Package Modelica.​Electrical.​QuasiStationary.​UsersGuide.​Overview
Overview

Information

The Modelica.Electrical.QuasiStationary library addresses the analysis of electrical circuits with purely sinusoidal voltages and currents. The main characteristics of the library are:

The main intention of this library is the modeling of quasi stationary behavior of single and multi phase AC circuits with fixed and variable frequency. Quasi stationary theory and applications can be found in [Dorf1993], [Burton1994], [Landolt1936], [Philippow1967], [Weyh1967], [Vaske1973].

Note

A general electrical circuit can be a DC circuit, an AC circuit with periodic sinusoidal or non-sinusoidal voltages and currents or a transient circuit without particular waveform of voltages and currents. Therefore a coupling model between a quasi stationary circuit and a general (transient) electrical circuit has to be designed carefully taking the specific application into account. As an example, you may look at the ideal AC DC converter, which is used in the rectifier example.

Extends from Modelica.​Icons.​Information (Icon for general information packages).

Package Contents

NameDescription
ACCircuitAC circuit
IntroductionIntroduction to phasors
PowerReal and reactive power
ReferenceSystemReference system

Class Modelica.​Electrical.​QuasiStationary.​UsersGuide.​Overview.​Introduction
Introduction to phasors

Information

The purely sinusoidal voltage

v=\sqrt{2}V_{\mathrm{RMS}}\cos(\omega t+\varphi_{v})

in the time domain can be represented by a complex rms phasor

\underline{v}=V_{\mathrm{RMS}}e^{j\varphi_{v}}.

For these quasi stationary phasor the following relationship applies:

\begin{displaymath} v=\mathrm{Re}(\sqrt{2}\underline{v}e^{j\omega t})\end{displaymath}

This equation is also illustrated in Fig. 1.

phasor_voltage.png
Fig. 1: Relationship between voltage phasor and time domain voltage

From the above equation it is obvious that for t = 0 the time domain voltage is v = cos(φv). The complex representation of the phasor corresponds with this instance, too, since the phasor is leading the real axis by the angle φv.

The explanation given for sinusoidal voltages can certainly also be applied to sinusoidal currents.

See also

AC circuit, Power, Reference system

Extends from Modelica.​Icons.​Information (Icon for general information packages).

Class Modelica.​Electrical.​QuasiStationary.​UsersGuide.​Overview.​ACCircuit
AC circuit

Information

A simple example of a series connection of a resistor, an inductor and a capacitor as depicted in Fig. 1 should be explained in the following. For various frequencies, the voltage drops across the resistor, the inductor and the capacitor should be determined.

resonance_circuit.png
Fig. 1: Series AC circuit of a resistor and an inductor at variable frequency

The voltage drop across the resistor

\underline{v}_{r}=R\underline{i}

and the inductor

\underline{v}_{l}=j\omega L\underline{i}

and the capacitor

\underline{v}_{l}=j\omega L\underline{i}

add up to the total voltage

\underline{v}=\underline{v}_{r}+\underline{v}_{l}

as illustrated in the phasor diagram of Fig. 2.

phasor_diagram.png
Fig. 2: Phasor diagram of a resistor and inductance series connection

Due to the series connection of the resistor, inductor and capacitor, the three currents are all equal:

img5.png

See also

Introduction, Power, Reference system

Extends from Modelica.​Icons.​Information (Icon for general information packages).

Class Modelica.​Electrical.​QuasiStationary.​UsersGuide.​Overview.​Power
Real and reactive power

Information

For periodic waveforms, the average value of the instantaneous power is real power P. Reactive power Q is a term associated with inductors and capacitors. For pure inductors and capacitors, real power is equal to zero. Yet, there is instantaneous power exchanged with connecting network.

The series resonance circuit which was also addressed in the AC circuit will be investigated.
Power of a resistor

The instantaneous voltage and current are in phase:

v_r.png
i_r.png

Therefore, the instantaneous power is

power_r.png

A graphical representation of these equations is depicted in Fig. 1

power_resistor.png
Fig. 1: Instantaneous voltage, current of power of a resistor

Real power of the resistor is the average of instantaneous power:

p_r.png

Power of an inductor

The instantaneous voltage leads the current by a quarter of the period:

v_l.png
i_l.png

Therefore, the instantaneous power is

power_l.png

A graphical representation of these equations is depicted in Fig. 2

power_inductor.png
Fig. 2: Instantaneous voltage, current of power of an inductor

Reactive power of the inductor is:

q_l.png

Power of a capacitor

The instantaneous voltage lags the current by a quarter of the period:

v_c.png
i_c.png

Therefore, the instantaneous power is

power_c.png

A graphical representation of these equations is depicted in Fig. 3

power_capacitor.png
Fig. 3: Instantaneous voltage, current of power of a capacitor

Reactive power of the capacitor is:

q_c.png

Complex apparent power

For an arbitrary component with two pins, real and reactive power can be determined by the complex phasors:

s.png

In this equation * represents the conjugate complex operator

See also

Introduction, AC circuit, Reference system

Extends from Modelica.​Icons.​Information (Icon for general information packages).

Class Modelica.​Electrical.​QuasiStationary.​UsersGuide.​Overview.​ReferenceSystem
Reference system

Information

The reference angle gamma:

Designing new components, the guidelines of the Modelica Specification dealing with Overconstrained Equation Operators for Connection Graphs have to be taken into account.

See also

Introduction, AC circuit, Power

Extends from Modelica.​Icons.​Information (Icon for general information packages).