Dwójniki

LP	Układ połączeń dwójnika, wykres wektorowy, charakterystyka	Impedancja	Zawada	Kąt fazowy φ, tg
1		$R$	$R$	$0$
2		$j\omega L$	$\omega L$	$+90^{\circ }$
3		${\frac {-j}{\omega C}}={\frac {1}{j\omega C}}$	${\frac {1}{\omega C}}$	$-90^{\circ }$
4		$R+j\omega L$	${\sqrt {R^{2}+(\omega L)^{2}}}$	$tg\varphi ={\frac {\omega L}{R}}$
5		$R-j{\frac {1}{\omega C}}$	${\sqrt {R^{2}+\left({\frac {1}{\omega L}}\right)^{2}}}$	$tg\varphi ={\frac {1}{\omega CR}}$
6		${\frac {R(\omega L)^{2}}{R^{2}+(\omega L)^{2}}}+j{\frac {R^{2}\omega L}{R^{2}+(\omega L)^{2}}}={\frac {R(\omega L)^{2}+jR^{2}\omega L}{R^{2}+(\omega L)^{2}}}$	${\frac {R\omega L}{\sqrt {R^{2}+(\omega L^{2})}}}={\frac {1}{\sqrt {\left({\cfrac {1}{R}}\right)^{2}+\left({\cfrac {1}{\omega L}}\right)^{2}}}}$	$tg\varphi ={\frac {R}{\omega L}}$
7		${\frac {R}{1+(R\omega C)^{2}}}-j{\frac {R^{2}\omega C}{1+(R\omega C)^{2}}}={\frac {R-jR^{2}\omega C}{1+(R\omega C)^{2}}}$	${\frac {R}{\sqrt {1+(\omega CR)^{2}}}}={\frac {1}{\sqrt {\left({\cfrac {1}{R}}\right)^{2}+(\omega C)^{2}}}}$	$tg\varphi =-R\omega C$
8		$j\left(\omega L={\frac {1}{\omega C}}\right)$	$\left|\omega L-{\frac {1}{\omega C}}\right|$ przy rezonansie = 0	$\varphi =+90^{\circ }$ przy rezonansie $\varphi =0$
9		$R+j\left(\omega -{\frac {1}{\omega C}}\right)$	${\sqrt {R^{2}+\left(\omega L-{\frac {1}{\omega C}}\right)^{2}}}$ przy rezonansie = R	$tg\varphi ={\frac {\omega L-{\cfrac {1}{\omega C}}}{R}}$ przy rezonansie $\varphi =0$
10		$R\left(\omega L-{\frac {1}{\omega C}}\right)^{2}+{\frac {R^{2}\left(\omega L-{\cfrac {1}{\omega C}}\right)}{R^{2}+\left(\omega L-{\cfrac {1}{\omega C}}\right)^{2}}}R^{2}+\left(\omega L-{\frac {1}{\omega C}}\right)^{2}$	${\frac {1}{\sqrt {\left({\cfrac {1}{R}}\right)^{2}+\left({\cfrac {1}{\omega L-{\cfrac {1}{\omega C}}}}\right)^{2}}}}$ przy rezonansie = 0	$tg\varphi ={\frac {R}{\omega L-{\cfrac {1}{\omega C}}}}$ przy rezonansie $=0$
11		$-j{\frac {1}{\omega C-{\cfrac {1}{\omega L}}}}={\frac {1}{j\left(\omega C-{\cfrac {1}{\omega L}}\right)}}$	$\left|{\frac {1}{\omega C-{\cfrac {1}{\omega L}}}}\right|$ przy rezonansie = ∞	$\varphi \neq 90^{\circ }$
12		${\frac {R}{1+R^{2}\left({\cfrac {1}{\omega L}}-\omega C\right)^{2}}}+j{\frac {R^{2}\left({\cfrac {1}{\omega L}}-\omega C\right)}{1+R^{2}\left({\cfrac {1}{\omega L}}-\omega C\right)^{2}}}={\frac {{\cfrac {1}{R}}+j\left({\cfrac {1}{\omega L}}-\omega C\right)}{\left({\cfrac {1}{R}}\right)^{2}+\left({\cfrac {1}{\omega L}}-\omega C\right)^{2}}}$	${\frac {R}{\sqrt {1+R^{2}\left({\cfrac {1}{\omega C}}-\omega C\right)^{2}}}}={\frac {1}{\sqrt {\left({\cfrac {1}{R}}\right)^{2}+\left({\cfrac {1}{\omega L}}-\omega C\right)^{2}}}}$ przy rezonansie = R	$tg\varphi =R\left({\frac {1}{\omega L}}-\omega C\right)$ przy rezonansie = 0
