Locus curve (system theory)

Locus as a line in the complex number plane

By a locus is understood in the system theory , the graphical representation of one of a real parameter dependent complex system size.

Mathematically, the locus is defined as follows:

The path described by a parameter-dependent complex pointer in the complex plane of numbers is called a locus. ${\displaystyle {\underline {z}}={\underline {z}}(t)}$

${\displaystyle {\underline {z}}=\mathrm {R} \mathrm {e} ({\underline {z}})+\mathrm {j} \mathrm {I} \mathrm {m} ({\underline {z}})=x(t)+\mathrm {j} y(t)}$

with the imaginary unit . The parameter is an element of a half-open, open or closed interval of real numbers. In the illustrated example applies: . ${\displaystyle \mathrm {j} }$ ${\displaystyle t}$ ${\displaystyle a\leq t\leq b}$

Locus curves are used in various technical disciplines, in particular in control engineering , communications engineering , high-frequency engineering , energy engineering and acoustics (or other applications in vibration theory). They are used to represent the properties or the behavior of a technical system such as a control or an electrical circuit with graphic means.

Typical examples of complex system variables that are represented by loci are

• the frequency response of the transfer factor (i.e. the ratio of the complex input and output variables) of linear time-invariant systems as a function of the frequency,
• Complex alternating current quantities (current, voltage) as well as impedances (complex resistances) and admittances (complex conductance values) of two-poles .

The parameter is often, but not mandatory, the frequency. Typical parameters in the theory of lines are, for example, the line length or the matching ratio . The impedance of a resistor , a coil or a capacitor at constant frequency is also given as a function of interfering (parasitic) component sizes (for example, a real coil not only has the desired inductance, but also a small ohmic resistance and a small capacitance).

Equations for the complex system size to be represented

In systems that consist of a finite number of concentrated components, the system size can be represented as a fractional rational function in the following form:

${\displaystyle {\underline {O}}(t)={\frac {{\underline {A}}+{\underline {B}}t+{\underline {C}}t^{2}+{\underline {D}}t^{3}+\dotsb }{{\underline {a}}+{\underline {b}}t+{\underline {c}}t^{2}+{\underline {d}}t^{3}+\dotsb }}}$.

Here is a real parameter, and and are complex quantities. The underscore indicates that they are complex. ${\displaystyle t}$ ${\displaystyle {\underline {A}},\dotsc ,{\underline {D}},\dotsc }$ ${\displaystyle {\underline {a}},\dotsc ,{\underline {d}},\dotsc }$

If the frequency is considered as the parameter, it is common to choose the angular frequency as the independent variable . In this case the following equation represents the system size: ${\displaystyle \omega =2\pi f}$

${\displaystyle H(\mathrm {j} \omega )={\frac {Y(\mathrm {j} \omega )}{X(\mathrm {j} \omega )}}={\frac {b_{0}+b_{1}(\mathrm {j} \omega )+\dotsb +b_{m}(\mathrm {j} \omega )^{m}}{a_{0}+a_{1}(\mathrm {j} \omega )+\dotsb +a_{n}(\mathrm {j} \omega )^{n}}}}$.

Because it always occurs together with the imaginary unit , it has become common practice, especially in control engineering, to specify the product as a parameter . The recorded unit makes it clear that these are complex quantities. The underscore can be omitted. ${\displaystyle \omega }$ ${\displaystyle \mathrm {j} }$ ${\displaystyle \mathrm {j} \omega }$ ${\displaystyle \mathrm {j} }$

Examples

Communications engineering

Circuit diagram of an RC low pass

Locus curve for the frequency response of an RC low pass . It represents the complex voltage transfer factor (complex quotient V of the sinusoidal output voltage to the input voltage)

Locus curves describe the transfer behavior of circuits that contain linear phase-shifting components (capacitors, coils) and are treated as imaginary reactances . Typical applications are resonant circuits or filters that ideally only allow electrical signals to pass through at certain frequencies or frequency ranges and otherwise block them; see for example low pass , high pass .

The frequency response of a low-pass filter (see illustration) has the following expression with the formula symbols for the quotient of the complex output signal and the complex input signal and for the angular frequency: ${\displaystyle {\underline {V}}}$ ${\displaystyle {\underline {u}}_{a}(\mathrm {j} \omega )}$ ${\displaystyle {\underline {u}}_{e}(\mathrm {j} \omega )}$ ${\displaystyle \omega }$

${\displaystyle {\underline {V}}(\mathrm {j} \omega )={\frac {{\underline {u}}_{a}(\mathrm {j} \omega )}{{\underline {u}}_{e}(\mathrm {j} \omega )}}}$.

With a low-pass filter as an RC element , the equation for the complex voltage transfer factor is:

${\displaystyle {\underline {V}}(\mathrm {j} \omega )={\frac {{\underline {u}}_{a}(\mathrm {j} \omega )}{{\underline {u}}_{e}(\mathrm {j} \omega )}}={\frac {1}{1+CR\mathrm {j} \omega }}}$.

