Problem 2 Three lossless transmission-line sections are used to connect two iden

Question

Problem 2 Three lossless transmission-line sections are used to connect two identical loads ZL-100 to a time-harmonic generator of internal impedance z,- 300 and open circuit voltage V,-100 V. The transmission lines have characteristic impedances Zol = 50, Zo2-50Q, and Zo-300 , lengths 11 = /2, 12 3A/2, and 13 /4, and are connected as shown below. Provide answers to z. #3 c2 #2 Z, a) Zi, the input impedance of line 1 b) Zi2, the input impedance of line 2 c) Z3, the input impedance of line3. d) The power (in watts) delivered to each Z

Explanation / Answer

Given that tranmission lines are lossless. That is it is terminated with impedance equal to characteristic impedance. A transmission line of finite length (lossless or lossy) that is terminated at one end with an impedance equal to the characteristic impedance appears to the source like an infinitely long transmission line and produces no reflections.

A transmission line has a distributed inductance on each line and a distributed capacitance between the two conductors. We will consider the line to have zero series resistance and the insulator to have infinite resistance (a zero conductance or perfect insulator). We will consider a “Lossy” line later in section 12 on page 25. Define L to be the inductance/unit length and C to be the capacitance/unit length. Consider a transmission line to be a pair of conductors divided into a number of cells with each cell having a small inductance in one line and having small capacitance to the other line. In the limit of these cells being very small, they can represent a distributed inductance with distributed capacitance to the other conductor. Consider one such cell corresponding to the components between position x and position x + x along the transmission line. 2 EQUATIONS FOR A “LOSSLESS” TRANSMISSION LINE 4 The small series inductance is L.x and the small parallel capacitance is C.x. Define the voltage and current to the right on the left side to be V and I. Define the voltage and current to the right on the right side to be V + V and I + I. We now can get two equations. 1. The current increment I between the left and right ends of the cell is discharging the capacitance in the cell. The charge on the cell’s capacitance = capacitance x voltage = C.x.V and so the current leaving the capacitance to provide I must be; I = t(Charge) = t(C.x.V ) The minus sign is due to the current leaving the capacitor. I = C.x. V t I x = C. V t Note the minus sign. 2. The voltage increment V between the left and right ends of the cell is due to the changing current through the cell’s inductance. (Lenz’s Law) V = Inductance. I t = x.L. I t V x = L. I t . Now take the limit of the cell being made very small so that the inductance and capacitance are uniformly distributed. The two equations then become I x = C. V t Equation 1. V x = L. I t Equation 2. Remember that L and C are the inductance/unit length measured, in Henries/meter and are the capacitance/unit length measured in Farads/meter. Differentiate equation 2 with respect to the distance x. 2 EQUATIONS FOR A “LOSSLESS” TRANSMISSION LINE 5 x ( V x ) = L. x ( I t ) 2V x2 = L. x ( I t ) x and t are independent variables and so the order of the partials can be changed. 2V x2 = L. t( I x ) Now substitute for I x from equation 1 above 2V x2 = L. t(C. V t ) 2V x2 = LC. 2V t2 Equation 3 This is usually called the Transmission Line Differential Equation. Notes • L and C are NOT just the inductance and the capacitance. They are both measured per unit length. • The Transmission Line Differential Equation 3 above does NOT have a minus sign. The Transmission Line Differential Equation 3 above is a normal 1 dimensional wave equation and is very similar to other wave equations in physics. From experience with such wave equations, we can try the normal solution of the form V = V (s) where s is a new variable s = x + ut. Substituting this into the two sides of the Transmission Line Differential Equation 3 above we get the two sides being 2V x2 and 1 u2 . 2V t2 Thus the form V (x + ut) can satisfy the Transmission Line Differential Equation 3 if and only if 1 u2 = LC Equation 4. Both roots of this satisfy the Equation 3. u = ± 1 LC The two roots give slightly different solutions and so, since the equation 3 is linear, any linear combination of the two solutions is a valid solution. Define u as the positive root u = + 1 LC Equation 5.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.