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
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.
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.
The equvalent circuit of dx on power transmission line: Two equations on pure sine signal: Distributed Parameter Networks 4.
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.
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
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.
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.
From the previous picture we suppose that the current formula is: Thus: Wave impedance Distributed Parameter Networks 9.
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:
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.
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.
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.
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.
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.
Current reflection factor is: Voltage reflection factor: Distributed Parameter Networks 16. Reflection factor at Z
The direct and reflected waves on complex plain: A and B are complex numbers: thus: Distributed Parameter Networks 17.
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.
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.