Schiefe Ebenen. Oben mit einem Neigungswinkel α von 45°, unten mit 22,5°. Die roten Pfeile symbolisieren von links nach rechts die Hangabtriebskraft , die Gewichtskraft und die Normalkraft .
R
e
i
b
u
n
g
s
k
o
e
f
f
i
z
i
e
n
t
=
R
e
i
b
u
n
g
s
k
r
a
f
t
N
o
r
m
a
l
k
r
a
f
t
μ
=
F
R
F
N
{\displaystyle Reibungskoeffizient={\frac {Reibungskraft}{Normalkraft}}\,\,\,\mu ={\frac {F_{R}}{F_{N}}}}
μ
0
=
H
a
f
t
r
e
i
b
u
n
g
s
k
o
e
f
f
i
z
i
e
n
t
e
n
{\displaystyle \mu _{\mathrm {0} }=Haftreibungskoeffizienten\,}
μ
=
G
l
e
i
t
r
e
i
b
u
n
g
s
k
o
e
f
f
i
z
i
e
n
t
e
n
{\displaystyle \mathrm {\mu } =Gleitreibungskoeffizienten\,}
μ
f
=
R
o
l
l
r
e
i
b
u
n
g
s
k
o
e
f
f
i
z
i
e
n
t
e
n
{\displaystyle \mu _{\mathrm {f} }=Rollreibungskoeffizienten\,}
F
N
=
F
G
−
F
⊥
{\displaystyle F_{N}=F_{G}-F_{\perp }}
F
⊥
=
F
⋅
s
i
n
α
{\displaystyle F_{\perp }=F\cdot sin\,\alpha }
F
R
=
μ
⋅
F
N
{\displaystyle F_{R}=\mu \cdot F_{N}}
F
k
=
F
R
=
F
⋅
c
o
s
α
{\displaystyle F_{\mathcal {k}}=F_{R}=F\cdot cos\,\alpha }
F
=
μ
⋅
(
F
G
−
F
⋅
s
i
n
α
)
c
o
s
α
{\displaystyle F={\frac {\mu \cdot (F_{G}-F\cdot sin\,\alpha )}{cos\,\alpha }}}
F
G
=
G
e
w
i
c
h
t
s
k
r
a
f
t
{\displaystyle F_{G}=Gewichtskraft\,}
F
N
=
N
o
r
m
a
l
k
r
a
f
t
{\displaystyle F_{N}=Normalkraft\,}
F
H
=
H
a
n
g
a
b
t
r
i
e
b
s
k
r
a
f
t
{\displaystyle F_{H}=Hangabtriebskraft\,}
F
R
=
R
e
i
b
u
n
g
s
k
r
a
f
t
{\displaystyle F_{R}=Reibungskraft\,}
F
G
=
m
⋅
g
{\displaystyle F_{G}=m\cdot g}
F
H
=
F
G
⋅
s
i
n
α
=
F
G
⋅
h
l
{\displaystyle F_{H}=F_{G}\cdot sin\,\alpha =F_{G}\cdot {\frac {h}{l}}}
F
N
=
F
G
⋅
c
o
s
α
=
F
G
⋅
b
l
{\displaystyle F_{N}=F_{G}\cdot cos\,\alpha =F_{G}\cdot {\frac {b}{l}}}
F
R
=
μ
⋅
F
N
=
μ
⋅
c
o
s
α
⋅
F
G
{\displaystyle F_{R}=\mu \cdot F_{N}=\mu \cdot cos\,\alpha \cdot F_{G}}
α
=
α
G
r
e
n
z
⇒
F
H
=
F
R
{\displaystyle \alpha =\alpha _{Grenz}\Rightarrow F_{H}=F_{R}}
F
G
⋅
s
i
n
α
=
μ
⋅
F
G
⋅
c
o
s
α
{\displaystyle F_{G}\cdot sin\,\alpha =\mu \cdot F_{G}\cdot cos\,\alpha }
s
i
n
α
=
μ
⋅
c
o
s
α
{\displaystyle sin\,\alpha =\mu \cdot cos\,\alpha }
μ
=
c
o
s
α
s
i
n
α
=
t
a
n
α
{\displaystyle \mu ={\frac {cos\,\alpha }{sin\,\alpha }}=tan\,\alpha }
F
=
F
H
−
F
R
