(a)Explain resonance frequency as applied in RLC series Circuit. (ii) Sketch a diagram to illustrate the variation of frequency, f, with the resistance, R, ...
Question 1 Report
(a)Explain resonance frequency as applied in RLC series Circuit.
(ii) Sketch a diagram to illustrate the variation of frequency, f, with the resistance, R, the capacitive reactance, X\(_c\) and the inductive reactance X\(_L\), in RLC series circuit.
(iii) Using the diagram drawn in (a)(ii) state whether the current in the circuit leads, lags or is in phase with the supply voltage when: (\(\alpha\)) f = f\(_o\); (\(\beta\)) f < f\(_o\) ; (\(\gamma\))f\(_o\); when f\(_o\) is the resonant frequency.
b)(i) Define mutual inductance.
(ii) The coil of an electric generator has 500 turns and 8.0cm diameter. If it rotates in a magnetic field of density 0.25T, calculate the angular speed when its peak voltage is 480V. [\(\pi\) = 3.142].
Answer Details
(a)(i) Resonance in a series RLC circuit
Resonance occurs at the resonant frequency \(f_o\), when the inductive reactance equals the capacitive reactance:
\[X_L=X_C\]
At this frequency, the inductive and capacitive effects cancel. Therefore, the circuit impedance is at its minimum value and is equal to the resistance \(R\). The current is consequently maximum.
\[f_o=\frac{1}{2\pi\sqrt{LC}}\]
(a)(ii) Variation of \(R\), \(X_L\), and \(X_C\) with frequency
The resistance \(R\) is constant as frequency changes. The inductive reactance increases with frequency:
\[X_L=2\pi fL\]
The capacitive reactance decreases as frequency increases:
\[X_C=\frac{1}{2\pi fC}\]
The point where the \(X_L\) and \(X_C\) curves meet is the resonant frequency \(f_o\).
(a)(iii) Phase relationship between current and supply voltage
- When \(f=f_o\), \(X_L=X_C\). The circuit is purely resistive, so the current is in phase with the supply voltage.
- When \(f<f_o\), \(X_C>X_L\). The circuit is net capacitive, so the current leads the supply voltage.
- When \(f>f_o\), \(X_L>X_C\). The circuit is net inductive, so the current lags the supply voltage.
The supplied reference answer reverses the lead/lag relationships away from resonance. In a capacitive circuit current leads voltage; in an inductive circuit current lags voltage.
(b)(i) Mutual inductance
Mutual inductance is the production of an induced e.m.f. in one coil when the current, and hence magnetic flux, in a nearby linked coil changes. Quantitatively, it is the ratio of induced e.m.f. in one coil to the rate of change of current in the other coil.
(b)(ii) Angular speed of the generator coil
For a rotating coil generator, the peak e.m.f. is:
\[E_0=NBA\omega\]
The coil diameter is \(8.0\,\text{cm}=0.080\,\text{m}\), so its radius is:
\[r=\frac{0.080}{2}=0.040\,\text{m}\]
Its area is:
\[A=\pi r^2=3.142(0.040)^2=5.027\times10^{-3}\,\text{m}^2\]
Substitute \(E_0=480\,\text{V}\), \(N=500\), and \(B=0.25\,\text{T}\):
\[\omega=\frac{E_0}{NBA}\]
\[\omega=\frac{480}{500\times0.25\times5.027\times10^{-3}}\]
\[\omega=\frac{480}{0.6284}=7.64\times10^2\,\text{rad s}^{-1}\]
Therefore, the angular speed is \(764\,\text{rad s}^{-1}\) (approximately).
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