_{1}=0}}
z
21
=
U
2
I
1
|
I
2
=
0
z
22
=
U
2
I
2
|
I
1
=
0
{\displaystyle z_{21}=\left.{\frac {U_{2}}{I_{1}}}\right|_{I_{2}=0}\quad z_{22}=\left.{\frac {U_{2}}{I_{2}}}\right|_{I_{1}=0}}
A
{\displaystyle A}
(
U
1
I
1
)
=
(
a
11
a
12
a
21
a
22
)
(
U
2
I
2
)
{\displaystyle {\begin{pmatrix}U_{1}\\I_{1}\end{pmatrix}}={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}{\begin{pmatrix}U_{2}\\I_{2}\end{pmatrix}}}
a
11
=
U
1
U
2
|
I
2
=
0
a
12
=
U
1
I
2
|
U
2
=
0
{\displaystyle a_{11}=\left.{\frac {U_{1}}{U_{2}}}\right|_{I_{2}=0}\quad a_{12}=\left.{\frac {U_{1}}{I_{2}}}\right|_{U_{2}=0}}
a
21
=
I
1
U
2
|
I
2
=
0
a
22
=
I
1
I
2
|
U
2
=
0
{\displaystyle a_{21}=\left.{\frac {I_{1}}{U_{2}}}\right|_{I_{2}=0}\quad a_{22}=\left.{\frac {I_{1}}{I_{2}}}\right|_{U_{2}=0}}
B
{\displaystyle B}
(
U
2
I
2
)
=
(
b
11
b
12
b
21
b
22
)
(
U
1
I
1
)
{\displaystyle {\begin{pmatrix}U_{2}\\I_{2}\end{pmatrix}}={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}{\begin{pmatrix}U_{1}\\I_{1}\end{pmatrix}}}
b
11
=
U
2
U
1
|
I
1
=
0
b
12
=
U
2
I
1
|
U
1
=
0
{\displaystyle b_{11}=\left.{\frac {U_{2}}{U_{1}}}\right|_{I_{1}=0}\quad b_{12}=\left.{\frac {U_{2}}{I_{1}}}\right|_{U_{1}=0}}
b
21
=
I
2
U
1
|
I
1
=
0
b
22
=
I
2
I
1
|
U
1
=
0
{\displaystyle b_{21}=\left.{\frac {I_{2}}{U_{1}}}\right|_{I_{1}=0}\quad b_{22}=\left.{\frac {I_{2}}{I_{1}}}\right|_{U_{1}=0}}
Преобразование параметров
{
U
1
=
h
11
I
1
+
h
12
U
2
I
2
=
h
21
I
1
+
h
22
U
2
{
I
1
=
y
11
U
1
+
y
12
U
2
I
2
=
y
21
U
1
+
y
22
U
2
.
{\displaystyle {\begin{cases}U_{1}=h_{11}I_{1}+h_{12}U_{2}\\I_{2}=h_{21}I_{1}+h_{22}U_{2}\end{cases}}\to ~~~{\begin{cases}I_{1}=y_{11}U_{1}+y_{12}U_{2}\\I_{2}=y_{21}U_{1}+y_{22}U_{2}\end{cases}}.}
I 1 :
{
I
1
=
1
h
11
U
1
h
12
h
11
U
2
I
2
=
h
21
I
1
+
h
22
U
2
.
{\displaystyle {\begin{cases}I_{1}={\frac {1}{h_{11}}}U_{1}-{\frac {h_{12}}{h_{11}}}U_{2}\\I_{2}=h_{21}I_{1}+h_{22}U_{2}\end{cases}}.}
{
I
1
=
1
h
11
U
1
h
12
h
11
U
2
I
2
=
h
21
(
1
h
11
U
1
h
12
h
11
U
2
)
+
h
22
U
2
.
