(1) Applied voltage : \[V=\sqrt{V_{R}^{2}+V_{L}^{2}}\]
(2) Impedance : \[Z=\sqrt{{{R}^{2}}+X_{L}^{2}}=\sqrt{{{R}^{2}}+{{\omega }^{2}}{{L}^{2}}}\]\[=\sqrt{{{R}^{2}}+4{{\pi }^{2}}{{\nu }^{2}}{{L}^{2}}}\]
(3) Current : \[i={{i}_{0}}\sin \,\left( \omega \,t-\varphi \right)\]
(4) Peak current \[{{i}_{0}}=\frac{{{V}_{0}}}{Z}=\frac{{{V}_{0}}}{\sqrt{{{R}^{2}}+X_{L}^{2}}}\]\[=\frac{{{V}_{0}}}{\sqrt{{{R}^{2}}+4{{\pi }^{2}}{{\nu }^{2}}{{L}^{2}}}}\]
(5) Phase difference : \[\varphi ={{\tan }^{-1}}\frac{{{X}_{L}}}{R}={{\tan }^{-1}}\frac{\omega L}{R}\]
(6) Power factor : \[\cos \varphi =\frac{R}{\sqrt{{{R}^{2}}+X_{L}^{2}}}\]
(7) Leading quantity : Voltage 
(1) Current : \[i={{i}_{0}}\sin \,\left( \omega \,t+\frac{\pi }{2} \right)\]
(2) Peak current : \[{{i}_{0}}=\frac{{{V}_{0}}}{{{X}_{C}}}={{V}_{0}}\omega \,C={{V}_{0}}(2\pi \nu \,C)\]
(3) Phase difference between voltage and current : \[\phi ={{90}^{o}}\](or \[-\frac{\pi }{2})\]
(4) Power factor : \[\cos \varphi =0\]
(5) Power : P = 0.
(6) Time difference : \[\text{TD}=\frac{T}{\text{4}}\]
(7) Phasor diagram : Current leads the voltage by \[\pi /2\]
(1) Current : \[i={{i}_{0}}\sin \,\left( \omega \,t-\frac{\pi }{2} \right)\]
(2) Peak current : \[{{i}_{0}}=\frac{{{V}_{0}}}{{{X}_{L}}}=\frac{{{V}_{0}}}{{{\omega }_{L}}}=\frac{{{V}_{0}}}{2\pi \nu L}\]
(3) Phase difference between
voltage and current \[\varphi ={{90}^{o}}\] (or \[+\frac{\pi }{2})\]
(4) Power factor : \[\cos \varphi =0\]
(5) Power : \[P=0\]
(6) Time difference : \[\text{T}\text{.D}\text{.}=\frac{T}{\text{4}}\]
(7) Phasor diagram : Voltage leads the current by \[\frac{\pi }{2}\]
(1) Current :\[i={{i}_{0}}\sin \omega \,t\]
(2) Peak current : \[{{i}_{0}}=\frac{{{V}_{0}}}{R}\]
(3) Phase difference between
voltage and current : \[\phi ={{0}^{o}}\]
(4) Power factor : \[\cos \varphi =1\]
(5) Power : \[P={{V}_{rms}}{{i}_{rms}}=\frac{{{V}_{0}}{{i}_{0}}}{2}\]
(6) Time difference : T.D. = 0
(7) Phasor diagram : Both are in same phase
| ac measurement | dc measurement |
| (1) All ac meters read r.m.s. value. | (1) All dc meters read average value |
| (2) All ac meters are based on heating effect of current. | (2) All dc meters are based on magnetic effect of current |
(3) Deflection in hot wire meters
\[\theta \propto i_{rms}^{2}\]
(non-linear scale)
|
(3) Deflection in dc meters
\[\theta \propto i\]
(Linear scale)
|
| Nature of wave form | Wave form | r.m.s. value | average value | Form factor \[{{R}_{f}}=\frac{r.m.s.\,value}{Average value}\] | Peak factor \[{{R}_{p}}=\frac{Peak value }{r.m.s.\,value}\] |
| Sinusoidal |  |
\[\frac{{{i}_{0}}}{\sqrt{2}}\] | \[\frac{2}{\pi }{{i}_{0}}\] | \[\frac{\pi }{2\sqrt{2}}=1.11\] | \[\sqrt{2}=1.41\] |
| Half wave rectified |  |
\[\frac{{{i}_{0}}}{2}\] | \[\frac{{{i}_{0}}}{\pi }\] | \[\frac{\pi }{2}=1.57\] | more...
