الأسبوع الثانى

1. الأسبوع الثانى 1

2. Transmission Line Theory 2

3. Introduction • Transmission line theory bridges the gap between field analysis and basic circuit theory, which is important in the analysis of microwave circuits and devices. • TL may be a considerable fraction of a wavelength, in size. • Thus a TL is a distributed parameter network, where voltages and currents can vary in magnitude and phase over its length.

5. The Lumped-element Circuit Model for Transmission Line There are magnetic field around this conductors and electric field between them. We have a conductance along the conductor. The dielectric between the two conductor produce a capacitance. These capacitances and conductance are not located at specific point but it distributed along the line; so this parameters are called the distributed parameters of the TL. Each inductance has a small resistance. There are small leakage current flow in the dielectric through parallel conductance.

6. The analysis used is the distributed circuit analysis. Studying the reactance effect as the transient time effect (the same phenomena). So in Tl; the view of the reactive components is very important. Because the resistance is very small; then the reactance and capacitance are very important wthen transfer the power through TL.

7. Presence of Electric and Magnetic Fields • Both Electric and Magnetic fields are present in the transmission lines • These fields are perpendicular to each other and to the direction of wave • Electric field is established by a potential difference between two conductors. • Implies equivalent circuit model must contain capacitor. • Magnetic field induced by current flowing on the line • Implies equivalent circuit model must contain inductor.

8. TL is schematically represented as a two-wire line (a). • The piece of line of infinitesimal length Δz can be modeled as a lumped-element circuit (b) , Where R, L, G, and C are per-unit-length quantities. (a) (b) R,L,C and G are called the primary constants of the line.

9. R=series resistance per unit length, for both conductors Ω/m. L=series inductance per unit length, for both conductors, in H/m. G=shunt conductance per unit length, in Ʊ/m. C=shunt capacitance per unit length, in F/m. • The series inductance L represents the total self-inductance of the two conductors, and the shunt capacitance C is due to the close proximity of the two conductors. • The series resistance R represents the resistance due to the finite conductivity of the individual conductors, and the shunt conductance G is due to dielectric loss in the material between the conductors. • R and G, therefore, represent loss. • A finite length of transmission line can be viewed as a cascade of sections of the form shown in fig.(b). • From the circuit of fig.(b), Kirchhoff’s law can be applied as:

10. The Lumped-element Circuit Model for Transmission Line Divide the T L into small elements. Distributed Elements This section represented by lumped elements as:

11. Each section of TLcan be represented by lumped circuit. When V is applied, some current I will flow in the circuit. ΔV ΔI

12. The impedance of this circuit is RΔx+jωlΔx. The changes in voltage and current can be written as: low freq. rang deals with the algebraic equations. At high freq. rang, the differential equations are used

13. Represent the characteristics of TL Because it contain the primary parameters of TL

14. Represent the characteristics of TL

15. All quantities have harmonic time variation Where Is the propagation constant

16. All quantities have harmonic time variation

17. At any location and any time; all of these voltages and currents have vary sinusoidal with a function of time as ejwt Where w is the angular frequency. We will solve these econd order diff. equations to get the variation in V and I Where ϒis constant for a specific TL then the solution of these equations can be written as:

18. r.m.s value Then., the instantaneous values of voltage and current can be determined by multiplied the solution of equations with jwt as: Instantaneous value ϒ is a complex value

19. Taken the complexity of ϒ into consideration Phase Amplitude dist. time

20. Voltage as a function of x for different time When t increased the wave shifted to +evx direction This is the traveling wave phenomena

21. For the second term of the solution - - This means that voltage and current will exist in the form of waves in the TL. In high frequency circuit analysis, the voltage and current visualize in the form of waves on the electrical circuits

22. Replace x by z Any two conductor system can be represented as

23. And Kirchhoff’s current law leads to: Dividing by Δz and taking the limits asΔz----0 gives the following differential equations Theseare the time domain form of the transmission line equations, also known as the telegrapher equations. Forsinusoidal steady-state condition, with cosine-based phasors, these Eqns can be simplified to:

24. Wave Propagation on a Transmission Line The two equations can be solved simultaneously to give wave equations for V(z) and I(z) : Traveling wave solutions can be found as where Is the complex propagation constant, which is a function of frequency. and The e−γz term represents wave propagation in the +z direction, and the eγz term represents wave propagation in the−z direction.

25. Applying to the voltage of gives the current on the line: Comparison with shows that a characteristic impedance, Z0, can be defined as

26. to relate the voltage and current on the line as follows: Then can be rewritten in the following form: