Étude de la qualité des harmoniques par un gradateur
On retrouve ce type de gradateur sur les simple variateur de lumière pour luminaire halogène le plus souvent. Le courant est mis en circulation à partir d'un certain instant de la sinusoïde et disparait naturellement à son passage à zéro, donc, pour une charge résistive, au passage à zéro de la tension.
La tension est définit par :
u
(
t
)
{\displaystyle u(t)}
u
(
t
)
=
U
2
sin
ω
t
{\displaystyle u(t)=U{\sqrt {2}}\sin \omega t}
Avec :
U = 230 V
ω
=
2
π
f
{\displaystyle \omega =2\pi f}
f = 50 Hz Le courant est définit par :
i
(
t
)
{\displaystyle i(t)}
i
(
t
)
=
{
0
si
k
π
<
θ
<
k
π
+
α
I
2
sin
(
ω
t
)
si
k
π
+
α
<
θ
<
(
k
+
1
)
π
{\displaystyle i(t)={\begin{cases}0&{\mbox{si }}k\pi <\theta <k\pi +\alpha \\I{\sqrt {2}}\sin \left(\omega t\right)&{\mbox{si }}k\pi +\alpha <\theta <\left(k+1\right)\pi \end{cases}}}
Avec :
θ
=
ω
t
{\displaystyle \theta =\omega t}
k
∈
Z
{\displaystyle k\in \mathbb {Z} }
U
=
R
×
I
{\displaystyle U=R\times I}
On peut écrit le courant :
i
(
t
)
=
I
0
+
∑
a
=
0
a
=
∞
I
a
2
sin
(
a
ω
t
−
φ
a
)
{\displaystyle i(t)=I_{0}+\sum _{a=0}^{a=\infty }I_{a}{\sqrt {2}}\sin \left(a\omega t-\varphi _{a}\right)}
Dans la suite de l'étude,
I
0
=
0
{\displaystyle I_{0}=0}
I
a
=
A
a
2
+
B
a
2
{\displaystyle I_{a}={\sqrt {A_{a}^{2}+B_{a}^{2}}}}
tan
φ
a
=
−
B
a
A
a
{\displaystyle \tan \varphi _{a}=-{\frac {B_{a}}{A_{a}}}}
Décomposition en série de Fourier du courant :
A
a
=
2
2
π
∫
0
2
π
[
i
(
t
)
sin
(
a
θ
)
]
d
θ
{\displaystyle A_{a}={\frac {2}{2\pi }}\int _{0}^{2\pi }\left[i(t)\sin \left(a\theta \right)\right]d\theta }
=
2
2
π
[
∫
0
π
(
2
2
sin
θ
×
sin
(
a
θ
)
)
d
θ
+
∫
π
2
π
(
2
2
sin
θ
×
sin
(
a
θ
)
)
d
θ
]
{\displaystyle ={\frac {2}{2\pi }}{\begin{bmatrix}\int _{0}^{\pi }\left(2{\sqrt {2}}\sin \theta \times \sin(a\theta )\right)d\theta \\+\int _{\pi }^{2\pi }\left(2{\sqrt {2}}\sin \theta \times \sin(a\theta )\right)d\theta \end{bmatrix}}}
=
2
2
π
[
∫
α
π
(
2
2
sin
θ
×
sin
(
a
θ
)
)
d
θ
+
∫
a
l
p
h
a
+
π
2
π
(
2
2
sin
θ
×
sin
(
a
θ
)
)
d
θ
]
{\displaystyle ={\frac {2}{2\pi }}{\begin{bmatrix}\int _{\alpha }^{\pi }\left(2{\sqrt {2}}\sin \theta \times \sin(a\theta )\right)d\theta \\+\int _{alpha+\pi }^{2\pi }\left(2{\sqrt {2}}\sin \theta \times \sin(a\theta )\right)d\theta \end{bmatrix}}}
=
I
2
2
π
{
∫
α
π
[
1
2
(
cos
(
θ
−
a
θ
)
−
cos
(
θ
+
a
θ
)
)
]
d
θ
+
∫
α
+
π
2
π
[
1
2
(
cos
(
θ
−
a
θ
)
−
cos
(
θ
+
a
θ
)
)
]
d
θ
}
{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[{\frac {1}{2}}\left(\cos(\theta -a\theta )-\cos(\theta +a\theta )\right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[{\frac {1}{2}}\left(\cos(\theta -a\theta )-\cos(\theta +a\theta )\right)\right]d\theta \end{Bmatrix}}}
A
a
=
I
2
2
π
{
∫
α
π
[
1
2
(
cos
(
(
1
−
a
)
θ
)
−
cos
(
(
1
+
a
)
θ
)
)
]
d
θ
+
∫
α
+
π
2
π
[
1
2
(
cos
(
(
1
−
a
)
θ
)
−
cos
(
(
1
+
a
)
θ
)
)
]
d
θ
}
{\displaystyle A_{a}={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[{\frac {1}{2}}\left(\cos \left((1-a)\theta \right)-\cos \left((1+a)\theta \right)\right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[{\frac {1}{2}}\left(\cos \left((1-a)\theta \right)-\cos \left((1+a)\theta \right)\right)\right]d\theta \end{Bmatrix}}}
∀
a
≠
1
{\displaystyle \forall a\neq 1}
A
a
=
I
2
2
π
{
[
sin
(
(
1
−
a
)
θ
)
1
−
a
]
α
π
−
[
sin
(
(
1
+
a
)
θ
)
1
+
a
]
α
π
+
[
sin
(
(
1
−
a
)
θ
)
1
−
a
]
α
+
π
2
π
−
[
sin
(
(
1
+
a
)
θ
)
1
+
a
]
α
+
π
2
π
}
{\displaystyle A_{a}={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {\sin \left((1-a)\theta \right)}{1-a}}\right]_{\alpha }^{\pi }\\-\left[{\frac {\sin \left((1+a)\theta \right)}{1+a}}\right]_{\alpha }^{\pi }\\+\left[{\frac {\sin \left((1-a)\theta \right)}{1-a}}\right]_{\alpha +\pi }^{2\pi }\\-\left[{\frac {\sin \left((1+a)\theta \right)}{1+a}}\right]_{\alpha +\pi }^{2\pi }\end{Bmatrix}}}
=
I
2
2
π
{
[
sin
(
(
1
−
a
)
π
)
−
sin
(
(
1
−
a
)
α
)