{\displaystyle F=F_{H}-F_{R}\,}
Fallzeit: t
Fallhöhe: h
Fallgeschwindigkeit: v(t)
t
=
v
g
{\displaystyle t={\frac {v}{g}}}
h
=
1
2
⋅
v
⋅
t
{\displaystyle h={\frac {1}{2}}\cdot v\cdot t}
v
=
g
⋅
t
{\displaystyle v=g\cdot t}
t
=
2
⋅
h
v
{\displaystyle t={\frac {2\cdot h}{v}}}
h
=
v
2
2
⋅
g
{\displaystyle h={\frac {v^{2}}{2\cdot g}}}
v
=
2
⋅
h
t
{\displaystyle v={\frac {2\cdot h}{t}}}
t
=
2
⋅
h
g
{\displaystyle t={\sqrt {\frac {2\cdot h}{g}}}}
h
=
1
2
⋅
g
⋅
t
2
{\displaystyle h={\frac {1}{2}}\cdot g\cdot t^{2}}
v
=
2
⋅
h
⋅
g
{\displaystyle v={\sqrt {2\cdot h\cdot g}}}
Sonderfall: senkrechter Wurf
v
(
t
)
=
v
0
−
g
⋅
t
{\displaystyle v(t)=v_{0}-g\cdot t}
Steiggeschwindigkeit
h
(
t
)
=
v
0
⋅
t
−
1
2
⋅
g
⋅
t
2
{\displaystyle h(t)=v_{0}\cdot t-{\frac {1}{2}}\cdot g\cdot t^{2}}
Steighöhe
v
0
X
=
v
0
⋅
cos
α
{\displaystyle v_{0_{X}}=v_{0}\cdot \cos \alpha }
v
0
Y
=
v
0
⋅
sin
α
{\displaystyle v_{0_{Y}}=v_{0}\cdot \sin \alpha }
v
X
=
v
0
X
=
v
0
⋅
cos
α
{\displaystyle v_{X}=v_{0_{X}}=v_{0}\cdot \cos \alpha }
v
Y
=
v
0
Y
−
v
F
a
l
l
=
v
0
⋅
sin
α
−
g
⋅
t
{\displaystyle v_{Y}=v_{0_{Y}}-v_{Fall}=v_{0}\cdot \sin \alpha -g\cdot t}
v
=
v
X
2
+
v
Y
2
{\displaystyle v={\sqrt {{v_{X}}^{2}+{v_{Y}}^{2}}}}
v
=
(
v
0
⋅
cos
α
)
2
+
(
v
0
⋅
sin
α
−
g
⋅
t
)
2
{\displaystyle v={\sqrt {(v_{0}\cdot \cos \alpha )^{2}+(v_{0}\cdot \sin \alpha -g\cdot t)^{2}}}}
v
=
v
0
2
−
2
⋅
g
⋅
h
{\displaystyle v={\sqrt {{v_{0}}^{2}-2\cdot g\cdot h}}}
t
s
=
v
0
⋅
sin
α
g
{\displaystyle t_{s}={\frac {v_{0}\cdot \sin \alpha }{g}}}
h
m
a
x
=
v
0
2
⋅
(
sin
α
)
2
2
⋅
g
{\displaystyle h_{max}={\frac {{v_{0}}^{2}\cdot (\sin \alpha )^{2}}{2\cdot g}}}
s
X
m
a
x
=
v
0
2
⋅
sin
2
α
g
{\displaystyle {s_{X}}_{max}={\frac {{v_{0}}^{2}\cdot \sin 2\alpha }{g}}}
Idealfall:
E
p
o
t
=
E
k
i
n
{\displaystyle E_{pot}=E_{kin}\,}
Realfall:
E
p
o
t
=
E
k
i
n
+
W
R
e
i
b
u
n
g
{\displaystyle E_{pot}=E_{kin}+W_{Reibung}\,}
E
p
o
t
−
E
n
e
r
g
i
e
d
e
r
L
a
g
e
−
p
o
t
e
n
t
i
e
l
l
e
E
n
e
r
g
i
e
{\displaystyle {E_{pot}}-EnergiederLage-potentielleEnergie\,}
E
k
i
n
−
E
n
e
r
g
i
e
d
e
r
B
e
w
e
g
u
n
g
−
k
i
n
e
t
i
s
c
h
e
E
n
e
r
g
i
e
{\displaystyle {E_{kin}}-EnergiederBewegung-kinetischeEnergie\,}
W
p
o
t
≡
E
