Complex service (electrical engineering)

The complex power (also complex apparent power ) is a computational variable in electrical engineering that combines the various performance parameters for alternating current into a value that is compatible with the symbolic method of the complex alternating current calculation . The name results from the fact that this value is a complex number .

motivation

The complex alternating current calculation is, as shown there, only suitable for linear arithmetic operations. It fails, for example, when multiplying complex voltage and complex current. In contrast to the instantaneous power and the apparent power , the corresponding product , like the product of the complex effective values, has no practical meaning. However, if you multiply a complex voltage by a conjugate complex current, the time-dependent parts and cancel each other out and only the mutual phase shift is included . That motivates the following definition. ${\displaystyle u(t)\cdot i(t)}$ ${\displaystyle U\cdot I}$ ${\displaystyle {\underline {u}}(t)\cdot {\underline {i}}(t)}$ ${\displaystyle {\underline {U}}\cdot {\underline {I}}}$ ${\displaystyle \mathrm {e} ^{\mathrm {j} \omega t}}$ ${\displaystyle \mathrm {e} ^{-\mathrm {j} \omega t}}$

definition

The product of the complex rms value of the voltage and the conjugate complex rms value of the current is defined as the complex power at a two- terminal network . Since the amplitudes are larger than the effective values, the complex power is therefore equal to half the product of the complex amplitude of the voltage and the conjugate of the complex amplitude of the current: ${\displaystyle {\underline {S}}}$ ${\displaystyle {\underline {U}}=U\cdot \mathrm {e} ^{\mathrm {j} \varphi _{u}}}$ ${\displaystyle {\underline {I}}^{\star }=I\cdot \mathrm {e} ^{-\mathrm {j} \varphi _{i}}}$ ${\displaystyle {\sqrt {2}}}$

${\displaystyle {\underline {S}}={\underline {U}}\cdot {\underline {I}}^{\star }={\frac {{\underline {\hat {u}}}\cdot {\underline {\hat {\imath }}}^{\star }}{2}}}$

Due to the detailed notation

${\displaystyle {\underline {S}}={\underline {U}}\cdot {\underline {I}}^{\star }=U\cdot \mathrm {e} ^{+\mathrm {j} \varphi _{u}}\cdot I\cdot \mathrm {e} ^{-\mathrm {j} \varphi _{i}}=U\cdot I\cdot \mathrm {e} ^{\mathrm {j} (\varphi _{u}-\varphi _{i})}=U\cdot I\cdot \left(\cos(\varphi _{u}-\varphi _{i})+\mathrm {j} \sin(\varphi _{u}-\varphi _{i})\right)}$

and the usual abbreviation for the phase shift and the division into real and imaginary parts ${\displaystyle \varphi =\varphi _{u}-\varphi _{i}}$

Power vector diagram for sinusoidal voltages and currents in the complex plane

${\displaystyle {\underline {S}}=S\cdot \mathrm {e} ^{\mathrm {j} \varphi }=S\cdot \left(\cos \varphi +\mathrm {j} \sin \varphi \right)=P+\mathrm {j} Q}$

it becomes clear how the three performance parameters commonly used in AC technology are related to complex performance:

• The real power is the real part of the complex power: ${\displaystyle P}$

${\displaystyle P=\mathrm {Re} \ {\underline {S}}=S\cdot \cos \varphi =U\cdot I\cdot \cos \varphi }$

• The (displacement) reactive power is the imaginary part of the complex power: ${\displaystyle Q}$

${\displaystyle Q=\mathrm {Im} \ {\underline {S}}=S\cdot \sin \varphi =U\cdot I\cdot \sin \varphi }$

• The apparent power is the amount of the complex power ${\displaystyle S}$

${\displaystyle S=\left|{\underline {S}}\right|=U\cdot I={\sqrt {P^{2}+Q^{2}}}}$

For example, if the voltage has the amplitude and the zero phase angle and the current has the amplitude and the zero phase angle , then one obtains the complex power ${\displaystyle {\hat {u}}=10\ {\text{V}}}$ ${\displaystyle \varphi _{u}=-30^{\circ }}$ ${\displaystyle {\hat {\imath }}=8\ {\text{A}}}$ ${\displaystyle \varphi _{i}=0^{\circ }}$

