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The reflection coefficient is defined as the ratio of reflected voltage to incident voltage: = Using this ratio in Eq 314 gives = ZL Zo ZL + Zo [316] Vref Vinc [315]

When the angle that a vector makes with the x-axis is larger than 90 , the sign of one or more components is negative, as shown in Figure 5-4



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You might be wondering why you need to resolve vectors into their components The answer C Cy is that doing this often makes adding vectors B By together much easier mathematically Two or more vectors (A, B, C, etc) may be added by first resolvA Ay ing each vector into its x- and y-components The x-components are added to form the x-component Ax Cx Bx of the resultant: Rx Ax Bx Cx Similarly, the y-components are added to form the y-component of the resultant: Ry Ay By Cy This process is illustrated graphically in Figure 5-5 Because Rx and Ry are at a right angle (90 ), the magnitude of the resultant vector can be calculated using the Pythagorean theorem, R2 Rx2 Ry2 To find the angle or direction of the resultant, recall that the tangent of the angle that the vector makes with the x-axis is given by the following





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R (R )

6 Given:

Figure 5-5 Rx is the sum of the x-components of A, B, and C Ry is the sum of the y-components The vector sum of Rx and Ry is the vector sum of A, B, and C

The angle of the resultant vector is equal to the inverse tangent of the quotient of the y-component divided by the x-component of the resultant vector

You can find the angle by using the tan 1 key on your calculator Note that when tan 0, most calculators give the angle between 0 and 90 , and when tan 0, the angle is reported to be between 0 and 90

Example 3-3 A 50- transmission line is connected to a 30- resistive load Calculate the reflection coefficient Solution: ZL Zo ZL + Zo (50 ) (30 ) (50 ) + (30 ) 20 = 025 80

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Use the following technique to solve problems for which you need to add or subtract vectors 1 Choose a coordinate system 2 Resolve the vectors into their x-components using Ax A cos , and their y-components using Ay A sin , where is the angle measured counterclockwise from the positive x-axis 3 Add or subtract the component vectors in the x-direction 4 Add or subtract the component vectors in the y-direction 5 Use the Pythagorean theorem, R Rx2 Ry2, to find the magnitude of the resultant vector 6 Use vector tan

3 class Stereo { void makeNoise() { assert true; } } 4 public class BoomBox2 extends Stereo { 5 public static void main(String[] args) { 6 new BoomBox2()go(args); 7 } 8 void go(String[] args) { 9 if(argslength > 0) makeNoise(); 10 if(args[0]equals("x")) Systemoutprint("x "); 11 if(args[0] == "x") Systemoutprintln("x2 "); 12 } }

Finding Your Way Home A GPS receiver indicates that your home is 150 km and 400 north of west, but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home

Draw the resultant vector, R, from your original location to your home Draw A, the known vector, and draw B, the unknown vector Known:

Unknown:

And (if the code compiles), the invocation:

Find the components of R Rx R cos (150 km) cos 1400 115 km Ry R sin (150 km) sin 1400 964 km Because A is due north, Ax

Transmission line responses 75 Example 3-4 In Example 3-3, the incident voltage is 3 V rms Calculate the reflected voltage Solution: If Vref = Vinc then Vref = Vinc = (025) (3 V) = 075 V The phase of the reflected signal is determined by the relationship of load impedance and transmission line characteristic impedance For resistive loads (Z = R + j0): if the ratio ZL/Zo is 10, then there is no reflection; if ZL/Zo is less than 10, then the reflected signal is 180 out of phase with the incident signal; if the ratio ZL/Zo is greater than 10 then the reflected signal is in phase with the incident signal In summary: Angle of reflection No reflection 180 0

1800 400

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