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Load Line Analysis#

Load line analysis is a technique for analyzing circuits which are reducible to a time-invariant linear source and a time-invariant non-linear one-port connected in parallel:

Load Line Circuit

We are interested in what current flows through the two one-ports and what the voltage across them is.

We begin by choosing the a reference direction for the voltage:

Load Line Analysis Voltage

Since we have a source, the standard is to use the active sign convention for it, but we keep the passive sign convention for the non-linear one-port:

Load Line Analysis Current

Definition: Operating Point

An operating point of this circuit is any pair \((v, i)\) of values for \(V\) and \(I\) such that

\[ (v, i) \in \mathcal{F}_{\text{non-linear}} \qquad \text{and} \qquad (v, -i) \in \mathcal{F}_{\text{source}} \]

An operating point is essentially any configuration of voltage and current at which both the source and the non-linear one-port can operate.

Info: Why the Minus?

The negative sign for the source is there because I-V characteristics are always given in terms of the passive sign convention. However, the standard for sources is to use the active sign convention when analyzing them in the context of a particular circuit, which is why we chose the current \(I\) to flow out of the positive terminal of the source. The I-V characteristic \(\mathcal{F}_{\text{source}}\) is thus given in terms of the negative of \(I\).

If we define \(I' \overset{\text{def}}{=} -I\), we would get the following equivalent description:

Reverse Current Line Load Analysis

There are three possibilities for the operating points:
- There may be no operating points. This means that the circuit cannot function. What exactly happens physically, however, is impossible to predict. The circuit might explode, it might do nothing or it might behave unexpectedly.
- There may be exactly one operating point. In this case, the circuit will settle into this point.
- There may be multiple operating point. In this case, the circuit will settle into one of these points, but it is not generally possible to predict which one.

Load line analysis is a technique to determine how many operating points there are and what they are.

Algorithm: Algebraic Load Line Analysis

If \(\mathcal{F}_{\text{non-linear}}\) and \(\mathcal{F}_{\text{source}}\) have implicit representations \(f_{\text{non-linear}}\) and \(f_{\text{source}}\), respectively, then the operating points of the circuit are precisely the solutions to the following system of equations:

\[ \left\vert \begin{aligned} f_{\text{non-linear}}(V, I) &= 0 \\ f_{\text{source}}(V, -I) &= 0 \end{aligned} \right. \]

Algorithm: Geometric Load Line Analysis

We can find the operating points graphically by looking at the graphs of the I-V characteristics \(\mathcal{F}_{\text{non-linear}}\) and \(\mathcal{F}_{\text{source}}\).

  1. Draw the graphs of the I-V characteristics \(\mathcal{F}_{\text{non-linear}}\) and \(\mathcal{F}_{\text{source}}\):

Load Line Analysis - I-V Characteristic

  • We use \(I'\) and \(V'\) for \(\mathcal{F}_{\text{source}}\) to make it clear that we take into account the active sign convention, i.e. that \(I'\) corresponds to \(-I\). We use the label \(V'\) for consistency, but \(V'\) is still the same as \(V\).
  1. Draw \(\mathcal{F}_{\text{source}}\) on the same graph as \(\mathcal{F}_{\text{non-linear}}\). When doing this, we need to reflect \(\mathcal{F}_{\text{source}}\) across the \(V\) axis, since \(I = -I'\).

Load Line Analysis Graphical

  • This reflected line is known as the load line. The points where it intersects \(\mathcal{F}_{\text{non-linear}}\) are precisely the operating points.