Sign Convention of \(V\) and \(I\)

Phasor Representation

Conversion of Time-domain to Phasor and Impedance Triangle

Powers: Instantaneous, Active, Reactive, and Complex

Relationship between the power quantities

Power Factor and its Significance

Advantage of Reactive Power Compensation

Convention for \(V\) and \(I\)

In linear circuit with sinusoidal \(V\) and \(I\) of frequency \(f\) applied for long so that steady state is reached, all \(V\) and \(I\) are at \(f=\omega/2\pi\)

Phasor converts \(v(t)\) and \(i(t)\) to complex variables \(\overrightarrow{V}\) and \(\overrightarrow{I}\)

\[\begin{aligned} v(t) & =\sqrt{2} V \cos \left(\omega t+\phi_{v}\right) \\ \Leftrightarrow \overrightarrow{V}& =V \angle \phi_{v} \\ i(t) & =\sqrt{2} I \cos \left(\omega t+\phi_{i}\right) \\ \Leftrightarrow \overrightarrow{I} & =I \angle \phi_{i} \end{aligned}\] Note: \(V\) and \(I\) are in RMS

To calculate \(i(t)\) use differential equation: \[R i(t)+L \frac{d i(t)}{d t}+\frac{1}{C} \int i(t) \cdot d t=\sqrt{2} V \cos (\omega t)\]

\[\begin{aligned} Z & =R+j X_{L}+j X_{c} ~~ =|Z|<\phi \\ X_{L} & =\omega L, ~~ \quad X_{c}=\left(\frac{1}{-\omega C}\right) \\ |Z| & =\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}} & \phi=\tan ^{-1}\left[\frac{\left(\omega L-\frac{1}{\omega C}\right)}{R}\right] \end{aligned}\] \(\checkmark\) Note: \(Z\) is complex but not a phasor, hence does not have time-domain expression

Instantaneous power \(p(t)=v(t)\cdot i(t)\) delivered by subcircuit-1 (Generator) is absorbed by subcircuit-2 (Load)

A negative value of \(p(t)\) reverses their role of delivering and absorbing power

Complex power (\(S\)), reactive power (\(Q\)), and power factor (pf) express how effectively active (average) power \(P\) transferred from one subcircuit to other

If \(v(t)\) and \(i(t)\) are in phase, \(p(t)\) pulsates at twice the steady-state \(f\)

\[\begin{aligned} p(t) & =\sqrt{2} V \cos \omega t \cdot \sqrt{2} I \cos \omega t \quad(i \text { in phase } \operatorname{with} v) \\ & = 2 V I \cos ^{2} \omega t=V I+V I \cos 2 \omega t \end{aligned}\] \(\ast\) \(\phi_{v}\) and \(\phi_{i}\) are assumed to be zero without loss of generality

In this case, for all times \(p(t) \geq 0\), and therefore power flows in only one direction from subcircuit 1 to 2

The average over one cycle of the second term of the RHS is zero, therefore average power is \(P= VI\)

Considering \(i(t)\) lags behind \(v(t)\) by \(\phi (t)\)

\(p(t)\) becomes -ve during a time-interval of \(\phi/\omega\) during each half-cycles as calculated by \[\begin{aligned} p(t)& =\sqrt{2} V \cos \omega t \cdot \sqrt{2} I \cos (\omega t-\phi) \\ & =V I \cos \phi+V I \cos (2 \omega t-\phi) \end{aligned}\]

Negative \(p(t)\) means power flow in opposite direction

This back and forth flow of power indicates real power is not transferred from one subcircuit to other

Avg. power \(P=VI\cos\phi\) is less than the previous case

\[\begin{aligned} S & =\overrightarrow{V}\overrightarrow{I}^{*} ~ *\rightarrow\text{conjugate}\\ & =V\angle\phi_{v}\cdot I\angle-\phi_{i}\\ & =VI\angle\left(\phi_{v}-\phi_{i}\right)\\ & =VI\angle\phi\\ & =P+jQ \end{aligned}\] where, \[\begin{array}{l} P=V I \cos \phi \\ Q=V I \sin \phi \end{array}\] \[\begin{array}{l} |S| =\sqrt{P^{2}+Q^{2}} \\ \phi =\tan^{-1}\left(\dfrac{Q}{P}\right) \end{array}\]

\(I\cos \phi\) is in phase with \(V\), and results in \(P\)

\(I\sin \phi\) is at \(90^\circ\) to \(V\), and results in \(Q\)

\[\begin{aligned}
P & =VI\cos\phi\\
\Rightarrow pf = \cos\phi & =\dfrac{P}{VI}
=\dfrac{\text{Real Power}}{\text{Apparaent Power}}\\
& =\dfrac{kW}{kVA} =\dfrac{R}{Z}
\end{aligned}\]

\[\begin{aligned} P & \rightarrow W(\text{Watts}); ~~ Q \rightarrow\text{VAR (Volt-Amperes Reactive)} \\ |S| & \rightarrow\text{VA (Volt-Amperes);} \\ \phi_{v},\phi_{i},\phi & \rightarrow\text{radians (+ve anticlockwise w.r.t real axis L to R)} \\ pf \rightarrow & \cos\phi = \text{ dimensionless, value between 0 to 1} \end{aligned}\]

Electrical usage cost is prop. to \(|S| = VI\)

Electrical insulation level and magnetic core size for a definite \(f\) depends on \(V\)

Conductor size depends on \(I\)

\(P\) represents (useful work \(+\) losses)

Desirable for \(Q = 0\) because it increases \(|S|\)

PF measure how effectively load draws \(P\)

Ideally \(P\) should be 1 (unity) i.e \(Q=0\)

Inductive load draws power at lagging pf (\(I\) lags \(V\))

Capacitive load draws power at leading pf (\(I\) leads \(V\))

\[\begin{aligned} \text{Total Real Power} & =\sum_{k}P_{k}=\sum_{k}I_{k}^{2}R_{k}\\ \text{Total Reactive Power} & =\sum_{k}Q_{k}=\sum_{k}I_{k}^{2}X_{k} \\ +Q & \rightarrow \text{lagging load (Inductive)} \\ -Q & \rightarrow \text{leading load (Capacitive)} \end{aligned}\]