{\displaystyle {\begin{cases}I_{1}={\frac {1}{h_{11}}}U_{1}-{\frac {h_{12}}{h_{11}}}U_{2}\\I_{2}=h_{21}\left({\frac {1}{h_{11}}}U_{1}-{\frac {h_{12}}{h_{11}}}U_{2}\right)+h_{22}U_{2}\end{cases}}.}
I
2
=
h
21
h
11
U
1
+
(
h
22
h
12
h
21
h
11
)
U
2
=
h
21
h
11
U
1
+
h
11
h
22
h
12
h
21
h
11
U
2
=
h
21
h
11
U
1
+
h
h
11
U
2
,
{\displaystyle I_{2}={\frac {h_{21}}{h_{11}}}U_{1}+\left(h_{22}-{\frac {h_{12}h_{21}}{h_{11}}}\right)U_{2}={\frac {h_{21}}{h_{11}}}U_{1}+{\frac {h_{11}h_{22}-h_{12}h_{21}}{h_{11}}}U_{2}={\frac {h_{21}}{h_{11}}}U_{1}+{\frac {\Delta _{h}}{h_{11}}}U_{2},}
h
=
h
11
h
22
h
12
h
21
.
{\displaystyle \Delta _{h}=h_{11}h_{22}-h_{12}h_{21}.}
{
I
1
=
1
h
11
U
1
h
12
h
11
U
2
I
2
=
h
21
h
11
U
1
+
h
h
11
U
2
.
{\displaystyle {\begin{cases}I_{1}={\frac {1}{h_{11}}}U_{1}-{\frac {h_{12}}{h_{11}}}U_{2}\\I_{2}={\frac {h_{21}}{h_{11}}}U_{1}+{\frac {\Delta _{h}}{h_{11}}}U_{2}\end{cases}}.}
y
11
=
1
h
11
;
y
12
=
h
12
h
11
;
y
21
=
h
21
h
11
;
y
22
=
h
h
11
;
{\displaystyle y_{11}={\frac {1}{h_{11}}};~~~y_{12}=-{\frac {h_{12}}{h_{11}}};~~~y_{21}={\frac {h_{21}}{h_{11}}};~~~y_{22}={\frac {\Delta _{h}}{h_{11}}};}
y
=
y
11
y
22
y
12
y
21
=
1
h
11
h
h
11
+
h
12
h
11
h
21
h
11
=
h
11
h
22
h
12
h
21
+
h
12
h
21
h
11
2
=
h
22
h
11
.
{\displaystyle \Delta _{y}=y_{11}y_{22}-y_{12}y_{21}={\frac {1}{h_{11}}}{\frac {\Delta _{h}}{h_{11}}}+{\frac {h_{12}}{h_{11}}}{\frac {h_{21}}{h_{11}}}={\frac {h_{11}h_{22}-h_{12}h_{21}+h_{12}h_{21}}{h_{11}^{2}}}={\frac {h_{22}}{h_{11}}}.}
H
{\displaystyle H}
Y
{\displaystyle Y}
Z
{\displaystyle Z}
G
{\displaystyle G}
A
{\displaystyle A}
H
{\displaystyle H}
h
11
=
1
/
y
11
{\displaystyle h_{11}=1/y_{11}}
h
12
=
y
12
/
y
11
{\displaystyle h_{12}=-y_{12}/y_{11}}
h
21
=
y
21
/
y
11
{\displaystyle h_{21}=y_{21}/y_{11}}
h
22
=
y
/
y
11
{\displaystyle h_{22}=\Delta _{y}/y_{11}}
h
=
y
22
/
y
11
{\displaystyle \Delta _{h}=y_{22}/y_{11}}
h
11
=
z
/
z
22
{\displaystyle h_{11}=\Delta _{z}/z_{22}}
h
12
=
z
12
/
z
22
{\displaystyle h_{12}=z_{12}/z_{22}}
h
21
=
z
21
/
z
22
{\displaystyle h_{21}=-z_{21}/z_{22}}
h
22
=
1
/
z
22
{\displaystyle h_{22}=1/z_{22}}
h
=
z
11
/
z
22
{\displaystyle \Delta _{h}=z_{11}/z_{22}}
h
11
=
g
22
/
g
{\displaystyle h_{11}=g_{22}/\Delta _{g}}
