Special Cases of Source and Load Impedance

Transcription

1 3/6/1 Special Cases of Source and oad present 1/ Special Cases of Source and oad Impedance et s look at specific cases of: I z 1.. and, V V z and then determe how they affect: z z 1. V. and abs.

2 3/6/1 Special Cases of Source and oad present / The first special case: the matched load I z j V V z z z Of course for this case where the load is matched to the transmission le we fd that and thus. As a result, the complex amplitude of the plus-wave simplifies to: V V e V e jβ jβ 1 1

3 3/6/1 Special Cases of Source and oad present 3/ The absorbed power is just the cident power ikewise, the absorbed/delivered power is simply that of the cident wave: abs V V 1 as the matched condition causes the reflected power to be zero ( )! Now, reconiz that the put impedance of the transmission le for this case will likewise be, we can alternatively express the absorbed power (without determ V!) : 1 V 1 V V Re abs

4 3/6/1 Special Cases of Source and oad present 4/ The nd special case: the conjuate match For this case, we fd the load takes on whatever value required to make. The put impedance is a conjuate match to the source impedance. V This is of course is a very important case! First, us the fact that: We can show that (trust me!) that the plus-wave amplitude simplifies to: V V e jβ 4Re

5 3/6/1 Special Cases of Source and oad present 5/ Yawn: the load absorbs the available power Not a particularly terest result, but now let s look at the absorbed power. abs 1 V 1 V 1 V Re 1 1 V 4Re V 8 R avl Re Re Re The absorbed power is of course the available power of the source. Sce the put impedance is a conjuate match to the source, the source is deliver enery at the hihest rate it possibly can!

6 3/6/1 Special Cases of Source and oad present 6/ But, the load still reflects power! Q: But if the power delivered is at a maximum, so too is the power absorbed by the load. I.E.: abs del avl If the power absorbed from the load is at a maximum, then the power reflected from the load must be at a mimum. The load must be absorb all the cident power; riht? A: You miht thk so and many eneers fact do thk so. To see why, consider the load But those eneers are correct! that mimizes the reflected power. We know this load must have one specific value the matched load.

7 3/6/1 Special Cases of Source and oad present 7/ If the load is then so is But, if the load is matched to the transmission le (i.e.,, then the put impedance will likewise be equal to :, = And, enerally speak, an put impedance of will not be equal to the complex conjuate of the source impedance:!!!! V del avl

8 3/6/1 Special Cases of Source and oad present 8/ ook closer at this result Recall that we just determed that for, the absorbed power is: abs V It can be shown that this value is less than the available power of the source: V V 8 R avl Q: Huh!? This makes no sense! After all, just look at the expression for absorbed power: abs V 1 Clearly, this value is maximized when (i.e., when )!!!

9 3/6/1 Special Cases of Source and oad present 9/ Remember, the load affects V + A: et s look closer at this result: abs V 1 Remember, we can rewrite this to show that the absorbed power is simply the difference between the power of the cident and reflected waves: V V V V V abs 1 abs

10 3/6/1 Special Cases of Source and oad present 1/ Absorbed power is maximized if the difference between cident and reflected is maximized! Of course, it is true that the load impedance affects the mus-wave amplitude V, and thus affects likewise the reflected wave power. Remember however, that the value of likewise affects the plus-wave amplitude V, and thus affects likewise the cident wave power! * Thus, the value of that mimizes will not enerally maximize! * ikewise the value of that maximizes will not enerally mimize. * Instead, the value of that maximizes the absorbed power abs is, by defition, the value that maximizes the difference. We fd that this ideal impedance! is the value that results the ideal case of

11 3/6/1 Special Cases of Source and oad present 11/ The third special case: the matched source I z j V V z For this case, we fd that V simplifies reatly: z z V V e V e V e j j j 1 V e j ook at what this says!