1
−
a
]
−
[
sin
(
(
1
+
a
)
π
)
−
sin
(
(
1
+
a
)
α
)
1
+
a
]
+
[
sin
(
(
1
−
a
)
2
π
)
−
sin
(
(
1
−
a
)
(
α
+
π
)
)
1
−
a
]
−
[
sin
(
(
1
+
a
)
2
π
)
−
sin
(
(
1
−
a
)
(
α
+
π
)
)
1
+
a
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {\sin \left((1-a)\pi \right)-\sin \left((1-a)\alpha \right)}{1-a}}\right]\\-\left[{\frac {\sin \left((1+a)\pi \right)-\sin \left((1+a)\alpha \right)}{1+a}}\right]\\+\left[{\frac {\sin \left((1-a)2\pi \right)-\sin \left((1-a)(\alpha +\pi )\right)}{1-a}}\right]\\-\left[{\frac {\sin \left((1+a)2\pi \right)-\sin \left((1-a)(\alpha +\pi )\right)}{1+a}}\right]\end{Bmatrix}}}
=
I
2
2
π
{
[
−
sin
(
(
1
−
a
)
α
)
1
−
a
]
−
[
−
sin
(
(
1
+
a
)
α
)
1
+
a
]
+
[
−
sin
(
(
1
−
a
)
(
α
+
π
)
)
1
−
a
]
−
[
−
sin
(
(
1
+
a
)
(
α
+
π
)
)
1
+
a
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {-\sin \left((1-a)\alpha \right)}{1-a}}\right]\\-\left[{\frac {-\sin \left((1+a)\alpha \right)}{1+a}}\right]\\+\left[{\frac {-\sin \left((1-a)(\alpha +\pi )\right)}{1-a}}\right]\\-\left[{\frac {-\sin \left((1+a)(\alpha +\pi )\right)}{1+a}}\right]\end{Bmatrix}}}
=
I
2
2
π
{
[
−
sin
(
(
1
−
a
)
α
)
−
sin
(
(
1
−
a
)
(
α
+
π
)
)
1
−
a
]
−
[
−
sin
(
(
1
+
a
)
α
)
+
sin
(
(
1
+
a
)
(
α
+
π
)
)
1
+
a
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {-\sin \left((1-a)\alpha \right)-\sin \left((1-a)(\alpha +\pi )\right)}{1-a}}\right]\\-\left[{\frac {-\sin \left((1+a)\alpha \right)+\sin \left((1+a)(\alpha +\pi )\right)}{1+a}}\right]\end{Bmatrix}}}
=
I
2
2
π
{
2
1
+
a
sin
[
(
(
1
+
a
)
α
)
+
(
(
1
+
a
)
(
α
+
π
)
)
2
]
cos
[
(
(
1
+
a
)
α
)
−
(
(
1
+
a
)
(
α
+
π
)
)
2
]
−
2
1
−
a
sin
[
(
(
1
−
a
)
α
)
+
(
(
1
−
a
)
(
α
+
π
)
)
2
]
cos
[
(
(
1
−
a
)
α
)
−
(
(
1
−
a
)
(
α
+
π
)
)
2
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}{\frac {2}{1+a}}\sin \left[{\frac {\left((1+a)\alpha \right)+\left((1+a)(\alpha +\pi )\right)}{2}}\right]\cos \left[{\frac {\left((1+a)\alpha \right)-\left((1+a)(\alpha +\pi )\right)}{2}}\right]\\-{\frac {2}{1-a}}\sin \left[{\frac {\left((1-a)\alpha \right)+\left((1-a)(\alpha +\pi )\right)}{2}}\right]\cos \left[{\frac {\left((1-a)\alpha \right)-\left((1-a)(\alpha +\pi )\right)}{2}}\right]\end{Bmatrix}}}
=
I
2
2
π
{
2
1
+
a
sin
[
α
+
a
α
+
α
+
π
+
a
α
+
a
π
2
]
cos
[
α
+
a
α
−
α
−
π
−
a
α
−
a
π
2
]
−
2
1
−
a
sin
[
α
−
a
α
+
α
+
π
−
a
α
−
a
π
2
]
cos
[
α
−
a
α
−
α
−
π
+
a
α
+
a
π
2
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}{\frac {2}{1+a}}\sin \left[{\frac {\alpha +a\alpha +\alpha +\pi +a\alpha +a\pi }{2}}\right]\cos \left[{\frac {\alpha +a\alpha -\alpha -\pi -a\alpha -a\pi }{2}}\right]\\-{\frac {2}{1-a}}\sin \left[{\frac {\alpha -a\alpha +\alpha +\pi -a\alpha -a\pi }{2}}\right]\cos \left[{\frac {\alpha -a\alpha -\alpha -\pi +a\alpha +a\pi }{2}}\right]\end{Bmatrix}}}
=
I
2
π
{
1
1
+
a
sin
[
α
(
2
+
2
a
)
+
π
(
1
+
a
)
2
]
cos
[
−
π
(
1
+
a
)
2
]
−
1
1
−
a
sin
[
α
(
2
−
2
a
)
+
π
(
1
−
a
)
2
]
cos
[
−
π
(
1
+
a
)
2
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\sin \left[{\frac {\alpha \left(2+2a\right)+\pi \left(1+a\right)}{2}}\right]\cos \left[{\frac {-\pi \left(1+a\right)}{2}}\right]\\-{\frac {1}{1-a}}\sin \left[{\frac {\alpha \left(2-2a\right)+\pi \left(1-a\right)}{2}}\right]\cos \left[{\frac {-\pi \left(1+a\right)}{2}}\right]\end{Bmatrix}}}
=
−
1
a
+
1
I
2
π
{
1
1
+
a
sin
[
α
(
2
+
2
a
)
+
π
(
1
+
a
)
2
]
−
1
1
−
a
sin
[
α
(
2
−
2
a
)
+
π
(
1
−
a
)
2
]
}
{\displaystyle =-1^{a+1}{\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\sin \left[{\frac {\alpha \left(2+2a\right)+\pi \left(1+a\right)}{2}}\right]\\-{\frac {1}{1-a}}\sin \left[{\frac {\alpha \left(2-2a\right)+\pi \left(1-a\right)}{2}}\right]\end{Bmatrix}}}
=
−
1
a
+
1
I
2
π
{
1
1
+
a
[
sin
(
α
(
1
+
a
)
)
cos
(
π
2
(
1
+
a
)
)
+
cos
(
α
(
1
+
a
)
)
sin
(
π
2
(
1
+
a
)
)
]
−
1
1
−
a
[
sin
(
α
(
1
−
a
)
)
cos
(
π
2
(
1
−
a
)
)
+
cos
(
α
(
1
−
a
)
)
sin
(
π
2
(
1
−
a
)
)
]
}