13		${\frac {R}{(R\omega C)^{2}+(\omega ^{2}LC-1^{2})}}-j\omega C{\frac {R^{2}-{\cfrac {L}{C}}+(\omega L)^{2}}{(R\omega C)^{2}+(\omega ^{2}LC-1)^{2}}}={\frac {R+j\omega (L-CR^{2}-\omega ^{2}L^{2}C)}{\omega ^{2}C^{2}\left[R^{2}+\left(\omega L-{\cfrac {1}{\omega C}}\right)^{2}\right]}}$	${\sqrt {\frac {R^{2}+(\omega L)^{2}}{(R\omega C)^{2}+(\omega ^{2}LC-1)^{2}}}}$ przy rezonansie: ${\frac {R}{CR}}={\frac {R^{2}+(\omega L)^{2}}{R}}$	$tg\varphi =-{\frac {\omega C(R^{2}+\omega ^{2}L^{2})-\omega L}{R}}=-{\frac {\omega C}{R}}\left(R^{2}+\omega ^{2}L^{2}-{\frac {L}{C}}\right)$ przy rezonansie $\varphi \approx 0$ jeżeli R jest małe
14		${\frac {R(\omega L)^{2}\cdot (\omega C)^{2}}{(R\omega C)^{2}+(\omega ^{2}LC-1)^{2}}}+j\omega L{\frac {(R\omega C)^{2}-(\omega ^{2}LC-1)^{2}}{(R\omega C)^{2}+(\omega ^{2}LC-1)^{2}}}$	$\omega L{\frac {1+(R\omega C)^{2}}{(R\omega C)^{2}+(\omega ^{2}LC-1)^{2}}}$ przy rezonansie: ${\frac {L}{CR}}$	$tg\varphi ={\frac {1}{R\omega L}}\left[\left(R^{2}{\frac {L}{C}}\right)+\left({\frac {1}{\omega C}}\right)^{2}\right]$ przy rezonansie $\varphi =0$ jeżeli R jest małe
15		$R+j{\frac {1}{{\cfrac {1}{\omega L}}-\omega C}}$	${\sqrt {R^{2}+\left({\frac {1}{\left({\cfrac {1}{\omega L}}-\omega C\right)}}\right)^{2}}}$	$tg\varphi ={\frac {1}{R\left({\cfrac {1}{\omega L}}-\omega C\right)}}$
16		${\frac {\left(\omega L_{1}-{\cfrac {1}{\omega C}}\right)\omega L_{2}}{\omega L_{1}-{\cfrac {1}{\omega C}}+\omega L_{2}}}$	$\left|{\frac {\left(\omega L_{1}-{\cfrac {1}{\omega C}}\right)\omega L_{2}}{\omega L_{1}-{\cfrac {1}{\omega C}}+\omega L_{2}}}\right|$	$\varphi =\pm 90^{\circ }$
17		$-j{\frac {\left(\omega L-{\cfrac {1}{\omega C_{1}}}\right){\cfrac {1}{\omega C_{2}}}}{\omega L-{\cfrac {1}{\omega C_{1}}}-{\cfrac {1}{\omega C_{2}}}}}$	$\left|{\frac {\left(\omega L-{\cfrac {1}{\omega C_{1}}}\right){\cfrac {1}{\omega C_{2}}}}{\omega L-{\cfrac {1}{\omega C_{1}}}-{\cfrac {1}{\omega C_{2}}}}}\right|$	$\varphi =\pm 90^{\circ }$
18		$j\left(\omega L-{\frac {1}{\omega C_{1}}}+{\cfrac {1}{{\cfrac {1}{\omega L_{2}}}-\omega C_{2}}}\right)$	$\left|\left(\omega L_{1}-{\cfrac {1}{\omega C_{1}}}+{\frac {1}{{\cfrac {1}{\omega L_{2}}}-\omega C_{2}}}\right)\right|$	$\varphi =\pm 90^{\circ }$