${\displaystyle {\underline {S}}=40\ {\text{VA}}\cdot \mathrm {e} ^{-\mathrm {j} \cdot 30^{\circ }}=40\ {\text{VA}}\cdot \left(\cos(-30^{\circ })+\mathrm {j} \cdot \sin(-30^{\circ })\right)\approx 34,64\ {\text{W}}-\mathrm {j} \cdot 20\ {\text{var}}}$

with the active power and the reactive power . If the current and voltage come from the consumer meter arrow system, this two-terminal pole is a capacitive ( ) consumer ( ). ${\displaystyle P=34,64\ {\text{W}}}$ ${\displaystyle Q=-20\ {\text{var}}}$ ${\displaystyle Q<0}$ ${\displaystyle P>0}$

Complex performance on passive two-pole systems

If the current strength and the impedance or admittance of a passive (linear) two-terminal network are given, then (assuming the consumer meter arrow system) also applies to the complex power ${\displaystyle {\underline {U}}={\underline {Z}}\cdot {\underline {I}}={\frac {\underline {I}}{\underline {Y}}}}$

${\displaystyle {\underline {S}}={\underline {Z}}\cdot I^{2}={\frac {I^{2}}{\underline {Y}}}={\frac {{\underline {Z}}\cdot {\hat {\imath }}^{2}}{2}}={\frac {{\hat {\imath }}^{2}}{2\ {\underline {Y}}}}}$

If, on the other hand, the voltage and the impedance or admittance are given, then also applies ${\displaystyle {\underline {I}}={\underline {Y}}\cdot {\underline {U}}={\frac {\underline {U}}{\underline {Z}}}}$

${\displaystyle {\underline {S}}={\underline {Y}}^{\star }\cdot U^{2}={\frac {U^{2}}{{\underline {Z}}^{\star }}}={\frac {{\underline {Y}}^{\star }\cdot {\hat {u}}^{2}}{2}}={\frac {{\hat {u}}^{2}}{2\ {\underline {Z}}^{\star }}}}$

This gives you for active, reactive and apparent power

${\displaystyle P=\mathrm {Re} \ {\underline {S}}=I^{2}\cdot \mathrm {Re} \ {\underline {Z}}=U^{2}\cdot \mathrm {Re} \ {\underline {Y}}={\frac {{\hat {\imath }}^{2}}{2}}\cdot \mathrm {Re} \ {\underline {Z}}={\frac {{\hat {u}}^{2}}{2}}\cdot \mathrm {Re} \ {\underline {Y}}}$

${\displaystyle Q=\mathrm {Im} \ {\underline {S}}=I^{2}\cdot \mathrm {Im} \ {\underline {Z}}=-U^{2}\cdot \mathrm {Im} \ {\underline {Y}}={\frac {{\hat {\imath }}^{2}}{2}}\cdot \mathrm {Im} \ {\underline {Z}}=-{\frac {{\hat {u}}^{2}}{2}}\cdot \mathrm {Im} \ {\underline {Y}}}$

${\displaystyle S=\left|{\underline {S}}\right|=I^{2}\cdot \left|{\underline {Z}}\right|=U^{2}\cdot \left|{\underline {Y}}\right|={\frac {U^{2}}{\left|{\underline {Z}}\right|}}={\frac {I^{2}}{\left|{\underline {Y}}\right|}}={\frac {{\hat {\imath }}^{2}}{2}}\cdot \left|{\underline {Z}}\right|={\frac {{\hat {u}}^{2}}{2}}\cdot \left|{\underline {Y}}\right|={\frac {{\hat {u}}^{2}}{2\ \left|{\underline {Z}}\right|}}={\frac {{\hat {\imath }}^{2}}{2\ \left|{\underline {Y}}\right|}}}$

