(Solved): 1. For the following signals, find the period, frequency, amplitude, time shift, and phase angle. a ...

1. For the following signals, find the period, frequency, amplitude, time shift, and phase angle. a) \( v_{1}(t)=200 \cos (360 \pi t)-200 \sin (360 \pi t)(\mathrm{V}) \) b) \( i_{1}(t)=-7 \cos (500 t)+2 \sin (500 t)(\mathrm{A}) \) 2. Write an expression for the sinusoid shown below. What are the phase angle and time shift of the waveform? 3. The following sinusoids can be written in the form \( a \cos (\omega t)+b \sin (\alpha t) \), where \( a \) and \( b \) are the Fourier coefficients. Find the Fourier coefficients, cyclic frequency and radian frequency of the following sinusoids: a) \( v_{1}(t)=80 \cos \left(120 \pi t+36.87^{\circ}\right)(\mathrm{V}) \) b) \( v_{2}(t)=30 \cos \left(15000 t-135^{\circ}\right)(\mathrm{V}) \) 4. Transform the following sinusoids into phasor form and draw the phasor diagram. Use the additive property of phasors to find \( v_{1}(t)+v_{2}(t) \). a) \( v_{1}(t)=-80 \sin (\omega t) \) (V) b) \( v_{2}(t)=50 \cos \left(\omega t-30^{\circ}\right)(\mathrm{V}) \) 5. Consider the following phasors: \[ \begin{array}{c} \mathbf{V}_{1}=100 e^{-j 60^{*}}(\mathrm{~V}), \omega=10^{3}(\mathrm{rad} / \mathrm{s}) \\ \mathbf{V}_{2}=120 e^{\mu 4^{*}}(\mathrm{~V}), \omega=10^{3}(\mathrm{rad} / \mathrm{s}) \\ \mathbf{I}_{1}=240 e^{-266^{*}}(\mathrm{~mA}), \omega=2 \pi \times 10^{4}(\mathrm{rad} / \mathrm{s}) \\ \mathbf{I}_{2}=-150 e^{-j 45^{*}}(\mathrm{~mA}), \omega=2 \pi \times 10^{4}(\mathrm{rad} / \mathrm{s}) \end{array} \] a) Find the waveform \( v_{3}(t)=v_{1}(t)-v_{2}(t) \) b) Find the waveform \( i_{3}(t)=2 i_{1}(t)+3 i_{2}(t) \) 6. For the following circuit, find the equivalent impedance, Z \( \mathrm{Z}_{\mathrm{e}} \), when \( \omega=100(\mathrm{krad} / \mathrm{s}) \). What two elements \( (R, L \), and/or \( C \) ) could be used to replace this circuit? 7. The circuit below is operating at \( 60(\mathrm{~Hz}) \). Find the equivalent impedance, \( Z_{e \text { es }} \), for this circuit.