h
12
=
g
12
/
g
{\displaystyle h_{12}=-g_{12}/\Delta _{g}}
h
21
=
g
21
/
g
{\displaystyle h_{21}=-g_{21}/\Delta _{g}}
h
22
=
g
11
/
g
{\displaystyle h_{22}=g_{11}/\Delta _{g}}
h
=
1
/
g
{\displaystyle \Delta _{h}=1/\Delta _{g}}
h
11
=
B
/
D
{\displaystyle h_{11}=B/D}
h
12
=
A
/
D
{\displaystyle h_{12}=\Delta _{A}/D}
h
21
=
1
/
D
{\displaystyle h_{21}=-1/D}
h
22
=
C
/
D
{\displaystyle h_{22}=C/D}
Y
{\displaystyle Y}
y
11
=
1
/
h
11
{\displaystyle y_{11}=1/h_{11}}
y
12
=
h
12
/
h
11
{\displaystyle y_{12}=-h_{12}/h_{11}}
y
21
=
h
21
/
h
11
{\displaystyle y_{21}=h_{21}/h_{11}}
y
22
=
h
/
h
11
{\displaystyle y_{22}=\Delta _{h}/h_{11}}
y
=
h
22
/
h
11
{\displaystyle \Delta _{y}=h_{22}/h_{11}}
y
11
=
z
22
/
z
{\displaystyle y_{11}=z_{22}/\Delta _{z}}
y
12
=
z
12
/
z
{\displaystyle y_{12}=-z_{12}/\Delta _{z}}
y
21
=
z
21
/
z
{\displaystyle y_{21}=-z_{21}/\Delta _{z}}
y
22
=
z
11
/
z
{\displaystyle y_{22}=z_{11}/\Delta _{z}}
y
=
1
/
z
{\displaystyle \Delta _{y}=1/\Delta _{z}}
y
11
=
g
/
g
22
{\displaystyle y_{11}=\Delta _{g}/g_{22}}
y
12
=
g
12
/
g
22
{\displaystyle y_{12}=g_{12}/g_{22}}
y
21
=
g
21
/
g
22
{\displaystyle y_{21}=-g_{21}/g_{22}}
y
22
=
1
/
g
22
{\displaystyle y_{22}=1/g_{22}}
y
=
g
11
/
g
22
{\displaystyle \Delta _{y}=g_{11}/g_{22}}
y
11
=
D
/
B
{\displaystyle y_{11}=D/B}
y
12
=
A
/
B
{\displaystyle y_{12}=-\Delta _{A}/B}
y
21
=
1
/
B
{\displaystyle y_{21}=-1/B}
y
22
=
A
/
B
{\displaystyle y_{22}=A/B}
Z
{\displaystyle Z}
z
11
=
h
/
h
22
{\displaystyle z_{11}=\Delta _{h}/h_{22}}
z
12
=
h
12
/
h
22
{\displaystyle z_{12}=h_{12}/h_{22}}
z
21
=
h
21
/
h
22
{\displaystyle z_{21}=-h_{21}/h_{22}}
z
22
=
1
/
h
22
{\displaystyle z_{22}=1/h_{22}}
z
=
h
11
/
h
22
{\displaystyle \Delta _{z}=h_{11}/h_{22}}
z
11
=
y
22
/
y
{\displaystyle z_{11}=y_{22}/\Delta _{y}}
z
12
=
y
12
/
y
{\displaystyle z_{12}=-y_{12}/\Delta _{y}}
z
21
=
y
21
/
y
{\displaystyle z_{21}=-y_{21}/\Delta _{y}}
z
22
=
y
11
/
y
{\displaystyle z_{22}=y_{11}/\Delta _{y}}
z
=
1
/
y
{\displaystyle \Delta _{z}=1/\Delta _{y}}
z
11
=
1
/
g
11
{\displaystyle z_{11}=1/g_{11}}
z
12
=
g
12
/
g
11
{\displaystyle z_{12}=-g_{12}/g_{11}}
z
21
=
g
21
/
g
11
{\displaystyle z_{21}=g_{21}/g_{11}}
z
22
=
g
/
g
11
