This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 72: 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 = ( −...
View
Full
Document
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.
 Spring '10
 Bedford

Click to edit the document details