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Physics · 2024 · 5 marks
CBSE 2024 · Region 5 · Set 2 · Q33
(a) (i) You are given three circuit elements X, Y and Z. They are connected one by one across a given ac source. It is found that V and I are in…
(a)(i)
You are given three circuit elements $\displaystyle \mathrm{X}, \mathrm{Y}$ and Z . They are connected one by one across a given ac source. It is found that V and $\displaystyle I$ are in phase for element $\displaystyle \mathrm{X} . V$ leads $\displaystyle I$ by $\displaystyle \left(\frac{\pi}{4}\right)$ for element $\displaystyle \mathrm{Y}$ while I leads V by $\displaystyle \left(\frac{\pi}{4}\right)$ for element $\displaystyle Z$. Identify elements $\displaystyle \mathrm{X}, \mathrm{Y}$ and $\displaystyle Z$.
(ii)
Establish the expression for impedance of circuit when elements $\displaystyle \mathrm{X}, \mathrm{Y}$ and Z are connected in series to an ac source. Show the variation of current in the circuit with the frequency of the applied ac source.
(iii)
In a series LCR circuit, obtain the conditions under which
(i)
impedance is minimum and (ii) wattless current flows in the circuit.
OR 33. (b) (i) Describe the construction and working of a transformer and hence obtain the relation for $\displaystyle \left(\frac{v_{\mathrm{s}}}{v_{\mathrm{p}}}\right)$ in terms of number of turns of primary and secondary.
(ii)
Discuss four main causes of energy loss in a real transformer.
Marking-scheme solution
(i)
X: Resistor
Y: real inductor (such that its reactance is equal to its resistance) / Inductor
Z: real capacitor (such that its reactance is equal to its resistance) / Capacitor
(ii)
From the figure:
$\displaystyle V_{m}^{2}=V_{R m}^{2}+\left(V_{C m}-V_{L m}\right)^{2}$
$\displaystyle V_{m}^{2}=\left(i_{m} R\right)^{2}+\left(i_{m} X_{C}-i_{m} X_{L}\right)^{2}$
Impedance $\displaystyle (Z)=\frac{V_{m}}{I_{m}}=\sqrt{R^{2}+\left(X_{C}-X_{L}\right)^{2}}$
(iii)
$\displaystyle Z=\sqrt{R^{2}+\left(X_{C}-X_{L}\right)^{2}}$
(i)
For the minimum value of impedance: $\displaystyle X_{C}=X_{L}$
(ii)
Average power consumed in an A.C. circuit over a cycle:
$\displaystyle P=V I \cos \phi$
For wattless current $\displaystyle P=0$
Since $\displaystyle V \neq 0$, $\displaystyle I \neq 0$
$\displaystyle \cos \phi=0$
i.e. $\displaystyle \phi=\frac{\pi}{2}$
(i)
Construction: A transformer consists of two sets of coils, insulated from each other. They are wound on a soft-iron core, either one on top of the other or on separate limbs of the core.
Working: When an alternating voltage is applied to the primary, the resulting current produces an alternating magnetic flux which links with the secondary and induces an e.m.f. in it.
For an ideal transformer the induced e.m.f. $\displaystyle \left(\varepsilon_{p}\right)$ in the primary coil for applied alternating voltage $\displaystyle \left(V_{P}\right)$:
$\displaystyle \varepsilon_{p}=V_{P}=-N_{P} \frac{d \phi}{d t}$ ------------------($\displaystyle 1$)
e.m.f. induced $\displaystyle \varepsilon_{S}$ in the secondary coil:
$\displaystyle \varepsilon_{S}=V_{S}=-N_{S} \frac{d \phi}{d t}$ -------------------($\displaystyle 2$)
From eq. ($\displaystyle 1$) and ($\displaystyle 2$):
$\displaystyle \frac{V_{S}}{V_{P}}=\frac{N_{S}}{N_{P}}$
(ii)
Energy losses (any four):
1. Flux leakage.
2. Resistance of windings / copper loss.
3. Eddy currents / iron loss.
4. Hysteresis.
5. Magnetostriction.
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