{\displaystyle =-1^{a+1}{\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}{\begin{bmatrix}\sin \left(\alpha (1+a)\right)\cos \left({\frac {\pi }{2}}(1+a)\right)\\+\cos \left(\alpha (1+a)\right)\sin \left({\frac {\pi }{2}}(1+a)\right)\end{bmatrix}}\\-{\frac {1}{1-a}}{\begin{bmatrix}\sin \left(\alpha (1-a)\right)\cos \left({\frac {\pi }{2}}(1-a)\right)\\+\cos \left(\alpha (1-a)\right)\sin \left({\frac {\pi }{2}}(1-a)\right)\end{bmatrix}}\end{Bmatrix}}}
=
−
1
a
+
1
I
2
π
{
1
1
+
a
[
sin
(
π
2
(
1
+
a
)
)
cos
(
α
(
1
+
a
)
)
]
−
1
1
−
a
[
sin
(
π
2
(
1
−
a
)
)
cos
(
α
(
1
−
a
)
)
]
}
{\displaystyle =-1^{a+1}{\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[\sin \left({\frac {\pi }{2}}(1+a)\right)\cos \left(\alpha (1+a)\right)\right]\\-{\frac {1}{1-a}}\left[\sin \left({\frac {\pi }{2}}(1-a)\right)\cos \left(\alpha (1-a)\right)\right]\end{Bmatrix}}}
A
a
=
{
?
si
a
=
1
0
si
a
=
2
k
+
1
avec
k
∈
N
−
I
2
π
{
cos
(
α
(
1
+
α
)
)
1
+
a
+
cos
(
α
(
1
+
α
)
)
1
−
a
}
si
a
=
4
k
avec
k
∈
N
I
2
π
{
cos
(
α
(
1
+
α
)
)
1
+
a
+
cos
(
α
(
1
+
α
)
)
1
−
a
}
si
a
=
4
k
−
2
avec
k
∈
N
∗
{\displaystyle A_{a}={\begin{cases}?&{\mbox{si }}a=1\\0&{\mbox{si }}a=2k+1{\mbox{ avec }}k\in \mathbb {N} \\-{\frac {I{\sqrt {2}}}{\pi }}\left\{{\frac {\cos \left(\alpha (1+\alpha )\right)}{1+a}}+{\frac {\cos \left(\alpha (1+\alpha )\right)}{1-a}}\right\}&{\mbox{si }}a=4k{\mbox{ avec }}k\in \mathbb {N} \\{\frac {I{\sqrt {2}}}{\pi }}\left\{{\frac {\cos \left(\alpha (1+\alpha )\right)}{1+a}}+{\frac {\cos \left(\alpha (1+\alpha )\right)}{1-a}}\right\}&{\mbox{si }}a=4k-2{\mbox{ avec }}k\in \mathbb {N} ^{*}\end{cases}}}
B
a
=
2
2
π
∫
0
2
π
[
i
(
θ
)
cos
(
a
θ
)
]
d
θ
{\displaystyle B_{a}={\frac {2}{2\pi }}\int _{0}^{2\pi }\left[i(\theta )\cos \left(a\theta \right)\right]d\theta }
=
1
π
{
∫
0
π
[
I
2
sin
θ
×
cos
(
a
θ
)
]
d
θ
+
∫
π
2
π
[
I
2
sin
θ
×
cos
(
a
θ
)
]
d
θ
}
{\displaystyle ={\frac {1}{\pi }}{\begin{Bmatrix}\int _{0}^{\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \\+\int _{\pi }^{2\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \end{Bmatrix}}}
=
1
π
{
∫
α
π
[
I
2
sin
θ
×
cos
(
a
θ
)
]
d
θ
+
∫
α
+
π
2
π
[
I
2
sin
θ
×
cos
(
a
θ
)
]
d
θ
}
{\displaystyle ={\frac {1}{\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \end{Bmatrix}}}
=
I
2
π
{
∫
α
π
[
1
2
(
sin
(
θ
+
a
θ
)
+
sin
(
θ
−
a
θ
)
)
]
d
θ
+
∫
α
+
π
2
π
[
1
2
(
sin
(
θ
+
a
θ
)
+
sin
(
θ
−
a
θ
)
)
]
d
θ
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[{\frac {1}{2}}\left(\sin(\theta +a\theta )+\sin(\theta -a\theta )\right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[{\frac {1}{2}}\left(\sin(\theta +a\theta )+\sin(\theta -a\theta )\right)\right]d\theta \end{Bmatrix}}}
=
I
2
π
{
∫
α
π
sin
[
(
1
+
a
)
θ
]
d
θ
+
∫
α
π
sin
[
(
1
−
a
)
θ
]
d
θ
+
∫
α
+
π
2
π
sin
[
(
1
+
a
)
θ
]
d
θ
+
∫
α
+
π
2
π
sin
[
(
1
−
a
)
θ
]
d
θ
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\sin \left[(1+a)\theta \right]d\theta \\+\int _{\alpha }^{\pi }\sin \left[(1-a)\theta \right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\sin \left[(1+a)\theta \right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\sin \left[(1-a)\theta \right]d\theta \end{Bmatrix}}}
∀
a
∈
N
−
{
1
}
{\displaystyle \forall a\in \mathbb {N} -\{1\}}
=
I
2
π
{
[
−
cos
(
1
+
a
)
θ
1
+
a
]
α
π
+
[
−
cos
(
1
−
a
)
θ
1
−
a
]
α
π
+
[
−
cos
(
1
+
a
)
θ
1
+
a
]
α
+
π
2
π
+
[
−
cos
(
1
−
a
)
θ
1
−
a
]
α
+
π
2
π
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {-\cos(1+a)\theta }{1+a}}\right]_{\alpha }^{\pi }\\+\left[{\frac {-\cos(1-a)\theta }{1-a}}\right]_{\alpha }^{\pi }\\+\left[{\frac {-\cos(1+a)\theta }{1+a}}\right]_{\alpha +\pi }^{2\pi }\\+\left[{\frac {-\cos(1-a)\theta }{1-a}}\right]_{\alpha +\pi }^{2\pi }\end{Bmatrix}}}
=
I
2
π
{
[
cos
(
(
1
+
a
)
α
)
−
cos
(
(
1
+
a
)
π
)
1
+
a
]
+
[
cos
(
(
1
+
a
)
(
α
+
π
)
)
−
cos
(
(
1
+
a
)
2
π
)
1
−
a
]
+
[
cos
(
(
1
+
a
)
(
α
+
π
)
)
−