p
o
t
=
m
⋅
g
⋅
h
{\displaystyle {W_{pot}}\equiv {E_{pot}}=m\cdot g\cdot h}
W
k
i
n
≡
E
k
i
n
=
1
2
⋅
m
⋅
v
2
{\displaystyle {W_{kin}}\equiv {E_{kin}}={\frac {1}{2}}\cdot m\cdot v^{2}}
W
s
p
a
n
=
F
⋅
s
2
F
=
C
⋅
s
{\displaystyle W_{span}={\frac {F\cdot s}{2}}\qquad F=C\cdot s}
W
s
p
a
n
=
1
2
⋅
C
⋅
s
2
[
C
]
=
N
m
{\displaystyle W_{span}={\frac {1}{2}}\cdot C\cdot s^{2}\qquad [C]={\frac {N}{m}}}
C = Federkonstante
S = Weg
W
=
F
⋅
s
[
W
]
=
N
m
=
k
g
⋅
m
s
2
⋅
m
=
J
=
W
s
{\displaystyle W=F\cdot s\qquad [W]=Nm={\frac {kg\cdot m}{s^{2}}}\cdot m=J=Ws}
k
W
h
≈
3
,
6
⋅
10
6
J
=
3
,
6
M
J
{\displaystyle kWh\approx 3,6\cdot 10^{6}J=3,6MJ}
W
=
F
‖
⋅
s
{\displaystyle W=F_{\|}\cdot s}
F
‖
=
F
⋅
cos
α
{\displaystyle F_{\|}=F\cdot \cos \alpha }
W
=
F
⋅
cos
α
⋅
s
{\displaystyle W=F\cdot \cos \alpha \cdot s}
W
R
=
m
⋅
g
⋅
μ
⋅
cos
α
⋅
l
{\displaystyle W_{R}=m\cdot g\cdot \mu \cdot \cos \alpha \cdot l}
W
R
=
F
R
⋅
l
{\displaystyle W_{R}=F_{R}\cdot l}
U
R
2
=
U
G
S
+
U
R
S
{\displaystyle U_{R_{2}}=U_{GS}+U_{RS}}
I
q
=
50...100
⋅
I
G
{\displaystyle I_{q}=50...100\cdot I_{G}}
R
2
=
U
R
2
I
q
{\displaystyle R_{2}={\frac {U_{R_{2}}}{I_{q}}}}
R
1
=
U
R
1
I
q
=
U
B
−
U
R
2
I
q
{\displaystyle R_{1}={\frac {U_{R_{1}}}{I_{q}}}={\frac {U_{B}-U_{R_{2}}}{I_{q}}}}
U
D
S
=
U
R
S
{\displaystyle U_{DS}=U_{RS}\,}
U
D
S
A
=
U
D
S
+
U
D
S
s
a
t
{\displaystyle U_{DS_{A}}=U_{DS}+U_{DS_{sat}}}
U
D
S
A
=
U
B
−
U
R
S
{\displaystyle U_{DS_{A}}=U_{B}-U_{RS}}
mit
U
D
S
s
a
t
=
|
U
p
|
−
|
U
G
S
|
{\displaystyle U_{DS_{sat}}=\vert U_{p}\vert -\vert U_{GS}\vert }
Festlegung des Stromes
I
S
{\displaystyle I_{S}\,}
im Arbeitspunkt
⇒
I
S
=
I
D
{\displaystyle \Rightarrow I_{S}=I_{D}\,}
U
D
S
+
U
D
S
s
a
t
=
U
B
−
U
R
S
{\displaystyle U_{DS}+U_{DS_{sat}}=U_{B}-U_{RS}}
U
D
S
+
U
R
S
=
U
B
−
U
D
S
s
a
t
{\displaystyle U_{DS}+U_{RS}=U_{B}-U_{DS_{sat}}}
2
⋅
U
R
S
=
U
B
−
U
D
S
s
a
t
{\displaystyle 2\cdot U_{RS}=U_{B}-U_{DS_{sat}}}
U
R
S
=
U
B
−
U
D
S
s
a
t
2
{\displaystyle U_{RS}={\frac {U_{B}-U_{DS_{sat}}}{2}}}
a
=
D
a
¨
m
p
f
u
n
g
s
m
a
s
s
α
=
D
a
¨
m
p
f
u
n
g
s
k
o
n
s
t
a
n
t
e
(
D
a
¨
m
p
f
u
n
g
s
k
e
n
n
w
e
r
t
)
l
=
L
a
¨
n
g
e
