 # AC Voltage Applied to a Resistor and Phasors

• If the current in a circuit changes its direction in every \tt \frac{T}{2} sec then the current is called ALTERNATING CURRENT (ac)
• The emf of a source which produces sinusoidally varying potential difference across its terminals is given as, e = eo sin wt
• The average current over a complete cycle is ZERO
• For symmetric waves, the average value for half cycle is measured
• \tt i_{ave}=\frac{2}{\pi}\ i_{0}=0.637\ i_{0}
• \tt e_{ave}=\frac{2}{\pi}\ e_{0}=0.637\ e_{0}
• \tt i_{rms}=\frac{i_{0}}{\sqrt{2}}\ i_{0}=0.707\ i_{0}
• \tt e_{rms}=\frac{e_{0}}{\sqrt{2}}\ i_{0}=0.707\ e_{0}
• If a d.c ammeter is connected to an a.c source then the reading is shown as zero.
• It is important to measure and specify rms values for a.c quantities .
• When a resistor is connected to an a.c source, the average power loss (P) \tt P=V_{rms}^2/R\ or\ P=I_{rms}V_{rms\ }\ or\ P=I_{rms}^2\ R
• Analogous value for resistance for an ideal inductor is LW. It is called Inductive reactance (XL)
• Analogous value for resistance for a capacitor is 1/CW. It is called capacitive reactance, XC.
• The REACTANCE is the total resistance offered by the circuit due to capacitor and inductor. It is equal to \tt X_{L}\sim X_{C}
• The total Resistance a circuit offers is called IMPEDENCE, Z.
\tt Z=\sqrt{\left(X_{L}-X_{C}\right)^2+R^{2}}
• VOLTAGE LEAD and CURRENT LEAD are the two phasor diagrams for LCR series circuits.
• • The peak current is given by \tt i_{0}=\frac{e_{0}}{Z} for AC voltage applied to inductor.
• The impedence is given by Z = LW (Pure inductor)
• The Phase difference between emf and current is given by the relation \tt \phi=\tan^{-1}\frac{\omega L}{Q}\Rightarrow\phi=\frac{\pi}{2}
• The instantaneous alternating current, i = io sin (wt + φ) when AC voltage is applied to the capacitor.
• The maximum current when AC voltage is applied to the capacitor is given by \tt i_{0}=\frac{e_{0}}{Z}
• The impedence when AC voltage is applied to capacitor is given by • When AC voltage is applied to capacitor, then the phase difference between emf and current is given by \tt \phi=\tan^{-1}\frac{1/c\omega }{0}\Rightarrow\phi=\frac{\pi}{2}
• When AC voltage is applied across L – R, the instantaneous alternating current is given as i = io sin (wt - φ) • When AC voltage is applied across L – R , the maximum current is \tt i_{0}=\frac{e_{0}}{Z}
• When AC voltage is applied across L – R, the impedance is \tt Z=\sqrt{R^{2}+L^{2}\omega^{2}}
• AC voltage is applied across L – R, then the phase difference between emf and current is \tt \phi=\tan^{-1}\frac{L\omega}{R}
• AC voltage applied across R – C series, then the instantaneous alternating current is, i = io sin (wt + φ) • AC voltage applied across R – C series, then the maximum current, \tt i_{0}=\frac{e_{0}}{Z}

### View the Topic in this video From 00:24 To 10:43

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1. The instantaneous value of alternating current at any instant of time t is given by
I = I0 sin ωt
where, I0 = peak value of alternating current.

2. Mean or average value of alternating current for first half cycle
I_{m} = \frac{2I_{0}}{\pi} = 0.637 I_{0}

3. Root mean square value of alternating current
I_{v} = I_{rms} = \frac{I_{0}}{\sqrt{2}} = 0.707 I_{0}

4.Root mean square value of alternating voltage
V_{rms} = \frac{V_{0}}{\sqrt{2}} = 0.707 V_{0}

5. Total resistance of a circuit offer Impedance of an AC circuit, Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}

6. The average power in an AC circuit,
P_{av} = V_{rms} \ i_{rms} \ \cos \theta