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

Three balanced three-phase voltages are given by the equations:

$V_a = 100\cos(60t), \quad V_b = 100\cos(60t - 120^\circ), \quad V_c = 100\cos(60t + 120^\circ)$

Show that their sum is zero by phasor addition.

Express $V(t)$ in the complex exponential form $V(t) = \text{Re}\{V e^{i40t}\}$ and state the complex number $V$ in the form $re^{i\theta}$.

Sum three sinusoidal voltages of the same frequency: $V_1=10\cos(100t),\quad V_2=10\cos(100t+120^\circ),\quad V_3=10\cos(100t+240^\circ)$ where $V_1, V_2, V_3$ are in volts and $t$ is in seconds.

Express the total voltage as $V(t)=A\cos(100t+B)$.

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

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 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 $V(t)=A\cos(40t+B).$

A sinusoid has phasor representation $V=12+5i$. Convert this to the time-domain form $v(t)=A\cos(40t+\phi)$ giving the values of $A$ and $\phi$, where $\phi$ is in degrees.

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 $V(t) = V_1(t) + V_2(t) = A\cos(50t + B).$

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

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 $A\cos(40t+B)$, where $A > 0$ and $-180^\circ < B \le 180^\circ$.