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Telecommunication Networks Distributed Parameter Networks

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Distributed Parameter Networks 1. The electric and magnetic power distribute homogeneously along the wire, but the current changes in time. d Lecher wire koaxial cable The electromagnetic wave passes on the wire. EM waves theory, but if d<<, we can examine the topic with quasi- stationary method. Generator Consumer

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Induction law for ABCD loop: Induction law for ABCD loop: The flux is proportional to the current: = Ldx i Loop equation for dx : Distributed Parameter Networks 2.

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Rearranging: On dx section charches accumulate, or accumulated charges disappear. This increases the difference between input and output current. On dx the change of charges in time: is the displacement current eltolási between the two wires Continuity equation : where u(x,t)Gdx is the leakage current Distributed Parameter Networks 3.

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The equvalent circuit of dx on power transmission line: Two equations on pure sine signal: Distributed Parameter Networks 4.

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The 2. equation differentiated by x and we put du/dx from 1. equation into 2. equation we get: Telegram equations Distributed Parameter Networks 5.

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We try to get a formula like this: substituting back : Considering a pozitive , we get: – propagation coefficient Distributed Parameter Networks 6. Attenuation factor Phase factor

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If a is considered This means a wave passing toward negative x direction on rate v The length of period is: Formula for wavelength and phase factor Wave passing on positive x is: Distributed Parameter Networks 7.

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The time dependence of the voltage measured in a given position on the wire is pure sinusoidal. At dx distance the amplitude decreases and the phase changes. Substituting voltage wave into Substituting voltage wave into equation, we get: equation, we get: Distributed Parameter Networks 8.

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From the previous picture we suppose that the current formula is: Thus: Wave impedance Distributed Parameter Networks 9.

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Substituting the negative direction wave we get: The general solution of the voltage wave is: or: Distributed Parameter Networks 10. The general solution of the current wave is:

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thus Thomson formula: where [L]=Henry and [C]=Farad [L]=Henry and [C]=Farad If we decrease the value of L and C towards 0, v does not reach c (speed of light). On idle wire the waves pass by c rate. Phase factor Ideal wire: rate és Distributed Parameter Networks 11.

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If we increase capacities with geometrical sizes the inductivity decreases and vice versa. Thus it is impossible to make a construction which can operate on higher rate than c. Two wires CapacityInduction Wave resistance Distributed Parameter Networks 12.

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In idle case the rate and wave impedance do not depend on frequency. If rate would depend on frequency, distortion would occure for example on pulse signal, because the spectra is: 0, 3 0, 5 0, … etc. and we would get different phase delays on different frequencies. On the cables with big attenuation the big delay time causes big problems: For idle wire: Distributed Parameter Networks 13.

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Let’s say: U 0 + =A and U 0 - =B and x=-l. We do not consider the time dependence case, thus: We examine the transmission line with Z at the end: IhIh IrIr UhUh UrUr Distributed Parameter Networks 14.

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Adding and substracting: Let’s count the value of A and B with U Z and I Z : If =0, it means U =U Z and I =I Z Distributed Parameter Networks 15.

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Current reflection factor is: Voltage reflection factor: Distributed Parameter Networks 16. Reflection factor at Z

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The direct and reflected waves on complex plain: A and B are complex numbers: thus: Distributed Parameter Networks 17.

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Where the two vectors are in phase voltage-maximum Where the phase difference is 180 o voltage-minimum If Z Z 0 along the line standing waves are self created If Z=Z 0, there are not standing waves Distributed Parameter Networks 18.

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Voltage standing wave: From the figure: Distance between two minimum or maximum values is /2 Distance between two minimum or maximum values is /2 Distance between maximum and minimum values is /4 Distance between maximum and minimum values is /4 or: Distributed Parameter Networks 19.

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