cos
(
(
1
+
a
)
2
π
)
1
+
a
]
+
[
cos
(
(
1
−
a
)
(
α
+
π
)
)
−
cos
(
(
1
−
a
)
2
π
)
1
−
a
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {\cos \left((1+a)\alpha \right)-\cos \left((1+a)\pi \right)}{1+a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-\cos \left((1+a)2\pi \right)}{1-a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-\cos \left((1+a)2\pi \right)}{1+a}}\right]\\+\left[{\frac {\cos \left((1-a)(\alpha +\pi )\right)-\cos \left((1-a)2\pi \right)}{1-a}}\right]\\\end{Bmatrix}}}
=
I
2
π
{
[
cos
(
(
1
+
a
)
α
)
−
(
−
1
)
a
+
1
1
+
a
]
+
[
cos
(
(
1
+
a
)
(
α
+
π
)
)
−
(
−
1
)
a
+
1
1
−
a
]
+
[
cos
(
(
1
+
a
)
(
α
+
π
)
)
−
1
1
+
a
]
+
[
cos
(
(
1
−
a
)
(
α
+
π
)
)
−
1
1
−
a
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {\cos \left((1+a)\alpha \right)-\left(-1\right)^{a+1}}{1+a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-\left(-1\right)^{a+1}}{1-a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-1}{1+a}}\right]\\+\left[{\frac {\cos \left((1-a)(\alpha +\pi )\right)-1}{1-a}}\right]\end{Bmatrix}}}
=
I
2
π
{
[
cos
(
(
1
+
a
)
α
)
+
(
−
1
)
a
+
cos
(
(
1
+
a
)
(
α
+
π
)
)
−
1
1
+
a
]
+
[
cos
(
(
1
−
a
)
α
)
+
(
−
1
)
a
+
cos
(
(
1
−
a
)
(
α
+
π
)
)
−
1
1
−
a
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {\cos \left((1+a)\alpha \right)+\left(-1\right)^{a}+\cos \left((1+a)(\alpha +\pi )\right)-1}{1+a}}\right]\\+\left[{\frac {\cos \left((1-a)\alpha \right)+\left(-1\right)^{a}+\cos \left((1-a)(\alpha +\pi )\right)-1}{1-a}}\right]\\\end{Bmatrix}}}
=
I
2
π
{
1
1
+
a
[
2
cos
(
(
1
+
a
)
α
+
(
1
+
a
)
(
α
+
π
)
2
)
cos
(
(
1
+
a
)
α
−
(
1
+
a
)
(
α
+
π
)
2
)
+
(
−
1
)
a
−
1
]
+
1
1
−
a
[
2
cos
(
(
1
−
a
)
α
+
(
1
−
a
)
(
α
+
π
)
2
)
cos
(
(
1
−
a
)
α
−
(
1
−
a
)
(
α
+
π
)
2
)
+
(
−
1
)
a
−
1
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[2\cos \left({\frac {(1+a)\alpha +(1+a)(\alpha +\pi )}{2}}\right)\cos \left({\frac {(1+a)\alpha -(1+a)(\alpha +\pi )}{2}}\right)+\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[2\cos \left({\frac {(1-a)\alpha +(1-a)(\alpha +\pi )}{2}}\right)\cos \left({\frac {(1-a)\alpha -(1-a)(\alpha +\pi )}{2}}\right)+\left(-1\right)^{a}-1\right]\end{Bmatrix}}}
=
I
2
π
{
1
1
+
a
[
2
cos
(
α
+
a
α
+
α
+
π
+
a
α
+
a
π
2
)
cos
(
α
+
a
α
−
α
−
π
−
a
α
−
a
π
2
)
+
(
−
1
)
a
−
1
]
+
1
1
−
a
[
2
cos
(
α
−
a
α
+
α
+
π
−
a
α
−
a
π
2
)
cos
(
α
−
a
α
−
α
−
π
+
a
α
+
a
π
2
)
+
(
−
1
)
a
−
1
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[2\cos \left({\frac {\alpha +a\alpha +\alpha +\pi +a\alpha +a\pi }{2}}\right)\cos \left({\frac {\alpha +a\alpha -\alpha -\pi -a\alpha -a\pi }{2}}\right)+\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[2\cos \left({\frac {\alpha -a\alpha +\alpha +\pi -a\alpha -a\pi }{2}}\right)\cos \left({\frac {\alpha -a\alpha -\alpha -\pi +a\alpha +a\pi }{2}}\right)+\left(-1\right)^{a}-1\right]\end{Bmatrix}}}
=
I
2
π
{
1
1
+
a
[
2
cos
(
(
2
α
+
π
)
(
1
+
a
)
2
)
cos
(
−
π
(
1
+
a
)
2
)
+
(
−
1
)
a
−
1
]
+
1
1
−
a
[
2
cos
(
(
2
α
+
π
)
(
1
−
a
)
2
)
cos
(
−
π
(
1
−
a
)
2
)
+
(
−
1
)
a
−
1
]
}
{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[2\cos \left({\frac {(2\alpha +\pi )(1+a)}{2}}\right)\cos \left({\frac {-\pi (1+a)}{2}}\right)+\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[2\cos \left({\frac {(2\alpha +\pi )(1-a)}{2}}\right)\cos \left({\frac {-\pi (1-a)}{2}}\right)+\left(-1\right)^{a}-1\right]\end{Bmatrix}}}
B
a
=
I
2
π
{
1
1
+
a
[
(
−
1
)
a
−
1
]
+
1
1
−
a
[
(
−
1
)
a
−
1
]
}
{\displaystyle B_{a}={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[\left(-1\right)^{a}-1\right]\end{Bmatrix}}}
B
a
=
{
?
si
a
=
1
0
si
a
=
2
k
+
1
avec
k
∈
N
∗
−
I
2
π
(
1
a
+
1
+
1
a
−
1
)
si
a
=
2
k
avec
k
∈
N
{\displaystyle B_{a}={\begin{cases}?&{\mbox{si }}a=1\\0&{\mbox{si }}a=2k+1{\mbox{ avec }}k\in \mathbb {N} ^{*}\\-{\frac {I{\sqrt {2}}}{\pi }}\left({\frac {1}{a+1}}+{\frac {1}{a-1}}\right)&{\mbox{si }}a=2k{\mbox{ avec }}k\in \mathbb {N} \end{cases}}}
