Chapter Questions
Complex numbers in polar form are often written as $r \angle \theta$, where $r$ is the modulus and $\theta$ is the argument, expressed in radians.Express $1+2 j$ in this way.
(a) Express the complex numbers $1+2 j,-1+2 j,-1-2 j$, and $1-2 j$ in polar form.(b) Simplify the following expressions, leaving them in Cartesian coordinates: $(1+2 j)(1-2 j),(1+2 j)^{2}, 1 /(1+2 j)^{2},(1+2 j)(1-2 j)$.
What happens to a complex number in the complex plane when it is (a) multiplied by $j$, (b) multiplied by $j^{2}$, (c) divided by $j$ ?
Find the real parts of the phasors $(1+3 j) \exp j(\omega t+2)$ and $[\exp j(\omega t+2)] /(1+3 j)$
Solve the following differential equation by means of phasors:$$2 \frac{d^{2} x}{d t^{2}}+3 \frac{d x}{d t}+4 x=5 \cos 6 t$$
Find the rms values for the waveforms shown in Fig. $2-5$.
A certain electric circuit draws a current of $1.00$ ampere rms when it is fed at 120 volts rms, 60 hertz. The current lags the voltage by $\pi / 4$ radian.(a) Express $V$ and $I$ in the form of phasors, and calculate the timeaveraged power dissipation.(b) Now calculate the power $V_{\mathrm{rm}} I_{\mathrm{rm}} \cos \theta$, where $\theta$ is $\pi / 4$.
Find the value of$$\frac{5.14 \exp j(\omega t+3)}{3.72 \exp j(\omega t+5)}$$in Cartesian form.
Show that$$\ln (a+b j)=\frac{1}{2} \ln \left(a^{2}+b^{2}\right)+j \arctan \frac{b}{a} \text {, }$$ if $a$ is positive. Then the point $a+b j$ lies in either the first or the fourth quadrant. If $a$ is not positive, be careful to use the proper angle (Sec. 2.1).
Show that the phasor $\boldsymbol{V}=V_{m}[\hat{\boldsymbol{x}} \exp j \omega t+\hat{\boldsymbol{y}} \exp j(\omega t-\pi / 2)]$ represents a vector of constant magnitude $V_{m}$ that rotates in the positive direction in the $x y$-plane at the angular velocity $\omega$.