Location curve

The representation of locus curves is a means of visualising the parameter-dependent change in complex variables. Impedance and admittance locus curves are useful for analysing the frequency-dependent behaviour of single-port networks. These curves can be used to represent the change in impedance or admittance as a function of (circular) frequency.

1 Definition of location curve

A locus curve is the graphical representation of a complex quantity \(\underline {z}\) as a function of a real parameter \(p\) in the complex plane[?, See ]: \begin {equation} \begin {aligned} \label {eq:def:ortskurve} \underline {z}(p) &= \Re \{\underline {z}(p)\} + \mathrm {j}\Im \{\underline {z}(p)\} &&\text {mit }\underline {z}\in \mathbb {C};\ p\in \mathbb {R} \end {aligned} \end {equation}Locus curves are used in many different areas to visualise possible states of a system. Figure 1 shows an example of the current locus curve of an asynchronous machine in the complex plane, also known as the Heyland circle or Ossanna circle. [SOURCE:XYZ] The complex stator current \(\underline {I}_s\) can be seen as a function of the slip \(s\). This form of representation is suitable, for example, for visualising different operating states. In the case of the asynchronous machine, these are motor, brake and generator operation.

In the example of the current locus curve of an asynchronous machine, various power values can also be determined graphically as shown in Figure 2. The examples serve to illustrate the definition and possible areas of application of locus curves, which is why we will refrain from deriving both examples.

Locus curves are well suited for visualising system states, variables and their influence on each other.

2 Relationship between pointer diagram and location curve

A pointer in the phasor diagram represents a complex quantity stationary in the complex plane. This form of representation serves to visualise phase shifts and magnitude ratios. Typical quantities for alternating current circuits include voltage, current, power and impedance.

Locus curves describe the path that a pointer travels when a parameter varies. The locus curve can thus be understood as a generalisation of a pointer in the pointer diagram. [?, Compare]Common parameters are frequency and component quantities. Common quantities are impedance and admittance.

Highlighting individual pointers with an indication of the varying parameter can clarify the course of the locus curve and facilitate interpretation. The rate and direction of change in position provide information about the dynamics of the system.

3 impedance locus curve

Impedance locus curves show the impedance of a system as a function of a parameter.

Example ?? shows an impedance locus curve for a variable component size.

Example ?? shows an impedance locus curve for variable frequency.

Impedance locus curve RL circuit, variable resistanceortskurve:z:rvarConstruction of the impedance locus curve of an RL series circuit for variable resistance \(R\). The circuit is shown in Figure 3.

The impedance of the RL element is given by:

\begin {equation*} \underline {Z} = R + \mathrm {j}\omega L \end {equation*}

If \(R\) is variable and described as (\(p\)-fold) multiples of a reference resistance \(R_0\) with \(p\) in the closed interval \([0,3]\), then the following applies:

\begin {equation*} R(p) = p \cdot R_0 \text {mit } p \in [0,3] \end {equation*}

Let \(\omega \) and \(L\) be constant, and let the reactance \(\omega L\) be equal to a reference reactance \(X_0\):

\begin {equation*} \omega L = X_0 \text {mit } \omega ,\ L = konst. \end {equation*}

Impedance \(\underline {Z}\) as a function of \(p\):

\begin {equation*} \underline {Z}(p) = p \cdot R_0 + \mathrm {j}X_0 \end {equation*}

Impedance locus curve RL element, variable frequencyortskurve:z:fvar

Construction of the impedance locus curve of the RL series connection in Figure 5 for variable frequency \(f\) and/or variable inductance \(L\). Setup as in Example ??, but with variable frequency \(\omega \).

Total impedance in general: \begin {align*} \underline {Z} &= R + \mathrm {j}\omega L \\ &= R + \mathrm {j}X \end {align*}

The total reactance \(X\) of the RL element is proportional to \(\omega \) and proportional to \(L\). This means that a change in \(\omega \) has the same effect as a change in \(L\) on the impedance of the RL element.

With \(X\) as a multiple \(p\) of a reference reactance \(X_0\) and \(R=R_0\), the following applies:

\begin {equation*} \underline {Z} = R_0 + p \cdot \mathrm {j}X_0 \end {equation*}

Shown at a complex level:

4 Admittance locus curve - Inversion of locus curves

Analogous to the impedance curve, an admittance curve describes the location curve of an admittance. The admittance \(\underline {Y}\) corresponds algebraically to the reciprocal of the impedance \(\underline {Z}\) and vice versa:

\begin {equation} \underline {Y} = \frac {1}{\underline {Z}} \leftrightarrow \underline {Z} = \frac {1}{\underline {Y}} \end {equation}

The conversion of impedance to admittance and vice versa is a special case of the Möbius transformation.

The following general properties of Möbius transformation therefore apply to the relationship between admittance and impedance locus curves: [?, Compare ]

• Angle preservation: \(\angle (\underline {Z_1},\underline {Z_2}) = \angle (\underline {Y_1},\underline {Y_2}) \qquad \text {with } \underline {Y}_i = \underline {Z}_i^{-1},\ i \in [1,2]\)
• Circle similarity: Circles and straight lines are mapped onto circles and straight lines

The formation of the reciprocal value corresponds in particular to inversion, an elementary type of Möbius transformation. An inversion of locus curves leads to the geometric transformations of circles and straight lines listed in Table 1. Figure 7 shows an example of the inversion of a lattice through the origin.

5 Local curves of basic circuits

Table 2 shows an overview of the impedance and admittance locality curves for the four basic circuits RL and RC elements in series and parallel connection.

Location curves for oscillatory LC and RLC elements differ from the location curves shown here in that the reactive component of the impedance or admittance can be either inductive or capacitive, depending on the frequency range, depending on which component predominates.

\(\vphantom {\bigg |}\)Basic-circuit	\(\mathbf {\underline {Z}}\)-Locus-curve	\(\mathbf {\underline {Y}}\)-Locus-curve