Antennifysiikkaa - Revision history

Mol: /* Transmission line */

2025-08-23T11:41:29Z

Transmission line

← Older revision		Revision as of 14:41, 23 August 2025
Line 49:		Line 49:
	Dipole transmissionLine1.png\|thumb\|The (long) transmission line is modeled as <math>Z_0</math>. If the frequency (wavelength) of the source is too large (small) compared to dimensions of the system, it need to be considered in more detailed. See also the svg file.		Dipole transmissionLine1.png\|thumb\|The (long) transmission line is modeled as <math>Z_0</math>. If the frequency (wavelength) of the source is too large (small) compared to dimensions of the system, it need to be considered in more detailed. See also the svg file.
	Dipole transmissionLine2.png\|thumb\|The physical realizations of the transmission lines are usually coaxial cables, twisted cables or twin lead cables.		Dipole transmissionLine2.png\|thumb\|The physical realizations of the transmission lines are usually coaxial cables, twisted cables or twin lead cables.
			Dipole differential transmissionline.png\|The system is analyzed as being differential short pieces. The conductance <math>G</math> is the conductance between the two wires, which exists because of the high frequency.
	</gallery>		</gallery>

Mol: /* Transmission line */

2025-08-23T11:22:47Z

Transmission line

← Older revision		Revision as of 14:22, 23 August 2025
Line 48:		Line 48:
	<gallery>		<gallery>
	Dipole transmissionLine1.png\|thumb\|The (long) transmission line is modeled as <math>Z_0</math>. If the frequency (wavelength) of the source is too large (small) compared to dimensions of the system, it need to be considered in more detailed. See also the svg file.		Dipole transmissionLine1.png\|thumb\|The (long) transmission line is modeled as <math>Z_0</math>. If the frequency (wavelength) of the source is too large (small) compared to dimensions of the system, it need to be considered in more detailed. See also the svg file.
			Dipole transmissionLine2.png\|thumb\|The physical realizations of the transmission lines are usually coaxial cables, twisted cables or twin lead cables.
	</gallery>		</gallery>

Mol: /* Transmission line */

2025-08-23T11:20:25Z

Transmission line

← Older revision		Revision as of 14:20, 23 August 2025
Line 45:		Line 45:

	=== Transmission line ===		=== Transmission line ===

			<gallery>
			Dipole transmissionLine1.png\|thumb\|The (long) transmission line is modeled as <math>Z_0</math>. If the frequency (wavelength) of the source is too large (small) compared to dimensions of the system, it need to be considered in more detailed. See also the svg file.
			</gallery>

	Transmission line		Transmission line

Mol: /* Transmission line */

2025-08-18T19:27:07Z

Transmission line

← Older revision		Revision as of 22:27, 18 August 2025
Line 81:		Line 81:
	= \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)}		= \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)}
	\left(		\left(
	1 - \Gamma e^{\phi_z + ~~\imath(~~2\gamma z)}		1 - \Gamma e^{\imath(\phi_z +2\gamma z)}
	\right)		\right)
	</math>		</math>

Mol: /* Transmission line */

2025-08-18T19:26:45Z

Transmission line

← Older revision		Revision as of 22:26, 18 August 2025
Line 74:		Line 74:

	<math>\frac{v^-}{v^+} = \Gamma e^{\imath \phi_z} = \frac{Z_L - Z_0}{Z_L + Z_0}</math>		<math>\frac{v^-}{v^+} = \Gamma e^{\imath \phi_z} = \frac{Z_L - Z_0}{Z_L + Z_0}</math>

			which gives

			<math>
			i(z,t) =
			= \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)}
			\left(
			1 - \Gamma e^{\phi_z + \imath(2\gamma z)}
			\right)
			</math>

Mol: /* Transmission line */

2025-08-18T19:24:41Z

Transmission line

← Older revision		Revision as of 22:24, 18 August 2025
Line 70:		Line 70:
	\right)		\right)
	</math>		</math>

			The fraction <math>\frac{v^-}{v^+}</math> is called reflection coefficient <math>\Gamma</math>

			<math>\frac{v^-}{v^+} = \Gamma e^{\imath \phi_z} = \frac{Z_L - Z_0}{Z_L + Z_0}</math>

Mol: /* Transmission line */

2025-08-18T19:21:49Z

Transmission line

← Older revision		Revision as of 22:21, 18 August 2025
Line 67:		Line 67:
	= \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)}		= \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)}
	\left(		\left(
	1 - \frac{v^-}{v^+}e^{\imath(-2\gamma z)}		1 - \frac{v^-}{v^+}e^{\imath(2\gamma z)}
	\right)		\right)
	</math>		</math>

