Impedance and admittance loci
An infinite straight line in the impedance plane becomes a finite circular arc in the admittance plane. With that single property, the behavior of a motor was predicted for decades with compass and straightedge, and the Smith chart comes out of it.
01What a locus is
The impedance of a circuit is a complex number: a point on the plane. If one of its parameters changes —the frequency, an adjustable resistor, the load on a motor—, that point moves and draws a curve. That curve is the locus of the impedance, and looking at it all at once teaches more than calculating a hundred isolated values: it shows at a glance the complete range of possible behaviors of the circuit.
What is remarkable is that, for the most common circuits, these loci are always of two kinds: straight lines or circles. And going from the impedance \( Z \) to the admittance \( Y = 1/Z \) turns one into the other.
02Series impedances: straight lines
In a series circuit the impedances add, and if only one of the terms changes, the point moves in a single direction.
| Series circuit | Parameter that varies | Locus of \( Z \) |
|---|---|---|
| RL | \( R \), from 0 to ∞ | Horizontal half-line at height \( \omega L \) |
| RL | \( \omega \), from 0 to ∞ | Vertical half-line going up, on \( \text{Re}\,Z = R \) |
| RC | \( \omega \), from 0 to ∞ | Vertical half-line rising from \( -j\infty \) up to \( R \) |
| RLC | \( \omega \), from 0 to ∞ | Complete vertical line: it crosses the real axis at resonance |
In the same way, in a parallel circuit it is the admittances that add, and they are the ones that draw straight lines.
03Inversion: from lines to circles
Taking the inverse of a complex number is a transformation with a precious geometric property: a straight line that does not pass through the origin becomes a circle that does pass through the origin, and vice versa. Thus, the vertical half-line \( Z = R + j\omega L \) of a series RL becomes, in admittance, a semicircle.
Since the current is \( \mathbf{I} = \mathbf{V}\,Y \), with a fixed voltage the tip of the current phasor traces that same circle, scaled. This is the circle diagram that electrical engineers used for decades to predict the behavior of an induction motor with a compass and straightedge, without a single complex multiplication.
Series RL circuit with \( R = 100 \) Ω. You move the frequency: on the left the point \( Z \) climbs its line; on the right, \( Y = 1/Z \) descends along its semicircle.
04This is where the Smith chart comes from
The Smith chart is the most famous application of this idea. Instead of inverting \( Z \), it transforms the normalized impedance with \( \Gamma = (z-1)/(z+1) \), another transformation of the same type —bilinear—, which also maps lines and circles to lines and circles. That is why the lines of constant resistance and constant reactance, which in the \( Z \)-plane are straight lines, appear on the chart as the families of circles that make it recognizable. Everything done with the chart —matching an antenna, calculating a stub— is working with transformed loci.
05Where it shows up in electronics
| Case | Locus in play |
|---|---|
| Impedance matching in RF | Moving along circles of the Smith chart by adding series and parallel reactances |
| Induction motor | The circle diagram of the current as the slip varies |
| Vector network analyzer | Shows the impedance of a device as a curve as a function of frequency |
| Nyquist criterion (control) | The locus of \( H(j\omega) \) and its position relative to the point −1 decides stability |
06In the lab
For the series RL of 100 Ω and 10 mH, calculate the frequency at which \( X_L = R \) and verify with the lab that there \( Y \) falls at the lowest point of the semicircle, with \( G = B = 5 \) mS.
Sketch by hand the locus of \( Y \) of a parallel RC as \( \omega \) varies, and from it deduce the locus of \( Z \) by inversion.
With a generator and an oscilloscope, measure the magnitude and phase of the current of a series RL at ten frequencies and plot the points on the complex plane. Check that they fall on a circle.
07Common mistakes
- Expecting a straight line in admittance for a series circuit. In series it is the impedances that are straight lines; their inverses are circles.
- Forgetting that the circle passes through the origin. A straight line that does not pass through the origin always inverts into a circle that does pass through it.
- Plotting the frequency as if it were an axis. In a locus the parameter is not seen on the axes: it is marked along the curve.
- Confusing the circle diagram with the Smith chart. They are relatives, but the chart uses a different transformation and is normalized to a characteristic impedance.
08Self-assessment
What locus does \( Z \) of a series RL describe as the frequency varies?
A vertical half-line on \( \text{Re}\,Z = R \), toward positive reactances.
What does that line become when you go to admittance?
A semicircle centered at \( 1/2R \) with radius \( 1/2R \), which passes through the origin.
What property does complex inversion have?
It transforms lines and circles into lines and circles; a straight line that does not pass through the origin becomes a circle that does.
What is the circle diagram of a motor?
The locus of its current as the load varies, which turns out to be a circle because the current is proportional to the admittance.
Why do circles appear on the Smith chart?
Because the bilinear transformation that defines it takes the lines of constant resistance and constant reactance to circles.
09Further reading
- Mac E. Van Valkenburg. Network Analysis. 3rd ed., Prentice Hall, 1974 (Spanish edition, “Análisis de redes”, Limusa). The classic treatment of loci and of inversion in the complex plane.
- Phillip H. Smith. Electronic Applications of the Smith Chart. 2nd ed., Noble Publishing, 1995. The book by the chart’s author: how the transformation of loci became a design tool.
- Stephen J. Chapman. Electric Machinery Fundamentals. 5th ed., McGraw-Hill, 2011 (Spanish edition, “Máquinas eléctricas”). The circle diagram of the induction motor, the most important historical application of this unit.