{\displaystyle z_{22}=\Delta _{g}/g_{11}}
z
=
g
22
/
g
11
{\displaystyle \Delta _{z}=g_{22}/g_{11}}
z
11
=
A
/
C
{\displaystyle z_{11}=A/C}
z
12
=
A
/
C
{\displaystyle z_{12}=\Delta _{A}/C}
z
21
=
1
/
C
{\displaystyle z_{21}=1/C}
z
22
=
D
/
C
{\displaystyle z_{22}=D/C}
G
{\displaystyle G}
g
11
=
h
22
/
h
{\displaystyle g_{11}=h_{22}/\Delta _{h}}
g
12
=
h
12
/
h
{\displaystyle g_{12}=-h_{12}/\Delta _{h}}
g
21
=
h
21
/
h
{\displaystyle g_{21}=-h_{21}/\Delta _{h}}
g
22
=
h
11
/
h
{\displaystyle g_{22}=h_{11}/\Delta _{h}}
g
=
1
/
h
{\displaystyle \Delta _{g}=1/\Delta _{h}}
g
11
=
y
/
y
22
{\displaystyle g_{11}=\Delta _{y}/y_{22}}
g
12
=
y
12
/
y
22
{\displaystyle g_{12}=y_{12}/y_{22}}
g
21
=
y
21
/
y
22
{\displaystyle g_{21}=-y_{21}/y_{22}}
g
22
=
1
/
y
22
{\displaystyle g_{22}=1/y_{22}}
g
=
y
11
/
y
22
{\displaystyle \Delta _{g}=y_{11}/y_{22}}
g
11
=
1
/
z
11
{\displaystyle g_{11}=1/z_{11}}
g
12
=
z
12
/
z
11
{\displaystyle g_{12}=-z_{12}/z_{11}}
g
21
=
z
21
/
z
11
{\displaystyle g_{21}=z_{21}/z_{11}}
g
22
=
z
/
z
11
{\displaystyle g_{22}=\Delta _{z}/z_{11}}
g
=
z
22
/
z
11
{\displaystyle \Delta _{g}=z_{22}/z_{11}}
g
11
=
C
/
A
{\displaystyle g_{11}=C/A}
g
12
=
A
/
A
{\displaystyle g_{12}=-\Delta _{A}/A}
g
21
=
A
/
A
{\displaystyle g_{21}=\Delta _{A}/A}
g
22
=
B
/
A
{\displaystyle g_{22}=B/A}
A
{\displaystyle A}
A
=
h
/
h
21
{\displaystyle A=-\Delta _{h}/h_{21}}
B
=
h
11
/
h
21
{\displaystyle B=-h_{11}/h_{21}}
C
=
h
22
/
h
21
{\displaystyle C=-h_{22}/h_{21}}
D
=
1
/
h
21
{\displaystyle D=-1/h_{21}}
A
=
y
22
/
y
21
{\displaystyle A=-y_{22}/y_{21}}
B
=
1
/
y
21
{\displaystyle B=-1/y_{21}}
C
=
y
/
y
21
{\displaystyle C=-\Delta _{y}/y_{21}}
D
=
y
11
/
y
21
{\displaystyle D=-y_{11}/y_{21}}
A
=
z
11
/
z
21
{\displaystyle A=z_{11}/z_{21}}
B
=
z
/
z
21
{\displaystyle B=\Delta _{z}/z_{21}}
C
=
1
/
z
21
{\displaystyle C=1/z_{21}}
D
=
z
22
/
z
21
{\displaystyle D=z_{22}/z_{21}}
A
=
1
/
g
21
{\displaystyle A=1/g_{21}}
B
=
g
22
/
g
21
{\displaystyle B=g_{22}/g_{21}}
C
=
g
11
/
g
21
{\displaystyle C=g_{11}/g_{21}}
D
=
g
/
g
21
{\displaystyle D=\Delta _{g}/g_{21}}
Преобразования схем in , Rout — входное и выходное сопротивления; KI , KU — коэффициенты усиления по току и напряжению.