(1) Peak value (\[{{i}_{0}}\] or \[{{V}_{0}}\]) : The maximum value of alternating quantity (i or V) is defined as peak value or amplitude.
(2) Mean square value \[\mathbf{(}\,\overline{{{V}^{\mathbf{2}}}}\,or \overline{\,{{i}^{2}}}\,\mathbf{)}\] : The average of square of instantaneous values in one cycle is called mean square value. It is always positive for one complete cycle. e.g. \[\overline{{{V}^{2}}}=\frac{1}{T}\int_{0}^{T}{{{V}^{2}}dt}=\frac{V_{0}^{2}}{2}\] or \[\overline{\,{{i}^{2}}}=\frac{i_{0}^{2}}{2}\]
(3) Root mean square (r.m.s.) value : Root of mean of square of voltage or current in an ac circuit for one complete cycle is called r.m.s. value. It is denoted by \[{{V}_{rms}}\] or \[{{i}_{rms}}\]
\[{{i}_{rms}}=\sqrt{\frac{i_{1}^{2}+i_{2}^{2}+...}{n}}=\sqrt{\overline{{{i}^{2}}}}=\sqrt{\frac{\int_{\,0}^{\,T}{{{i}^{2}}dt}}{\int_{\,0}^{\,T}{dt}}}=\frac{{{i}_{0}}}{\sqrt{2}}\]\[=0.707\,{{i}_{0}}=70.7%\,\,{{i}_{0}}\]
Similarly \[{{V}_{rms}}=\frac{{{V}_{0}}}{\sqrt{2}}=0.707\,{{V}_{0}}=70.7%\,\]of \[{{V}_{0}}\]
\[\left[ \langle {{\sin }^{2}}(\omega \,t)\rangle =\langle {{\cos }^{2}}(\omega \,t)\rangle =\frac{1}{2} \right]\]
(i) The r.m.s. value of alternating current is also called virtual value or effective value.
(ii) In general when values of voltage or current for alternating circuits are given, these are r.m.s. value.
(iii) ac ammeter and voltmeter are always measure r.m.s. value. Values printed on ac circuits are r.m.s. values.
(iv) In our houses ac is supplied at 220 V, which is the r.m.s. value of voltage. It's peak value is \[\sqrt{2}\times 200=311V.\]
(v) r.m.s. value of ac is equal to that value of dc, which when passed through a resistance for a given time will produce the same amount of heat as produced by the alternating current when passed through the same resistance for same time.
(4) Mean or Average value (\[{{i}_{av}}\] or \[{{V}_{av}}\]) : The average value of alternating quantity for one complete cycle is zero.
The average value of ac over half cycle (\[t=0\] to \[T/2\])
\[{{i}_{av}}=\frac{\int_{\,0}^{\,T/2}{\,i\,dt}}{\int_{\,0}^{\,T/2}{dt}}=\frac{2{{i}_{0}}}{\pi }=0.637{{i}_{0}}=63.7%\] of \[{{i}_{0}}\],
Similarly \[{{V}_{av}}=\frac{2{{V}_{0}}}{\pi }=0.637{{V}_{0}}=63.7%\] of \[{{V}_{0}}\].
(5) Peak to peak value : It is equal to the sum of the magnitudes of positive and negative peak values
\[\therefore \] Peak to peak value \[={{V}_{0}}+{{V}_{0}}=2{{V}_{0}}\]
\[=2\sqrt{2}\,{{V}_{rms}}=2.828\,{{V}_{rms}}\]
(6) Form factor and peak factor : The ratio of r.m.s. value of ac to it's average during half cycle is defined as form factor. The ratio of peak value and r.m.s. value is called peak factor
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