p
(
t
)
=
u
(
t
)
×
i
(
t
)
{\displaystyle p(t)=u(t)\times i(t)}
P
=
p
(
t
)
¯
{\displaystyle P={\overline {p(t)}}}
=
1
2
π
∫
0
2
π
[
p
(
t
)
]
d
t
{\displaystyle ={\frac {1}{2\pi }}\int _{0}^{2\pi }\left[p(t)\right]dt}
=
2
2
π
∫
0
π
[
u
(
t
)
×
i
(
t
)
]
d
t
{\displaystyle ={\frac {2}{2\pi }}\int _{0}^{\pi }\left[u(t)\times i(t)\right]dt}
=
1
π
∫
α
π
[
u
(
t
)
×
i
(
t
)
]
d
t
{\displaystyle ={\frac {1}{\pi }}\int _{\alpha }^{\pi }\left[u(t)\times i(t)\right]dt}
=
1
π
∫
α
π
[
(
U
2
sin
θ
)
×
(
I
2
sin
θ
)
]
d
θ
{\displaystyle ={\frac {1}{\pi }}\int _{\alpha }^{\pi }\left[\left(U{\sqrt {2}}\sin \theta \right)\times \left(I{\sqrt {2}}\sin \theta \right)\right]d\theta }
=
2
U
I
π
∫
α
π
[
sin
2
θ
]
d
θ
{\displaystyle ={\frac {2UI}{\pi }}\int _{\alpha }^{\pi }\left[\sin ^{2}\theta \right]d\theta }
=
2
U
I
π
∫
α
π
[
1
−
cos
(
2
θ
)
2
]
d
θ
{\displaystyle ={\frac {2UI}{\pi }}\int _{\alpha }^{\pi }\left[{\frac {1-\cos \left(2\theta \right)}{2}}\right]d\theta }
=
U
I
π
[
θ
−
sin
(
2
θ
)
2
]
α
π
{\displaystyle ={\frac {UI}{\pi }}\left[\theta -{\frac {\sin \left(2\theta \right)}{2}}\right]_{\alpha }^{\pi }}
=
U
I
π
[
π
−
sin
(
2
π
)
2
−
α
−
sin
(
2
α
)
2
]
{\displaystyle ={\frac {UI}{\pi }}\left[\pi -{\frac {\sin \left(2\pi \right)}{2}}-\alpha -{\frac {\sin \left(2\alpha \right)}{2}}\right]}
P
=
U
I
π
[
π
−
α
+
sin
(
2
α
)
2
]
{\displaystyle P={\frac {UI}{\pi }}\left[\pi -\alpha +{\frac {\sin \left(2\alpha \right)}{2}}\right]}
I
e
f
f
=
∑
a
=
1
a
=
∞
(
I
a
2
)
{\displaystyle I_{eff}={\sqrt {\sum _{a=1}^{a=\infty }\left(I_{a}^{2}\right)}}}
T
H
D
=
∑
a
=
2
a
=
∞
(
I
a
I
1
)
2
=
∑
a
=
2
a
=
∞
I
a
2
I
1
=
I
H
M
I
1
{\displaystyle THD={\sqrt {\sum _{a=2}^{a=\infty }\left({\frac {I_{a}}{I_{1}}}\right)^{2}}}={\frac {\sqrt {\sum _{a=2}^{a=\infty }{I_{a}}^{2}}}{I_{1}}}={\frac {I_{HM}}{I_{1}}}}
D
F
=
∑
a
=
2
a
=
∞
I
a
2
I
e
f
f
=
I
H
M
I
e
f
f
{\displaystyle DF={\frac {\sqrt {\sum _{a=2}^{a=\infty }{I_{a}}^{2}}}{I_{eff}}}={\frac {I_{HM}}{I_{eff}}}}
S
2
=
(
∑
a
=
1
∞
U
a
2
)
×
(
∑
a
=
1
∞
I
a
2
)
{\displaystyle S^{2}=\left(\sum _{a=1}^{\infty }U_{a}^{2}\right)\times \left(\sum _{a=1}^{\infty }I_{a}^{2}\right)}
F
P
=
P
S
{\displaystyle FP={\frac {P}{S}}}
cos
φ
=
P
1
S
1
{\displaystyle \cos \varphi ={\frac {P_{1}}{S_{1}}}}
La tension est définit par :
u
(
t
)
{\displaystyle u(t)}
u
(
t
)
=
U
2
sin
ω
t
{\displaystyle u(t)=U{\sqrt {2}}\sin \omega t}
Avec :
U = 230 V
ω
=
2
π
f
{\displaystyle \omega =2\pi f}
f = 50 Hz Le courant est définit par :
i
(
t
)
{\displaystyle i(t)}
i
(
t
)
=
{
I
2
sin
(
ω
t
)
si
k
π
<
θ
<
k
π
+
α
0
si
k
π
+
α
<
θ
<
(
k
+
1
)
π
{\displaystyle i(t)={\begin{cases}I{\sqrt {2}}\sin \left(\omega t\right)&{\mbox{si }}k\pi <\theta <k\pi +\alpha \\0&{\mbox{si }}k\pi +\alpha <\theta <\left(k+1\right)\pi \end{cases}}}
Avec :
θ
=
ω
t
{\displaystyle \theta =\omega t}
k
∈
Z
{\displaystyle k\in \mathbb {Z} }
U
=
R
×
I
{\displaystyle U=R\times I}
[
π
2
−
α
;
π
2
+
α
]
+
[
3
π
2
−
α
;
3
π
2
−
α
]
{\displaystyle \left[{\frac {\pi }{2}}-\alpha ;{\frac {\pi }{2}}+\alpha \right]+\left[{\frac {3\pi }{2}}-\alpha ;{\frac {3\pi }{2}}-\alpha \right]}
[ modifier | modifier le wikicode ]
[
0
;
π
2
−
α
]
+
[
π
2
+
α
;
3
π
2
−
α
]
+
[
3
π
2
+
α
;
2
π
]
{\displaystyle \left[0;{\frac {\pi }{2}}-\alpha \right]+\left[{\frac {\pi }{2}}+\alpha ;{\frac {3\pi }{2}}-\alpha \right]+\left[{\frac {3\pi }{2}}+\alpha ;2\pi \right]}
[ modifier | modifier le wikicode ] 
![{\displaystyle A_{a}={\frac {2}{2\pi }}\int _{0}^{2\pi }\left[i(t)\sin \left(a\theta \right)\right]d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c79dd32da133a9ee1cf250309479317005f97c59)
![{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[{\frac {1}{2}}\left(\cos(\theta -a\theta )-\cos(\theta +a\theta )\right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[{\frac {1}{2}}\left(\cos(\theta -a\theta )-\cos(\theta +a\theta )\right)\right]d\theta \end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4beb8d8682576c60242cafad32967a75d0b6e93d)
