Question Type 8: Applying complex numbers in the context of sinusoidal functions Exercises

Express the sinusoidal voltage $V(t)=10\cos(40t)$ as a complex phasor $V= A\angle\phi$.

Convert the voltage $V(t)=5\cos(40t+90^\circ)$ into its complex exponential form $V(t)=\Re\{V\,e^{j40t}\}$ and state the phasor $V$.

Three balanced three-phase voltages are $V_a=100\cos(60t),\quad V_b=100\cos(60t-120^\circ),\quad V_c=100\cos(60t+120^\circ).$ Show that their sum is zero by phasor addition.

Given two sinusoidal voltages $V_1(t)=10\cos(40t),\quad V_2(t)=20\cos(40t+30^\circ),$ find the resulting voltage in the form $V(t)=A\cos(40t+B).$

Compute the sum $V(t)=10\cos(40t+350^\circ)+10\cos(40t-10^\circ)$ and express it as $A\cos(40t+B)$.

Two voltages are given by $V_1=20\cos(40t+170^\circ),\quad V_2=20\cos(40t-170^\circ).$ Find the resultant voltage $V(t)=V_1+V_2$ in the form $A\cos(40t+B)$.

Derive an expression for the sum $V(t)=V_m\cos(\omega t+\alpha)+V_m\cos(\omega t-\alpha)$ in the form $V(t)=A\cos(\omega t+\phi)$ and find $A$ in terms of $V_m$ and $\alpha$.

Given $V_1(t)=15\cos(50t-20^\circ)\quad\text{and}\quad V_2(t)=25\cos(50t+40^\circ),$ determine the amplitude $A$ and phase $B$ of $V(t)=V_1(t)+V_2(t)=A\cos(50t+B).$

Sum three sinusoidal voltages of the same frequency: $V_1=10\cos(100t),\quad V_2=10\cos(100t+120^\circ),\quad V_3=10\cos(100t+240^\circ).$ Express the total voltage as $V(t)=A\cos(100t+B)$.

A sinusoid has phasor representation $V=12+j5$. Convert this to time-domain form $v(t)=A\cos(40t+\phi),$ giving $A$ and $\phi$ in degrees.

Given phasors $V_1=5\angle30^\circ$, $V_2=12\angle(-45^\circ)$, and $V_3=8\angle60^\circ$ at $\omega=40\,$rad/s, find the time-domain sum $v(t)=\Re\{(V_1+V_2+V_3)e^{j40t}\}$ in the form $A\cos(40t+B)$.

Express the difference $V(t)=30\cos(40t+45^\circ)-50\cos(40t-15^\circ)$ in the form $A\cos(40t+B)$.