In series $L C R$ circuit, the plot of $I_{\max }$ versus $\omega$ is shown in figure. Find the bandwidth and mark in the figure.
Consider the diagram .
Bandwidth $=\omega_2-\omega_1$
where $\omega_1$ and $\omega_2$ corresponds to frequencies at which magnitude of current is $\frac{1}{\sqrt{2}}$ times of maximum value.
$$I_{\mathrm{rms}}=\frac{I_{\mathrm{max}}}{\sqrt{2}}=\frac{1}{\sqrt{2}} \approx 0.7 \mathrm{~A}$$
Clearly from the diagram, the corresponding frequencies are $0.8 \mathrm{~rad} / \mathrm{s}$ and $1.2 \mathrm{~rad} / \mathrm{s}$.
$$\Delta \omega=\text { Bandwidth }=1 \cdot 2-0.8=0.4 \mathrm{~rad} / \mathrm{s} $$
The alternating current in a circuit is described by the graph shown in figure. Show rms current in this graph.
$$\begin{aligned} I_{\mathrm{rms}} & =\mathrm{rms} \text { current } \\ & =\sqrt{\frac{1^2+2^2}{2}}=\sqrt{\frac{5}{2}}=1.58 \mathrm{~A} \approx 1.6 \mathrm{~A} \end{aligned}$$
The rms value of the current $\left(I_{\mathrm{rms}}\right)=1.6 \mathrm{~A}$ is indicated in the graph.
How does the sign of the phase angle $\phi$, by which the supply voltage leads the current in an L-C-R series circuit, change as the supply frequency is gradually increased from very low to very high values.
$$\begin{aligned} &\text { The phase angle ( } \phi \text { ) by which voltage leads the current in L-C-R series circuit is given by }\\ &\begin{aligned} & \tan \phi=\frac{X_L-X_C}{R}=\frac{2 \pi v L-\frac{1}{2 \pi v C}}{R} \\ & \tan \phi < 0\left(\text { for } \nu< \nu_0\right) \\ & \tan \phi > 0\left(\text { for } \nu > \nu_0\right) \\ & \tan \phi=0 \quad\left(\text { for } \nu=\nu_0=\frac{1}{2 \pi \sqrt{2 C}}\right) \end{aligned} \end{aligned}$$
A device ' $X$ ' is connected to an AC source. The variation of voltage, current and power in one complete cycle is shown in figure.
(a) Which curve shows power consumption over a full cycle?
(b) What is the average power consumption over a cycle?
(c) Identify the device $X$.
(a) We know that
$$\text { Power }=P=V I$$
that is curve of power will be having maximum amplitude, equals to multiplication of amplitudes of voltage $(V)$ and current $(I)$ curve. So, the curve will be represented by $A$.
(b) As shown by shaded area in the diagram, the full cycle of the graph consists of one positive and one negative symmetrical area.
Hence, average power over a cycle is zero.
(c) As the average power is zero, hence the device may be inductor ( $L$ ) or capacitor (C) or the series combination of $L$ and $C$.
Both alternating current and direct current are measured in amperes. But how is the ampere defined for an alternating current?
For a Direct Current (DC),
$$1 \text { ampere = } 1 \text { coulomb } / \mathrm{sec}$$
An AC current changes direction with the source frequency and the attractive force would average to zero. Thus, the AC ampere must be defined in terms of some property that is independent of the direction of current. Joule's heating effect is such property and hence it is used to define rms value of AC.