Chap7_Practice

# Chap7_Practice - Problem 7.2 Express the current waveform i...

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Unformatted text preview: Problem 7.2 Express the current waveform i ( t ) = − . 2cos ( 6 π × 10 9 t + 60 ◦ ) mA in standard cosine form and then determine the following: (a) Its amplitude, frequency, and phase angle. (b) i ( t ) at t = . 1 ns. Solution: i ( t ) = − . 2cos ( 6 π × 10 9 t + 60 ◦ ) (mA) = . 2cos ( 6 π × 10 9 t + 60 ◦ − 180 ◦ ) (mA) = . 2cos ( 6 π × 10 9 t − 120 ◦ ) (mA) . (a) amplitude = . 2 mA; f = 3 × 10 9 Hz = 3 GHz; φ = − 120 ◦ . (b) i ( . 1 ns ) = . 2cos ( 6 π × 10 9 × . 1 × 10 − 9 − 120 ◦ ) (mA) = . 2cos ( . 6 π − 120 ◦ ) (mA) = . 2cos ( 108 ◦ − 120 ◦ ) (mA) = . 196 mA . Problem 7.8 A multiplier circuit has two input ports, designated v 1 and v 2 , and one output port whose voltage v out is equal to the product of v 1 and v 2 . Assume v 1 = 10cos2 π f 1 t V , v 2 = 10cos2 π f 2 t V , (a) Obtain an expression for v out in terms of the sum and difference frequencies, f s = f 1 + f 2 and f d = f 1 − f 2 . (b) Plot its waveform over the time interval [0, 2 s], given that f 1 = 3 Hz and f 2 = 2 Hz. Solution: (a) v out ( t ) = v 1 ( t ) · v 2 ( t ) = ( 10cos2 π f 1 t )( 10cos2 π f 2 t ) = 100cos2 π f 1 t cos2 π f 2 t . Using the identity cos x cos y = 1 2 [ cos ( x + y )+ cos ( x − y )] , v out ( t ) = 100 2 { cos [ 2 π ( f 1 + f 2 ) t ]+ cos [ 2 π ( f 1 − f 2 ) t ] } = 50 ( cos2 π f s t + cos2 π f d t ) . (b) For f 1 = 3 Hz and f 2 = 2 Hz, v out ( t ) = 50 ( cos10 π t + cos2 π t ) (V) . The plot of v out ( t ) is shown in Fig. P7.8. 4 V 2 V- 2 V- 4 V 1 0.8 0.6 0.4 0.2 v 1 v out t (s) v 2 Figure P7.8: Plots of v 1 ( t ) , v 2 ( t ) and v out ( t ) . Section 7-2: Complex Numbers Problem 7.10 Express the following complex numbers in polar form: (a) z 1 = 3 + j 4 (b) z 2 = − 6 + j 8 (c) z 3 = − 6 − j 4 (d) z 4 = j 2 (e) z 5 = ( 2 + j ) 2 (f) z 6 = ( 3 − j 2 ) 3 (g) z 7 = ( −...
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## This note was uploaded on 02/21/2011 for the course MSE 231 taught by Professor Bedford during the Spring '10 term at University of Michigan.

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Chap7_Practice - Problem 7.2 Express the current waveform i...

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