As an example, one obtains the reactive power of a capacitor with the capacitance at a voltage with the amplitude and the angular frequency ${\displaystyle C}$ ${\displaystyle {\hat {u}}}$ ${\displaystyle \omega }$

${\displaystyle Q=-{\frac {{\hat {u}}^{2}\cdot \omega \ C}{2}}}$

Specifically, at and on a capacitor with a capacity of a reactive power of . ${\displaystyle 230\ {\text{V}}}$ ${\displaystyle 50\ {\text{Hz}}}$ ${\displaystyle 100\ {\text{nF}}}$ ${\displaystyle Q\approx -1,66\ {\text{var}}}$

Relationship of the complex performance with the instantaneous performance

The flow of energy into a two-pole is described by the instantaneous value of the electrical power and is the product of the real instantaneous values ​​of voltage and current. When using the cosine notation for the real signals, ${\displaystyle p(t)}$

${\displaystyle p(t)=u(t)\cdot i(t)=\mathrm {Re} \ {\underline {u}}\;\cdot \;\mathrm {Re} \ {\underline {i}}}$

Because of the relationship for complex numbers one can write ${\displaystyle \mathrm {Re} \ {\underline {a}}={\frac {1}{2}}({\underline {a}}+{\underline {a}}^{\star })}$

${\displaystyle p(t)={\frac {1}{4}}({\underline {u}}+{\underline {u}}^{\star })\cdot ({\underline {i}}+{\underline {i}}^{\star })={\frac {1}{4}}({\underline {u}}\;{\underline {i}}^{\star }+{\underline {u}}^{\star }\;{\underline {i}}\ +{\underline {u}}\;{\underline {i}}+{\underline {u}}^{\star }\;{\underline {i}}^{\star })}$

and formulate it again as a real part after rearranging

${\displaystyle p(t)={\frac {1}{2}}\mathrm {Re} \ ({\underline {u}}\ {\underline {i}}^{\star }+{\underline {u}}\ {\underline {i}})={\frac {1}{2}}\mathrm {Re} \left({\hat {u}}\ {\hat {\imath }}\ \mathrm {e} ^{\mathrm {j} (\varphi _{u}-\varphi _{i})}+{\hat {u}}\ {\hat {\imath }}\ \mathrm {e} ^{\mathrm {j} (2\omega t+\varphi _{u}+\varphi _{i})}\right)}$

and introduce complex rms values

${\displaystyle p(t)=\mathrm {Re} \left({\underline {U}}\ {\underline {I}}^{\star }+{\underline {U}}\ {\underline {I}}\ \mathrm {e} ^{\mathrm {j} 2\omega t}\right)=\mathrm {Re} \left({\underline {U}}\ {\underline {I}}^{\star }+{\underline {U}}\ {\underline {I}}^{\star }\ \mathrm {e} ^{\mathrm {j} 2(\omega t+\varphi _{i})}\right)=\mathrm {Re} \left[{\underline {U}}\ {\underline {I}}^{\star }\left(1+\mathrm {e} ^{\mathrm {j} 2(\omega t+\varphi _{i})}\right)\right]}$

This ultimately results in the fundamental relationship between instantaneous performance and complex performance:

${\displaystyle p(t)=\mathrm {Re} \left[{\underline {S}}\cdot \left(1+\mathrm {e} ^{\mathrm {j} 2(\omega t+\varphi _{i})}\right)\right]}$

This makes it possible to analyze the instantaneous power without using the addition theorems of the circular functions .

Analysis of the current performance

By forming the real part of the product

${\displaystyle p(t)=\mathrm {Re} \left({\underline {S}}\right)\cdot \mathrm {Re} \left(1+\mathrm {e} ^{\mathrm {j} 2(\omega t+\varphi _{i})}\right)-\mathrm {Im} \left({\underline {S}}\right)\cdot \mathrm {Im} \left(1+\mathrm {e} ^{\mathrm {j} 2(\omega t+\varphi _{i})}\right)}$

and then actually executes the real or imaginary part formation, the formula known from classical power calculation (using the cosine notation for the real signals) finally results