12 3/6/1 Special Cases of Source and oad present 1/ This is the answer reardless of the load! It says that the cident wave this case is dependent of the load attached at the other end (the value is nowhere to be found)! Thus, for the one case wave., we fact can consider V z as be the source And then the reflected wave V z is the causal result of this stimulus. Specifically, for this case we can write directly the plus-wave without know anyth about the load : j βz 1 V jβ j βz jβ z V z V e V e e e

13 3/6/1 Special Cases of Source and oad present 13/ This blue box equation will come quite handy! Therefore, the value at the load (i.e., z ) is: V jβ V z e And the value of the plus-wave at the source (i.e., z ) is: V z V This last result is very important. Remember this result, it will come quite handy later the course!

14 3/6/1 Special Cases of Source and oad present 14/ All I can say is: wow! Now consider another really important result. Recall the power associated with the plus-wave is: V Insert the simplified expression for V (i.e., when ): j V Ve V 8 8 Wow! Q: I don t see why this is wow. Am I miss someth? A: Apparently you are. ook closer at the above result.

15 3/6/1 Special Cases of Source and oad present 15/ Do I have to expla everyth? Remember, this result is for the special case where the source impedance is a real value, numerically equal to : j Therefore: R Re The source resistance is numerically equal to transmission le characteristic impedance. Thus, the power of the plus-wave can be alternatively written as: V V Wow! 8 8R Q: This result looks vauely familiar; haven t we seen this before? A: Sih. This result is the available power of the source!!!!!!!

16 3/6/1 Special Cases of Source and oad present 16/ The cident power is the available power (wow)! Therefore: V V avl Wow! 8 8R For the special case j, the power associated with the plus- wave (i.e., the cident wave) is equal to the available power avl of the source. Q: Wow! Does this mean that the power absorbed by the load is equal to the available power avl of the source? A: Absolutely not!

17 3/6/1 Special Cases of Source and oad present 17/ But some of this cident power is reflected (Doh!) Remember, only if does the power absorbed by the load equal the available power. * abs avl * Mak j causes the power of the cident wave to be equal to the available power. * But, the value of the load impedance determes how much of that cident power is absorbed and how much ets reflected.

18 3/6/1 Special Cases of Source and oad present 18/ Q: Hey, that s riht! The really special case! If the load is matched (i.e., j ), then wouldn t none of the cident power be reflected (i.e., ). And so wouldn t all the cident (i.e., available!) power be absorbed by the load? abs avl avl A: That s exactly correct! Q: But you said earlier that mimiz the reflected power would not result maximiz the absorbed power. Aren t you contradict yourself? A: Not at all.

19 3/6/1 Special Cases of Source and oad present 19/ A conjuate match! Generally speak, mimiz the reflected power (i.e., ) will not result maximiz the absorbed power. However, we are consider the special case where also! I z V V z z z Thus, maximum power transfer will occur if there is a conjuate match. Note for this special case, mean that put impedance must be equal to for maximum power delivery. Of course, the put impedance matched (i.e., )! will be equal to when the load impedance is

20 3/6/1 Special Cases of Source and oad present / ee-zeros everywhere Thus, many ways, the case: (i.e., both source and load impedances are numerically equal to ) is ideal. I z V V z z z et s summarize this ideal case : * A conjuate match occurs ( ), and so all the available power from the source is absorbed by the load: del avl abs

21 3/6/1 Special Cases of Source and oad present 1/ It s just so simple; you hardly deserve credit for this class * The cident (plus) wave is dependent of the load impedance: V jβz V z e V z e V z V jβ * And the power associated with this cident wave is equal to the available power of the source: V avl 8 V * The reflected (mus) wave is zero: V z * So of course, the power associated with the reflected wave is likewise zero.

22 3/6/1 Special Cases of Source and oad present / oad AND source is matched it s a very ood th * This aa allows us to verify that all available power is absorbed by the load: abs avl avl * Fally, the total current and voltae simplifies nicely: V V jβ z V z V z jβ z V z V z V z e I z e I z V V z z z