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.
A coil of 0.01 H inductance and $1 \Omega$ resistance is connected to 200 V , $50 \mathrm{~Hz} A \mathrm{C}$ supply. Find the impedance of the circuit and time lag between maximum alternating voltage and current.
$$\begin{aligned} \text { Given, inductance } L & =0.01 \mathrm{H} \\ \text { resistance } R & =1 \Omega, \text { voltage }(V)=200 \mathrm{~V} \\ \text { and } \quad \text { frequency }(f) & =50 \mathrm{~Hz} \end{aligned}$$
$$\begin{aligned} \text{Impedance of the circuit}\quad Z & =\sqrt{R^2+X_L^2}=\sqrt{R^2+(2 \pi f L)^2} \\ & =\sqrt{1^2+(2 \times 3.14 \times 50 \times 0.01)^2} \end{aligned}$$
$$\text{or}\quad Z=\sqrt{10.86}=3.3 \Omega$$
$$\begin{aligned} \tan \phi & =\frac{\omega L}{R}=\frac{2 \pi f L}{R}=\frac{2 \times 3.14 \times 50 \times 0.01}{1}=3.14 \\ \phi & =\tan ^{-1}(3.14) \approx 72 \Upsilon \end{aligned}$$
Phase difference
$$\phi=\frac{72 \times \pi}{180} \mathrm{rad}$$
Time lag between alternating voltage and current
$$\Delta t=\frac{\phi}{\omega}=\frac{72 \pi}{180 \times 2 \pi \times 50}=\frac{1}{250} \mathrm{~s}$$
A 60 W load is connected to the secondary of a transformer whose primary draws line voltage. If a current of 0.54 A flows in the load, what is the current in the primary coil? Comment on the type of tansformer being used.
Given, $P_S=60 \mathrm{~W}, I_S=0.54 \mathrm{~A}$
Current in the primary $I_p=$ ?
Taking line voltage as 220 V.
We can write Since,
$$\begin{array}{ll} \Rightarrow & P_L=60 \mathrm{~W}, I_L=0.54 \mathrm{~A} \\ \Rightarrow & V_L=\frac{60}{0.54}=110 \mathrm{~V} .\quad\text{.... (i)} \end{array}$$
Voltage in the secondary $\left(E_S\right)$ is less than voltage in the primary $\left(E_P\right)$.
Hence, the transformer is step down transformer.
Since, the transformation ratio
$r=\frac{V_s}{V_p}=\frac{I_p}{I_s}$
$$\begin{array}{lr} \text { Substituting the values, } & \frac{110 \mathrm{~V}}{220 \mathrm{~V}}=\frac{I_p}{0.54 \mathrm{~A}} \\ \text { On solving } & I_p=0.27 \mathrm{~A} \end{array}$$