# If a and b are the max and min amplitude of amplitude modulated wave , write the expression for the modulation index

## Answers

**Answer:Modulation index for an AM wave:
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**The modulation index of an amplitude modulated wave is defined as the ratio of the amplitude of modulating signal (Em) to the amplitude of carrier wave (Ec) i.e.,
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**Amplitude modulation index, ma =begin mathsize 14px style straight E subscript straight m over straight E subscript straight c end style
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**For a modulated wave,
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**ma=begin mathsize 14px style fraction numerator straight E subscript max minus straight E subscript min over denominator straight E subscript max plus straight E subscript min end fraction end style
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**Em < Ec, to avoid distortion.
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**Modulation index for an AM wave for which the maximum amplitude is 'a' while the minimum amplitude id 'b':
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**Modulation index, begin mathsize 14px style straight a subscript straight m equals straight E subscript straight m over straight E subscript straight c end style ... (1)
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**Maximum amplitude of modulated wave, a = Ec + Em ... (2)
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**Mimimum amplitude of modulated wave, b = Ec - Em ... (3)
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**From eqn (2) and (3), we get,
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**begin mathsize 14px style straight E subscript straight c equals fraction numerator straight a plus straight b over denominator 2 end fraction space comma space space straight E subscript straight m equals fraction numerator straight a minus straight b over denominator 2 end fraction therefore From space left parenthesis 1 right parenthesis comma space we space get comma Modulation space index comma space straight a subscript straight m equals straight E subscript straight m over straight E subscript straight c equals fraction numerator begin display style fraction numerator left parenthesis straight a minus straight b right parenthesis over denominator 2 end fraction end style over denominator begin display style fraction numerator left parenthesis straight a plus straight b right parenthesis over denominator 2 end fraction end style end fraction equals fraction numerator left parenthesis straight a minus straight b right parenthesis over denominator left parenthesis straight a plus straight b right parenthesis end fraction end style**