The numerator polynomial is reduced to . ${\displaystyle 1}$

The locus of the transfer  factor fulfills the circular equation of a circle with the radius R  = 0.5 around the point M = 0.5+ 0 j, because the following applies:

${\displaystyle \left|{\underline {V}}-{\frac {1}{2}}\right|=\left|{\frac {1}{1+j\omega CR}}-{\frac {1}{2}}\right|=\left|{\frac {1-j\omega CR}{2\cdot (1+j\omega CR)}}\right|={\frac {1}{2}}}$

The PT ₁ element that occurs in control engineering can be understood as a combination of an RC low-pass filter with the time constant and a frequency-independent amplifier with the gain factor. ${\displaystyle T=R\cdot C}$ ${\displaystyle K}$

${\displaystyle H(\mathrm {j} \omega )={\frac {K}{1+T\mathrm {j} \omega }}}$

Control engineering

Locus of the frequency response of a PT ₂ -member (K = 1, d <1)

The locus of the frequency response used in control technology is also called the Nyquist diagram . Harry Nyquist has formulated a stability criterion for regulations with the help of this locus .

The locus of the frequency response is drawn and used both for individual components and for component groups up to the complete chain of the cut control loop . The curve for a PT ₂ element (amplifier with 2nd order delay) is shown.

The frequency response of this element is the following expression with the gain factor, the attenuation factor and the time constant : ${\displaystyle K}$ ${\displaystyle d}$ ${\displaystyle T}$

${\displaystyle H(\mathrm {j} \omega )={\frac {Y(\mathrm {j} \omega )}{X(\mathrm {j} \omega )}}={\frac {K}{1+2dT\mathrm {j} \omega -T^{2}\omega ^{2}}}}$

Electrical Power Engineering

Locus curve of impedance Z ( series connection of inductance jωL and variable ohmic resistance R (p))

In energy technology , the frequency of the current is constant, which is why locus curves are used to represent and investigate transmission ratios that vary with a parameter other than frequency. The values ​​of ohmic resistances, coils and capacitors come into question as variable values ​​in the system. Most often the complex impedance (quotient of complex voltage and complex current ) or the complex conductance (quotient of complex current and complex voltage) is represented. ${\displaystyle u}$ ${\displaystyle i}$

The complex equation for impedance is connected to the parameter (in R  =  p  · R ₀ ) and the sign of the impedance (see illustration), the following expression: ${\displaystyle p}$ ${\displaystyle {\underline {Z}}}$

${\displaystyle {\underline {Z}}(p)=L\mathrm {j} \omega +R_{0}p}$

The denominator polynomial is reduced to . ${\displaystyle 1}$

The creation of loci

The relationships that can be represented with loci can be determined by measuring the magnitude and phase, and the curves can be drawn point by point with the pairs of measured values ​​in the complex plane. The first and third of the figures show that loci often have a simple geometric shape and can be deduced from a few pairs of measured values.

This fact also makes it possible to specify such simple locus curves (straight lines, circles, parabolas) purely theoretically, which can be sufficient in particular for qualitative considerations. Their inversions also have simple geometric shapes.

Inversion of loci

The inversion of locus curves is important, for example, when calculating the reciprocal value for calculating the conductance from the impedance ${\displaystyle {\underline {Y}}}$ ${\displaystyle {\underline {Z}}}$

${\displaystyle {\underline {Y}}={\frac {1}{\underline {Z}}}}$

It is a special case of Möbius transformation and can be carried out graphically in simple cases using the following basic rules and the inversion of individual points.

original locus	inverted locus
Straight through the origin	Straight through the origin
Especially not through the origin	Circle through the origin
Circle through the origin	Especially not through the origin
Circle not through the origin	Circle not through the origin

See also

• Nyquist diagram
• Complex AC bill
• Phase response
• Smith Chart
• Hodograph

literature

• Wilfried Weißgerber: Electrical engineering for engineers 2, 5 loci . 6th edition. Vieweg + Teubner, 2007, ISBN 978-3-8348-0191-3 .
• Heinz Unbehauen: Control engineering 1 . 14th edition. Vieweg + Teubner, 2007, ISBN 978-3-384-80230-9 , pp. 80-86 .
• Gert Hagmann : Fundamentals of electrical engineering . 11th edition, Wiebelsheim 2005, ISBN 3-89104-687-1 .

Individual evidence

1. ^ Lothar Papula: Mathematical formula collection: For engineers and natural scientists. Vieweg, Wiesbaden 2006, ISBN 978-3-8348-0156-2 ( limited preview in the Google book search)
2. ↑ Wilfried Weißgerber: Electrical engineering for engineers 2. 5 loci. 6th edition. Vieweg + Teubner, 2007, ISBN 978-3-8348-0191-3 .
3. ↑ for example in Jan Lunze: Regelstechnik 1. Springer, 2007, ISBN 978-3-540-70790-5 , p. 224.