![{\displaystyle A_{a}={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[{\frac {1}{2}}\left(\cos \left((1-a)\theta \right)-\cos \left((1+a)\theta \right)\right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[{\frac {1}{2}}\left(\cos \left((1-a)\theta \right)-\cos \left((1+a)\theta \right)\right)\right]d\theta \end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66b71d8a85ddfa51e57ade23276c2e83fe1104c4)
![{\displaystyle A_{a}={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {\sin \left((1-a)\theta \right)}{1-a}}\right]_{\alpha }^{\pi }\\-\left[{\frac {\sin \left((1+a)\theta \right)}{1+a}}\right]_{\alpha }^{\pi }\\+\left[{\frac {\sin \left((1-a)\theta \right)}{1-a}}\right]_{\alpha +\pi }^{2\pi }\\-\left[{\frac {\sin \left((1+a)\theta \right)}{1+a}}\right]_{\alpha +\pi }^{2\pi }\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4ce8f149a62ffbe8b4421548da46f56d8db8fa)
![{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {\sin \left((1-a)\pi \right)-\sin \left((1-a)\alpha \right)}{1-a}}\right]\\-\left[{\frac {\sin \left((1+a)\pi \right)-\sin \left((1+a)\alpha \right)}{1+a}}\right]\\+\left[{\frac {\sin \left((1-a)2\pi \right)-\sin \left((1-a)(\alpha +\pi )\right)}{1-a}}\right]\\-\left[{\frac {\sin \left((1+a)2\pi \right)-\sin \left((1-a)(\alpha +\pi )\right)}{1+a}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/667c9ae2a7af590d5d8bc9e2976c57666780aba6)
![{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {-\sin \left((1-a)\alpha \right)}{1-a}}\right]\\-\left[{\frac {-\sin \left((1+a)\alpha \right)}{1+a}}\right]\\+\left[{\frac {-\sin \left((1-a)(\alpha +\pi )\right)}{1-a}}\right]\\-\left[{\frac {-\sin \left((1+a)(\alpha +\pi )\right)}{1+a}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/921c329927113927c97b4123cf3cef078ab8171f)
![{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}\left[{\frac {-\sin \left((1-a)\alpha \right)-\sin \left((1-a)(\alpha +\pi )\right)}{1-a}}\right]\\-\left[{\frac {-\sin \left((1+a)\alpha \right)+\sin \left((1+a)(\alpha +\pi )\right)}{1+a}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01aac46627f8cc966832a507d578d92ac48b42f1)
![{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}{\frac {2}{1+a}}\sin \left[{\frac {\left((1+a)\alpha \right)+\left((1+a)(\alpha +\pi )\right)}{2}}\right]\cos \left[{\frac {\left((1+a)\alpha \right)-\left((1+a)(\alpha +\pi )\right)}{2}}\right]\\-{\frac {2}{1-a}}\sin \left[{\frac {\left((1-a)\alpha \right)+\left((1-a)(\alpha +\pi )\right)}{2}}\right]\cos \left[{\frac {\left((1-a)\alpha \right)-\left((1-a)(\alpha +\pi )\right)}{2}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abb20c7ebefc124d9b854769aabd3559b77cddad)
![{\displaystyle ={\frac {I{\sqrt {2}}}{2\pi }}{\begin{Bmatrix}{\frac {2}{1+a}}\sin \left[{\frac {\alpha +a\alpha +\alpha +\pi +a\alpha +a\pi }{2}}\right]\cos \left[{\frac {\alpha +a\alpha -\alpha -\pi -a\alpha -a\pi }{2}}\right]\\-{\frac {2}{1-a}}\sin \left[{\frac {\alpha -a\alpha +\alpha +\pi -a\alpha -a\pi }{2}}\right]\cos \left[{\frac {\alpha -a\alpha -\alpha -\pi +a\alpha +a\pi }{2}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11053d27290d8e4451bfeea8723d536c8bf685bb)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\sin \left[{\frac {\alpha \left(2+2a\right)+\pi \left(1+a\right)}{2}}\right]\cos \left[{\frac {-\pi \left(1+a\right)}{2}}\right]\\-{\frac {1}{1-a}}\sin \left[{\frac {\alpha \left(2-2a\right)+\pi \left(1-a\right)}{2}}\right]\cos \left[{\frac {-\pi \left(1+a\right)}{2}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81b4dab4e9306a5fcfa9e8bc45550ddabb457838)
