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Simplification of complex expression

  1. Nov 9, 2017 #1
    1. The problem statement, all variables and given/known data

    For the expression:

    $$E=\frac{E_{0}}{2}\left(\exp\left[\frac{j\pi V}{2V_{\pi}}\right]+j\exp\left[-\frac{j\pi V}{2V_{\pi}}\right]\right)$$

    I want to show that if ##V=m(t)-\frac{V_{\pi}}{2}##, then ##|E|^2## can be written as:

    $$|E|^2=\frac{E^2_{0}}{2}\left(1-\cos\left(\frac{\pi m(t)}{V_{\pi}}\right)\right). \tag{1}$$

    Note: here ##j^2=-1##.

    2. Relevant equations


    3. The attempt at a solution

    Substituting:

    $$E(t)=\frac{E_{0}}{2}\left(\exp\left[\frac{j\pi}{2V_{\pi}}\left(m(t)-\frac{V_{\pi}}{2}\right)\right]+j\exp\left[-\frac{j\pi}{2V_{\pi}}\left(m(t)-\frac{V_{\pi}}{2}\right)\right]\right)$$

    $$=\frac{E_{0}}{2}\left(\exp\left[j\left(\frac{\pi}{2V_{\pi}}m(t)-\frac{\pi}{4}\right)\right]+j\exp\left[-j\left(\frac{\pi}{2V_{\pi}}m(t)-\frac{\pi}{4}\right)\right]\right)$$

    Multiplying by the complex conjugate:

    ##|E(2)|^{2}=\left(\frac{E_{0}}{2}\right)^{2}\left(\exp\left[j\left(\frac{\pi}{2V_{\pi}}m(t)-\frac{\pi}{4}\right)\right]+j\exp\left[-j\left(\frac{\pi}{2V_{\pi}}m(t)-\frac{\pi}{4}\right)\right]\right).\left(\exp\left[j\left(\frac{\pi}{2V_{\pi}}m(t)-\frac{\pi}{4}\right)\right]-j\exp\left[-j\left(\frac{\pi}{2V_{\pi}}m(t)-\frac{\pi}{4}\right)\right]\right)##

    $$|E(2)|^{2}=\underline{\left(\frac{E_{0}}{2}\right)^{2}\left(\exp\left[j\left(\frac{\pi m(t)}{V_{\pi}}-\frac{\pi}{2}\right)\right]+\exp\left[-j\left(\frac{\pi m(t)}{V_{\pi}}-\frac{\pi}{2}\right)\right]\right)}.$$

    Writing this explicitly in terms of trigonometric functions:

    ##=\left(\frac{E_{0}}{2}\right)^{2}\left[\left(\cos\left(\frac{\pi m(t)}{V_{\pi}}\right)+j\sin\left(\frac{\pi m(t)}{V_{\pi}}\right)\right)\left(\underbrace{\cos\left(-\frac{\pi}{2}\right)+j\sin\left(-\frac{\pi}{2}\right)}_{-j}\right)+\left(\cos\left(-\frac{\pi m(t)}{V_{\pi}}\right)+j\sin\left(-\frac{\pi m(t)}{V_{\pi}}\right)\right)\underbrace{\left(\cos\left(\frac{\pi}{2}\right)+j\sin\left(\frac{\pi}{2}\right)\right)}_{j}\right]##

    $$=\left(\frac{E_{0}}{2}\right)^{2}\left[-j\cos\left(\frac{\pi m(t)}{V_{\pi}}\right)+\sin\left(\frac{\pi m(t)}{V_{\pi}}\right)+j\cos\left(\frac{\pi m(t)}{V_{\pi}}\right)+\sin\left(\frac{\pi m(t)}{V_{\pi}}\right)\right]$$

    $$=\boxed{\frac{E_{0}^{2}}{2}\sin\left(\frac{\pi m(t)}{V_{\pi}}\right)}\stackrel{?}{=}\frac{E_{0}^{2}}{2}\left(1-\cos\left(\frac{\pi m(t)}{V_{\pi}}\right)\right)$$

    If we had sin2, then we might have been able to use the half-angle formula. But I am not sure what to do here.

    So, how can I get from ##\frac{E_{0}^{2}}{2}\sin\left(\frac{\pi m(t)}{V_{\pi}}\right)## to equation (1)? Did I make a mistake somewhere? :confused:

    Any help is greatly appreciated.
     
  2. jcsd
  3. Nov 9, 2017 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    There are sign errors here. What is the complex conjugate of ##e^{jx}##?
     
  4. Nov 9, 2017 #3

    Charles Link

    User Avatar
    Homework Helper

    Suggestion to make it much simpler: Let ## m'=\frac{\pi m}{2V_{\pi}} ##. Also write ## je^{jx} ## as ## e^{j (\pi/2)} e^{jx} ##. A few minutes of work including correctly taking complex conjugates should get you the result.
     
  5. Nov 9, 2017 #4
    I see. I made a mistake taking the complex conjugate of the expression. So I used:

    $$\overline{\exp\left[j\left(\frac{\pi m}{2V_{\pi}}-\frac{\pi}{4}\right)\right]+j\exp\left[-j\left(\frac{\pi m}{2V_{\pi}}-\frac{\pi}{4}\right)\right]}=\exp\left[-j\left(\frac{\pi m}{2V_{\pi}}-\frac{\pi}{4}\right)\right]-j\exp\left[j\left(\frac{\pi m}{2V_{\pi}}-\frac{\pi}{4}\right)\right]$$

    and I got the correct result. Thank you so much for the suggestions.
     
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