Mol: /* Transmission line */

2025-08-18T19:20:32Z

Transmission line

← Older revision		Revision as of 22:20, 18 August 2025
Line 54:		Line 54:
	If a transmission line has a length greater than about 10% of a wavelength, then the line length will noticeably affect the circuit's impedance. The solutions to the above equations is the sum of forward and backward traveling (reflected) waves:		If a transmission line has a length greater than about 10% of a wavelength, then the line length will noticeably affect the circuit's impedance. The solutions to the above equations is the sum of forward and backward traveling (reflected) waves:
	<math>		<math>
	v(z,t) = v^+ e^{-\alpha z} e^{\imath \omega t - \gamma z} + v^- e^{\alpha z} e^{\imath \omega t +\gamma z}		v(z,t) = v^+ e^{-\alpha z} e^{\imath( \omega t - \gamma z )} + v^- e^{\alpha z} e^{\imath (\omega t +\gamma z) }
	</math>		</math>
	and if we assume that <math>\alpha=0</math> we have the telegraphers equations https://en.wikipedia.org/wiki/Telegrapher's_equations		and if we assume that <math>\alpha=0</math> we have the telegraphers equations https://en.wikipedia.org/wiki/Telegrapher's_equations

	<math>		<math>
	v(z,t) = v^+ e^{\imath \omega t - \gamma z} + v^- e^{\imath \omega t +\gamma z}		v(z,t) = v^+ e^{\imath (\omega t - \gamma z)} + v^- e^{\imath (\omega t +\gamma z)}
	</math>		</math>
	and a similar for <math>i=i(z,t)</math>. If we replace $i$ by Ohm law, we get		and a similar for <math>i=i(z,t)</math>. If we replace $i$ by Ohm law, we get

	<math>		<math>
	i(z,t) = \frac{v^+}{Z_0} e^{\imath \omega t - \gamma z} + \frac{v^-}{Z_0} e^{\imath \omega t +\gamma z}		i(z,t) = \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z) } + \frac{v^-}{Z_0} e^{\imath (\omega t +\gamma z)}
	= \frac{v^+}{Z_0} e^{\imath \omega t - \gamma z}		= \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)}
	\left(		\left(
	1 - \frac{v^-}{v^+}e^{\imath(-2\gamma z)}		1 - \frac{v^-}{v^+}e^{\imath(-2\gamma z)}

Mol: /* Transmission line */

2025-08-18T19:18:45Z

Transmission line

← Older revision		Revision as of 22:18, 18 August 2025
Line 61:		Line 61:
	v(z,t) = v^+ e^{\imath \omega t - \gamma z} + v^- e^{\imath \omega t +\gamma z}		v(z,t) = v^+ e^{\imath \omega t - \gamma z} + v^- e^{\imath \omega t +\gamma z}
	</math>		</math>
	and a similar for <math>i=i(z,t)</math>		and a similar for <math>i=i(z,t)</math>. If we replace $i$ by Ohm law, we get

	<math>		<math>
Line 67:		Line 67:
	= \frac{v^+}{Z_0} e^{\imath \omega t - \gamma z}		= \frac{v^+}{Z_0} e^{\imath \omega t - \gamma z}
	\left(		\left(
	1 - \frac{v^-}{v^-}e^{\imath(-2\gamma z)}		1 - \frac{v^-}{v^+}e^{\imath(-2\gamma z)}
	\right)		\right)
	</math>		</math>

Mol: /* Transmission line */

2025-08-18T19:17:45Z

Transmission line

← Older revision		Revision as of 22:17, 18 August 2025
Line 57:		Line 57:
	</math>		</math>
	and if we assume that <math>\alpha=0</math> we have the telegraphers equations https://en.wikipedia.org/wiki/Telegrapher's_equations		and if we assume that <math>\alpha=0</math> we have the telegraphers equations https://en.wikipedia.org/wiki/Telegrapher's_equations

	<math>		<math>
	v(z,t) = v^+ e^{\imath \omega t - \gamma z} + v^- e^{\imath \omega t +\gamma z}		v(z,t) = v^+ e^{\imath \omega t - \gamma z} + v^- e^{\imath \omega t +\gamma z}
	</math>		</math>
	and a similar for <math>i=i(z,t)</math>. The characteristic impedance is		and a similar for <math>i=i(z,t)</math>

			<math>
			i(z,t) = \frac{v^+}{Z_0} e^{\imath \omega t - \gamma z} + \frac{v^-}{Z_0} e^{\imath \omega t +\gamma z}
			= \frac{v^+}{Z_0} e^{\imath \omega t - \gamma z}
			\left(
			1 - \frac{v^-}{v^-}e^{\imath(-2\gamma z)}
			\right)
			</math>


			The characteristic impedance is

	Thus we have		Thus we have
Line 67:		Line 79:
	</math>		</math>
	and similar for the current. The constant <math>\gamma^2 = (R+\imath \omega L)(G+\imath \omega C)</math>.		and similar for the current. The constant <math>\gamma^2 = (R+\imath \omega L)(G+\imath \omega C)</math>.


	<math>		<math>