${\displaystyle p(t)=P\cdot \left(1+\cos {2(\omega t+\varphi _{i})}\right)-Q\cdot \sin {2(\omega t+\varphi _{i})}}$

Instantaneous power (curve 1) as a superposition of reactive power generating oscillation (curve 2) and active power generating oscillation (curve 3) at the concrete zero phase angles and ${\displaystyle p(t)}$ ${\displaystyle \varphi _{u}=50^{\circ }}$ ${\displaystyle \varphi _{i}=90^{\circ }}$

Their components can be interpreted as follows:

1. The sum of both components is the total instantaneous power (marked as curve 1 in the diagram). It oscillates with twice the basic frequency around its mean value, which is equal to the real power , and has an amplitude in the size of the apparent power . ${\displaystyle p(t)}$ ${\displaystyle P}$ ${\displaystyle S=\left|{\underline {S}}\right|}$
2. The right-hand component of the instantaneous power (marked as curve 2 in the diagram) is also alternating with twice the basic frequency. Their mean over time is the same and their amplitude is the same as the amount of reactive power . The energy represented by this power flows "always alternately in the same amount" between the generator and load two-pole back and forth in both directions and can therefore have no effect on average over time. Based on their definition, assuming the consumer counting arrow system, positive reactive power is generally caused by inductances and negative reactive power by capacities. In practice, it is said that inductors “draw reactive power” and capacities “deliver reactive power”. ${\displaystyle 0}$ ${\displaystyle |Q|}$
3. The left component of the instantaneous power (marked as curve 3 in the diagram) consists (because of ) of " oscillations" (sinusoidally oscillating at twice the basic frequency), which rise from to and thus have twice the level of the active power, whereby their mean value over time (im Diagram drawn with dashed lines) is also the same . The energy represented by this power always flows in the same direction and can therefore (on average over time) have an "actual effect" in the load two-pole (e.g. heating or mechanical work). The real power is therefore the amplitude of the "actually effective" component of the instantaneous power and at the same time its mean value over time. Since the active power is lower by a factor of the apparent power, it is called the active factor (sometimes also power factor ). A negative real power indicates a "rearward (opposite to the reference direction) energy transport". This case occurs when the two-pole system acts as an (active) generator in the consumer counting arrow system or a passive two-pole system is described in the generator counting arrow system. ${\displaystyle 2\cos ^{2}\alpha =1+\cos 2\alpha }$ ${\displaystyle \cos ^{2}}$ ${\displaystyle 0}$ ${\displaystyle 2\;P}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle \cos \varphi }$ ${\displaystyle \cos \varphi }$

On the basis of this analysis and the graphical representation, the following statements can also be made about active and reactive energy:

• The active energy that flows into or is given off by the two-terminal network during a period has the amount . ${\displaystyle T}$ ${\displaystyle W_{P}=|P|\cdot T}$
• The area of ​​a half-oscillation of the reactive power curve represents the “total energy oscillating back and forth” . Since the area of ​​a half sine wave is known , this results in energy . ${\displaystyle W_{Q}}$ ${\displaystyle 2}$ ${\displaystyle W_{Q}=|Q|\cdot {\frac {T}{2\pi }}={\frac {|Q|}{\omega }}}$

For example, for the “pendulum energy” of a capacitor with the capacitance at a voltage with the amplitude and the angular frequency corresponding to the reactive power specified above ${\displaystyle C}$ ${\displaystyle {\hat {u}}}$ ${\displaystyle \omega }$

${\displaystyle W_{Q}={\frac {{\hat {u}}^{2}\cdot \omega \ C}{2\ \omega }}={\frac {{\hat {u}}^{2}\cdot C}{2}}}$

But that is precisely the known energy of a capacitor charged to the voltage . ${\displaystyle {\hat {u}}}$

literature

• Klaus Lunze : Theory of AC circuits . 8th edition. Verlag Technik GmbH, Berlin 1991, ISBN 3-341-00984-1 .
• Reinhold Paul: Electrical engineering 2 - networks . Springer-Verlag, Berlin Heidelberg New York 1994, ISBN 3-540-55866-7 .