{\displaystyle {\begin{aligned}a&={\mathsf {D{\ddot {a}}mpfungsmass}}\\\alpha &={\mathsf {D{\ddot {a}}mpfungskonstante\,(D{\ddot {a}}mpfungskennwert)}}\\l&={\mathsf {L{\ddot {a}}nge}}\,\end{aligned}}}
a
=
20
⋅
l
g
(
I
1
I
2
)
[
a
]
=
d
B
{\displaystyle a=20\cdot lg\left({\frac {I_{1}}{I_{2}}}\right)\qquad [a]=dB\,}
1
d
B
=
0
,
115
N
p
{\displaystyle 1dB=0,115\,Np\,}
1
N
p
=
8
,
69
d
B
{\displaystyle 1Np=8,69\,dB\,}
α
=
a
l
[
α
]
=
d
B
k
m
{\displaystyle \alpha ={\frac {a}{l}}\qquad [\alpha ]={\frac {dB}{km}}\,}
U
2
=
U
1
⋅
e
−
α
⋅
l
=
U
1
⋅
e
−
a
I
2
=
I
1
⋅
e
−
α
⋅
l
=
I
1
⋅
e
−
a
{\displaystyle {\begin{alignedat}{2}U_{2}&=U_{1}\cdot \mathrm {e} ^{-\alpha \cdot l}\,&=&\,U_{1}\cdot \mathrm {e} ^{-a}\\I_{2}&=I_{1}\cdot \mathrm {e} ^{-\alpha \cdot l}\,&=&\,I_{1}\cdot \mathrm {e} ^{-a}\end{alignedat}}}
Wenn
Z
L
{\displaystyle Z_{L}\,}
reell ist gilt:
P
2
=
U
2
⋅
I
2
P
2
=
U
1
⋅
e
−
a
⋅
I
1
⋅
e
−
a
P
2
=
U
1
⋅
I
1
⋅
e
−
2
a
P
2
=
P
1
⋅
e
−
2
a
−
2
a
=
ln
P
2
P
1
a
=
−
1
2
⋅
ln
P
2
P
1
⇒
a
=
1
2
⋅
ln
P
1
P
2
b
z
w
.
a
=
10
⋅
lg
P
1
P
2
{\displaystyle {\begin{aligned}P_{2}&=U_{2}\cdot I_{2}\\P_{2}&=U_{1}\cdot \mathrm {e} ^{-a}\cdot I_{1}\cdot \mathrm {e} ^{-a}\\P_{2}&=U_{1}\cdot I_{1}\cdot \mathrm {e} ^{-2a}\\P_{2}&=P_{1}\cdot \mathrm {e} ^{-2a}\\-2a&=\ln {\frac {P_{2}}{P_{1}}}\\a&=-{1 \over 2}\cdot \ln {\frac {P_{2}}{P_{1}}}\quad \Rightarrow \quad a\,=\,{1 \over 2}\cdot \ln {\frac {P_{1}}{P_{2}}}\qquad {\mathsf {bzw.}}\qquad a\,=\,10\cdot \lg {\frac {P_{1}}{P_{2}}}\end{aligned}}}
Spannungspegel:
U
x
:
b
e
l
i
e
b
i
g
e
S
p
a
n
n
u
n
g
i
n
d
e
r
P
e
g
e
l
s
t
r
e
c
k
e
{\displaystyle U_{x}{\mathsf {:beliebige\ Spannung\ in\ der\ Pegelstrecke}}}
U
1
:
B
e
z
u
g
s
g
r
o
¨
s
s
e
a
m
A
n
f
a
n
g
d
e
r
S
t
r
e
c
k
e
{\displaystyle U_{1}{\mathsf {:Bezugsgr{\ddot {o}}sse\ am\ Anfang\ der\ Strecke}}}
P
U
r
=
ln
(
U
x
U
1
)
[
N
p
]
P
U
r
=
20
⋅
lg
(
U
x
U
1
)
[
d
B
]
{\displaystyle P_{U_{r}}=\ln \left({\frac {U_{x}}{U_{1}}}\right)\qquad \left[Np\right]\qquad \qquad \qquad P_{U_{r}}=20\cdot \lg \left({\frac {U_{x}}{U_{1}}}\right)\qquad \left[dB\right]}
Strompegel:
P
I
r
=
ln
(
I
x
I
1
)
[
N
p
]
P
I
r
=
20
⋅
lg
(
I
x
I
1
)
[
d
B
]