R
i
n
=
U
1
I
1
;
R
o
u
t
=
U
2
I
2
;
K
I
=
I
2
I
1
;
K
U
=
U
2
U
1
.
{\displaystyle R_{in}={\frac {U_{1}}{I_{1}}};\qquad R_{out}={\frac {U_{2}}{I_{2}}};\qquad K_{I}={\frac {I_{2}}{I_{1}}};\qquad K_{U}={\frac {U_{2}}{U_{1}}}.}
Схема
H
{\displaystyle H}
Y
{\displaystyle Y}
Z
{\displaystyle Z}
G
{\displaystyle G}
R
i
n
=
h
11
+
h
R
1
+
h
22
R
{\displaystyle R_{in}={\frac {h_{11}+\Delta _{h}R}{1+h_{22}R}}}
R
o
u
t
=
h
11
+
r
h
+
h
22
r
{\displaystyle R_{out}={\frac {h_{11}+r}{\Delta _{h}+h_{22}r}}}
K
I
=
h
21
1
+
h
22
R
{\displaystyle K_{I}={\frac {h_{21}}{1+h_{22}R}}}
K
U
=
h
21
R
h
11
+
h
R
{\displaystyle K_{U}={\frac {-h_{21}R}{h_{11}+\Delta _{h}R}}}
R
i
n
=
1
+
y
22
R
y
11
+
y
R
{\displaystyle R_{in}={\frac {1+y_{22}R}{y_{11}+\Delta _{y}R}}}
R
o
u
t
=
1
+
y
11
r
y
22
+
y
r
{\displaystyle R_{out}={\frac {1+y_{11}r}{y_{22}+\Delta _{y}r}}}
K
I
=
y
21
y
11
+
y
R
{\displaystyle K_{I}={\frac {y_{21}}{y_{11}+\Delta _{y}R}}}
K
U
=
y
21
R
1
+
y
22
R
{\displaystyle K_{U}={\frac {-y_{21}R}{1+y_{22}R}}}
R
i
n
=
z
+
z
11
R
z
22
+
R
{\displaystyle R_{in}={\frac {\Delta _{z}+z_{11}R}{z_{22}+R}}}
R
o
u
t
=
z
+
z
22
r
z
22
+
r
{\displaystyle R_{out}={\frac {\Delta _{z}+z_{22}r}{z_{22}+r}}}
K
I
=
z
21
z
22
+
R
{\displaystyle K_{I}={\frac {-z_{21}}{z_{22}+R}}}
K
U
=
z
21
R
z
+
z
11
R
{\displaystyle K_{U}={\frac {z_{21}R}{-\Delta _{z}+z_{11}R}}}
R
i
n
=
g
22
+
R
g
+
g
11
R
{\displaystyle R_{in}={\frac {g_{22}+R}{\Delta _{g}+g_{11}R}}}
R
o
u
t
=
g
22
+
g
r
1
+
g
11
r
{\displaystyle R_{out}={\frac {g_{22}+\Delta _{g}r}{1+g_{11}r}}}
K
I
=
g
21
g
+
g
11
R
{\displaystyle K_{I}={\frac {-g_{21}}{\Delta _{g}+g_{11}R}}}
K
U
=
g
21
R
g
22
+
R
{\displaystyle K_{U}={\frac {g_{21}R}{g_{22}+R}}}