![{\displaystyle =-1^{a+1}{\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\sin \left[{\frac {\alpha \left(2+2a\right)+\pi \left(1+a\right)}{2}}\right]\\-{\frac {1}{1-a}}\sin \left[{\frac {\alpha \left(2-2a\right)+\pi \left(1-a\right)}{2}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5b65a323f4ba9be421084c53109d75b43af10e)
![{\displaystyle =-1^{a+1}{\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[\sin \left({\frac {\pi }{2}}(1+a)\right)\cos \left(\alpha (1+a)\right)\right]\\-{\frac {1}{1-a}}\left[\sin \left({\frac {\pi }{2}}(1-a)\right)\cos \left(\alpha (1-a)\right)\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5eb1b202be07337159ad42153ce0dd27045147e)
![{\displaystyle B_{a}={\frac {2}{2\pi }}\int _{0}^{2\pi }\left[i(\theta )\cos \left(a\theta \right)\right]d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/700fe642ac5d03ffac8c5e970b1726c6ff66d539)
![{\displaystyle ={\frac {1}{\pi }}{\begin{Bmatrix}\int _{0}^{\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \\+\int _{\pi }^{2\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03d2f1c75090c73e683dc9a90e07178d6cb704f6)
![{\displaystyle ={\frac {1}{\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[I{\sqrt {2}}\sin \theta \times \cos \left(a\theta \right)\right]d\theta \end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d9ad9e426b186267ff4a7e463eb10252e4cde1)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\left[{\frac {1}{2}}\left(\sin(\theta +a\theta )+\sin(\theta -a\theta )\right)\right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\left[{\frac {1}{2}}\left(\sin(\theta +a\theta )+\sin(\theta -a\theta )\right)\right]d\theta \end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9557af328398a4b644ae35ae4278936e306c5d7)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\int _{\alpha }^{\pi }\sin \left[(1+a)\theta \right]d\theta \\+\int _{\alpha }^{\pi }\sin \left[(1-a)\theta \right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\sin \left[(1+a)\theta \right]d\theta \\+\int _{\alpha +\pi }^{2\pi }\sin \left[(1-a)\theta \right]d\theta \end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9e8dae23dd06f91625a4a694b3c1e500d4e9ba)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {-\cos(1+a)\theta }{1+a}}\right]_{\alpha }^{\pi }\\+\left[{\frac {-\cos(1-a)\theta }{1-a}}\right]_{\alpha }^{\pi }\\+\left[{\frac {-\cos(1+a)\theta }{1+a}}\right]_{\alpha +\pi }^{2\pi }\\+\left[{\frac {-\cos(1-a)\theta }{1-a}}\right]_{\alpha +\pi }^{2\pi }\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/130c2b98df3020d39b6220804ac421bd785b5154)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {\cos \left((1+a)\alpha \right)-\cos \left((1+a)\pi \right)}{1+a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-\cos \left((1+a)2\pi \right)}{1-a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-\cos \left((1+a)2\pi \right)}{1+a}}\right]\\+\left[{\frac {\cos \left((1-a)(\alpha +\pi )\right)-\cos \left((1-a)2\pi \right)}{1-a}}\right]\\\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120ae00750964af1045208f8c996d74fc346f572)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {\cos \left((1+a)\alpha \right)-\left(-1\right)^{a+1}}{1+a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-\left(-1\right)^{a+1}}{1-a}}\right]\\+\left[{\frac {\cos \left((1+a)(\alpha +\pi )\right)-1}{1+a}}\right]\\+\left[{\frac {\cos \left((1-a)(\alpha +\pi )\right)-1}{1-a}}\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dacd8323ed5c2cc70c1fa995a38ceb4f7d6eaa8e)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}\left[{\frac {\cos \left((1+a)\alpha \right)+\left(-1\right)^{a}+\cos \left((1+a)(\alpha +\pi )\right)-1}{1+a}}\right]\\+\left[{\frac {\cos \left((1-a)\alpha \right)+\left(-1\right)^{a}+\cos \left((1-a)(\alpha +\pi )\right)-1}{1-a}}\right]\\\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39bf8a508750679fc4ed9cf0972f99f3296276fa)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[2\cos \left({\frac {(1+a)\alpha +(1+a)(\alpha +\pi )}{2}}\right)\cos \left({\frac {(1+a)\alpha -(1+a)(\alpha +\pi )}{2}}\right)+\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[2\cos \left({\frac {(1-a)\alpha +(1-a)(\alpha +\pi )}{2}}\right)\cos \left({\frac {(1-a)\alpha -(1-a)(\alpha +\pi )}{2}}\right)+\left(-1\right)^{a}-1\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a91e40af2154fba09b073ea75016c0586fcb7ae8)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[2\cos \left({\frac {\alpha +a\alpha +\alpha +\pi +a\alpha +a\pi }{2}}\right)\cos \left({\frac {\alpha +a\alpha -\alpha -\pi -a\alpha -a\pi }{2}}\right)+\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[2\cos \left({\frac {\alpha -a\alpha +\alpha +\pi -a\alpha -a\pi }{2}}\right)\cos \left({\frac {\alpha -a\alpha -\alpha -\pi +a\alpha +a\pi }{2}}\right)+\left(-1\right)^{a}-1\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21b0973d8d3f9e3ff7c2e45a5bcd6e0c6cc2ed8b)
![{\displaystyle ={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[2\cos \left({\frac {(2\alpha +\pi )(1+a)}{2}}\right)\cos \left({\frac {-\pi (1+a)}{2}}\right)+\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[2\cos \left({\frac {(2\alpha +\pi )(1-a)}{2}}\right)\cos \left({\frac {-\pi (1-a)}{2}}\right)+\left(-1\right)^{a}-1\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b38e31154066b86ccbe94081d29b7ac54c018362)
![{\displaystyle B_{a}={\frac {I{\sqrt {2}}}{\pi }}{\begin{Bmatrix}{\frac {1}{1+a}}\left[\left(-1\right)^{a}-1\right]\\+{\frac {1}{1-a}}\left[\left(-1\right)^{a}-1\right]\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f47a71e222dee1eeb988e6773c02b563e0c5ad6d)
![{\displaystyle ={\frac {1}{2\pi }}\int _{0}^{2\pi }\left[p(t)\right]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74c9cbc74d8aa9dca82cd6901a00d8754e69b800)
![{\displaystyle ={\frac {2}{2\pi }}\int _{0}^{\pi }\left[u(t)\times i(t)\right]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ddd33a1fb69bf72d7a429dccc34b8a016da916)
![{\displaystyle ={\frac {1}{\pi }}\int _{\alpha }^{\pi }\left[u(t)\times i(t)\right]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b134a8d420a601c16353523f7ab7b4e189c5842)
![{\displaystyle ={\frac {1}{\pi }}\int _{\alpha }^{\pi }\left[\left(U{\sqrt {2}}\sin \theta \right)\times \left(I{\sqrt {2}}\sin \theta \right)\right]d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/faa26b078128a621d4ce79cce737e1ee47b813f7)
![{\displaystyle ={\frac {2UI}{\pi }}\int _{\alpha }^{\pi }\left[\sin ^{2}\theta \right]d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a80559d3a25fa093f46a1abf337c9ed223f9dd82)
![{\displaystyle ={\frac {2UI}{\pi }}\int _{\alpha }^{\pi }\left[{\frac {1-\cos \left(2\theta \right)}{2}}\right]d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c855171c3315211a817ade86ac61e993be96f827)
![{\displaystyle ={\frac {UI}{\pi }}\left[\theta -{\frac {\sin \left(2\theta \right)}{2}}\right]_{\alpha }^{\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fec8c0c8aa8fc393457261a376932ec6d9ffecf)
![{\displaystyle ={\frac {UI}{\pi }}\left[\pi -{\frac {\sin \left(2\pi \right)}{2}}-\alpha -{\frac {\sin \left(2\alpha \right)}{2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54d0c9c085dfe1b289e2d795b2b51b7a338e3206)
![{\displaystyle P={\frac {UI}{\pi }}\left[\pi -\alpha +{\frac {\sin \left(2\alpha \right)}{2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3585461026b5f73ed89d808baebe3c99a8fa2e87)
![{\displaystyle \left[{\frac {\pi }{2}}-\alpha ;{\frac {\pi }{2}}+\alpha \right]+\left[{\frac {3\pi }{2}}-\alpha ;{\frac {3\pi }{2}}-\alpha \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be25d28301d6101ec0a089151fb746e75b497fff)
![{\displaystyle \left[0;{\frac {\pi }{2}}-\alpha \right]+\left[{\frac {\pi }{2}}+\alpha ;{\frac {3\pi }{2}}-\alpha \right]+\left[{\frac {3\pi }{2}}+\alpha ;2\pi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/256ad1f6f73c31ccedbe476ea59e79e65ff69d6c)