Review of Chapter 2 Plus Matlab Examples 2 2 Power in single phase circuits Let v t and i t be defined as v t Vm cos t v and i t I m cos t i then the instantaneous power is give by p t v t i t Vm I m cos t v cos t i Vm I m cos 2 t v i cos v i 2 Vrms I rms cos 2 t v v i cos v i V I cos 2 t v cos v i sin 2 t v sin v i cos v i p t V I cos 1 cos 2 t v V I sin sin 2 t v where v i The angle is called the power factor angle and is the angle between the voltage and current at a particular 2 port Note the power factor is given by cos Note that the cosine is an even function thus the sign of is lost For an inductive circuit memory helper ELI voltage leads the current the carrent lags the voltage and we call this a lagging power factor because the current lags the voltage For a capacitive circuit memory helper ICE the current leads the voltage and we call this a leading power factor Thus we always say clearly if the power factor is lagging inductive or leading capacitive The lag or lead is the angle by which the current lags or leads the voltage V V It is also noted that is the impedance angle thus Z v i The I I instantaneous power can be expressed in two parts p t pR t p X t thus pR t V I cos 1 cos 2 t v and p X t V I sin sin 2 t v It is common to define P and Q as follows 1 P avg p t avg pR t V I cos Q V I sin Thus in terms of P and Q the instantaneous power can be expressed as p t P 1 cos 2 t v Q sin 2 t v Thus P is the average value of the cosine terms counting DC as a zero frequency cosine term while Q is the magnitude of the sine term It is noted that besides the DC term power has twice line frequency components Example 2 1 A supply voltage v t 100 cos t is applied across a load whose impedance is given by Z 1 25 60 Determine i t and p t Use Matlab to plot i t v t p t pR t and p X t over an interval from 0 to 2 The Matlab program is shown below Vm 100 thetav 0 Voltage amplitude and phase angle Z 1 25 gama 60 Impedance magnitude and phase angle thetai thetav gama Current phase angle in degree theta thetav thetai pi 180 Degree to radian Im Vm Z Current amplitude wt 0 05 2 pi wt from 0 to 2 pi v Vm cos wt Instantaneous voltage i Im cos wt thetai pi 180 Instantaneous current p v i Instantaneous power V Vm sqrt 2 I Im sqrt 2 rms voltage and current P V I cos theta Average power Q V I sin theta Reactive power S P j Q Complex power pr P 1 cos 2 wt thetav Eq 2 6 px Q sin 2 wt thetav Eq 2 8 PP P ones 1 length wt Average power with length w for plot xline zeros 1 length wt generates a zero vector wt 180 pi wt converting radian to degree subplot 2 2 1 plot wt v wt i wt xline grid title v t Vm coswt i t Im cos wt num2str thetai xlabel wt degree subplot 2 2 2 plot wt p wt xline grid title p t v t i t xlabel wt degree subplot 2 2 3 plot wt pr wt PP wt xline grid title pr t Eq 2 6 xlabel wt degree subplot 2 2 4 plot wt px wt xline grid title px t Eq 2 8 xlabel wt degree subplot 111 2 S 2 0000e 003 3 4641e 003i v t Vm coswt i t Im cos wt 60 100 50 4000 0 2000 50 0 100 0 100 200 300 wt degree pr t Eq 2 6 400 2000 4000 4000 3000 2000 2000 0 1000 2000 0 2 3 p t v t i t 6000 0 100 200 300 wt degree 400 4000 0 100 200 300 wt degree px t Eq 2 8 0 100 200 300 wt degree 400 400 Complex power It is observed that the term VI gives the result N B v i VI V I cos j V I sin which is identical to S VI P jQ where S is the complex power 2 V 2 OTHER FORMS for S S R I jX I Z I Z 2 2 3 For example for an inductive load the current lags the voltage and the phasor diagrams would be as shown below V I S Q P Phasor diagrams V I and the power triangle for an inductive load And for a capacitive load the current would lead the voltage and the phasor diagrams would be as shown below I V P Q S Phasor diagrams V I and the power triangle for a capacitive load leading power factor angle 2 4 Complex power balance The total value of S for a circuit is the SUM of S for the components of S Example 2 2 Three impedances in parallel are suppleid by a source V 1200 0 V where the impedances are given by Z1 60 j 0 Z 2 6 j12 and Z 3 30 j 30 Find the power absorbed by each load and the total complex power As a check compute the power supplied by the source 4 Matlab program follows V 1200 Z1 60 Z2 6 j 12 Z3 30 j 30 I1 V Z1 I2 V Z2 I3 V Z3 S1 V conj I1 S2 V conj I2 S3 V conj I3 S S1 S2 S3 Sv V conj I1 I2 I3 S1 24000 S2 4 8000e 004 S3 2 4000e 004 S 9 6000e 004 Sv 9 6000e 004 9 6000e 004i 2 4000e 004i 7 2000e 004i 7 2000e 004i It is observed that the complex power is conserved 5 2 5 Power factor correction Adding a capacitor usually in parallel with an inductive load can improve the power factor Example 2 3 Two loads Z1 100 j 0 and Z 2 10 j 20 are connected across a 200 V 60 Hz source a Find the total real and reactive power the power factor at the source and the total current b Find the capacitance of the capacitor connected across the loads to improve the overall power factor to 0 8 lagging Q S QC d P Example 2 3 The Matlab program follows V 200 Z1 100 Z2 10 j 20 I1 V Z1 I2 V Z2 S1 V conj I1 S2 V conj I2 I …