{\displaystyle P_{I_{r}}=\ln \left({\frac {I_{x}}{I_{1}}}\right)\qquad \quad \left[Np\right]\qquad \qquad \qquad P_{I_{r}}=20\cdot \lg \left({\frac {I_{x}}{I_{1}}}\right)\qquad \left[dB\right]}
Leistungspegel:
P
r
=
1
2
⋅
ln
(
P
x
P
1
)
[
N
p
]
P
r
=
10
⋅
lg
(
P
x
P
1
)
[
d
B
]
{\displaystyle P_{r}={1 \over 2}\cdot \ln \left({\frac {P_{x}}{P_{1}}}\right)\qquad \left[Np\right]\qquad \qquad \qquad P_{r}=10\cdot \lg \left({\frac {P_{x}}{P_{1}}}\right)\qquad \left[dB\right]}
U
0
=
0
,
775
V
Z
L
=
R
0
=
600
Ω
(
A
n
p
a
s
s
u
n
g
)
I
0
=
1
,
29
m
A
P
0
=
1
m
W
{\displaystyle {\begin{aligned}U_{0}&=0,775\mathrm {V} \\Z_{L}&=R_{0}=600\Omega \qquad {\mathsf {(Anpassung)}}\\I_{0}&=1,29\mathrm {mA} \\P_{0}&=1\mathrm {mW} \end{aligned}}}
U
0
=
0
,
775
V
→
[
P
u
]
=
d
B
μ
P
0
=
1
m
W
→
[
P
]
=
d
B
m
{\displaystyle {\begin{alignedat}{3}U_{0}&=0,775\mathrm {V} &\rightarrow &\left[P_{u}\right]&=&\mathrm {dB} \mu \\P_{0}&=1\mathrm {mW} &\rightarrow &\left[P\right]&=&\mathrm {dBm} \end{alignedat}}}
U
0
=
1
μ
V
R
0
=
75
Ω
{\displaystyle {\begin{aligned}U_{0}&=1\mu \mathrm {V} \\R_{0}&=75\Omega \end{aligned}}}
U
0
=
1
μ
V
→
[
P
u
]
=
d
B
μ
V
{\displaystyle U_{0}=1\mu \mathrm {V} \rightarrow \left[P_{u}\right]=\mathrm {dB} \mu \mathrm {V} }
P
0
=
1
W
{\displaystyle {\begin{aligned}P_{0}&=1\mathrm {W} \end{aligned}}}
P
0
=
1
W
→
[
P
]
=
d
B
W
{\displaystyle P_{0}=1\mathrm {W} \rightarrow \left[P\right]=\mathrm {dB} \mathrm {W} }
P
u
=
ln
U
x
U
0
[
N
p
]
P
u
=
20
⋅
lg
U
x
U
0
[
d
B
]
P
i
=
ln
I
x
I
0
[
N
p
]
P
i
=
20
⋅
lg
I
x
I
0
[
d
B
]
P
=
1
2
⋅
ln
P
x
P
0
[
N
p
]
P
=
10
⋅
lg
P
x
P
0
[
d
B
]
{\displaystyle {\begin{alignedat}{4}P_{u}&=&\ln {\frac {U_{x}}{U_{0}}}\qquad &\left[Np\right]&\qquad \qquad \qquad P_{u}&=&20\cdot \lg {\frac {U_{x}}{U_{0}}}\qquad &\left[dB\right]\\P_{i}&=&\ln {\frac {I_{x}}{I_{0}}}\qquad &\left[Np\right]&\qquad \qquad \qquad P_{i}&=&20\cdot \lg {\frac {I_{x}}{I_{0}}}\qquad &\left[dB\right]\\P&=&{1 \over 2}\cdot \ln {\frac {P_{x}}{P_{0}}}\qquad &\left[Np\right]&\qquad \qquad \qquad P&=&10\cdot \lg {\frac {P_{x}}{P_{0}}}\qquad &\left[dB\right]\\\end{alignedat}}}
Hierunter versteht man Differenz aus Anfangs- und Endpegel eines Systems
a
=
P
u
e
−
P
u
a
{\displaystyle a=P_{u_{e}}-P_{u_{a}}\,}