Spring 2018

ECE 110

Course Notes

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Suppose you want to investigate the devices that you find in your lab kit. You build a simple test circuit by connecting a voltage source $V_x$, a resistor $R_1$ and the circuit element you want to test, all in series. (The resistor is there to prevent a short circuit in case the element is another voltage source.) If you measure the voltage drop $V$ across the element and the current $I$ through it, you have a data point $(V, I)$ about the element's behavior.

Figure 1

Fig. 1: Test circuit. The voltage source $V_x$ and the resistor $R_1$ form the fixed part of the test circuit and we say they comprise the source subcircuit. The element being tested is interchangeable and we say it forms the load subcircuit. Typically, the source subcircuit delivers electrical energy to the load subcircuit, but this is not always the case. $V$ is the voltage drop from terminal $a$ to terminal $b$ and $I$ is the current from $a$ to $b$ through the load subcircuit.

Notice that $V$ and $I$ are in standard labeling with respect to the load subcircuit. But they are in nonstandard labeling in relation to the source subcircuit because $I$ flows from terminal $b$ to terminal $a$ through that part of the circuit. The source subcircuit in Fig. 1 is called a Thévenin equivalent subcircuit.

If you vary $V_x$ in the test circuit in Fig. 1, then the measured data point $(V, I)$ varies as well. In this way, you can trace a plot of the current-voltage relationship called the I-V characteristic of the element. When the element is in the load subcircuit, we plot the I-V characteristic with $V$ and $I$ in standard labeling.

Element Schematic I-V Characteristic Explanation
Resistor   A resistor $R$ satisfies Ohm's law $I=V/R$, so its I-V characteristic goes through the origin and has slope $1/R$.
Voltage
Source
A voltage source $V_s$ maintains a fixed voltage drop and can allow any current, so its I-V characteristic is a vertical line at $V=V_s$.
Current
Source
A current source $I_s$ maintains a fixed current and can allow any voltage drop, so its I-V characteristic is a horizontal line at $I=-I_s$. Note that there is a negative sign because the current arrow labels on $I$ and $I_s$ are in opposite directions.
Short
Circuit
A short circuit is a direct connection between two terminals. The short circuit maintains zero voltage drop and can allow any current, so its I-V characteristic is a vertical line at $V=0$. Notice that a short circuit behaves identically to a zero voltage source with $V_s=0\text{ V}$.
Open
Circuit
An open circuit is the absence of a connection between two terminals. The open circuit maintains zero current and can allow any voltage drop, so its I-V characteristic is a horizontal line at $I=0$. Notice that an open circuit behaves identically to a zero current source with $I_s=0\text{ A}$.
Table 1: I-V characteristics of basic circuit elements.

Resistors and sources have linear I-V characteristics, but we will see that other circuit elements have nonlinear ones. For example, a diode's I-V characteristic has a bend.

The load subcircuit may consist of more than one element in series and/or parallel combinations. The load I-V characteristic can be obtained from the elements' I-V characteristics using KVL and KCL. In fact, if the load subcircuit consists of sources and resistors only, the load I-V characteristic must be linear, as illustrated by the examples below.

Subcircuit   Schematic I-V Characteristic Explanation
Voltage
Source and
Resistor
in Series
KVL implies that $V=V_R+V_s$, so the load I-V characteristic is the resistor I-V characteristic shifted to the right by $V_s$ (i.e. along the $V$-axis pointing right by an amount $+V_s$).
Current
Source and
Resistor
in Parallel
KCL implies that $I=I_R-I_s$, so the load I-V characteristic is the resistor I-V characteristic shifted down by $I_s$ (i.e. along the $I$-axis pointing up by an amount $-I_s$).
Table 2: I-V characteristics of combinations of circuit elements in load subcircuit.

If a combination of elements is in the source subcircuit instead of the load subcircuit, then $V$ and $I$ are in nonstandard labeling. This difference occurs because the current arrow label of $I$ points out of the top terminal of the source subcircuit but into the top terminal of the load subcircuit. Therefore, the source I-V characteristic is a vertically flipped copy of the equivalent load I-V characteristic.

Subcircuit   Schematic I-V Characteristic Explanation
Voltage
Source and
Resistor
in Series
(Source)
The source I-V characteristic is a line with slope $-1/R$ and $V$-intercept of $V_s$. It is flipped vertically compared to the equivalent load I-V characteristic.
Current
Source and
Resistor
in Parallel
(Source)
The source I-V characteristic is a line with slope $-1/R$ and $I$-intercept of $I_s$. It is flipped vertically compared to the equivalent load I-V characteristic.
Table 3: I-V characteristics of combinations of circuit elements in source subcircuit.

We extend the formula for power to the case of nonstandard labeling by reversing its sign to account for switching the relative orientation of the voltage polarity and current arrow labels.

P = \begin{cases}
VI, & \text{under standard labeling,} \\
{-}VI, & \text{under nonstandard labeling.}
\end{cases} \label{IVC-PVI}

The I-V plane spans the axes on which an I-V characteristic is plotted. In Table 4 below, we use equation $\eqref{IVC-PVI}$ to identify the sign of $P$ in each quadrant under both standard and nonstandard labelings. We then determine whether a subcircuit operating in a given quadrant is absorbing or delivering power, depending on whether it is a load or source subcircuit.

Labeling Subcircuit Type Sign of $P$ by I-V Plane Quadrant
Nonstandard Labeling Source Subcircuit
Table 4: Signs of $P$ on the I-V plane.

The I-V characteristic of the load subcircuit represents all the $(V,I)$ values at which it can operate. Likewise, the source subcircuit's I-V characteristic represents all of its possible $(V,I)$ values. Therefore, when you plot both I-V characteristics together on the same I-V plane, their point of intersection is the $(V,I)$ value at which both subcircuits operate together. For this reason, the intersection is called the operating point.

We can use this insight to analyze a circuit graphically. Finding the operating point in this way is called the load line method. (Confusingly, the source I-V characteristic is known as the load line because it intersects the load I-V characteristic.) Table 5 below shows two examples for the test circuit in Fig. 1 with different load subcircuits, a resistor and current source, respectively.

Circuit   Schematic Load Line Method Plot
Fig. 1 with Resistor as Load Subcircuit
Fig. 1 with Current Source as Load Subcircuit
Table 5: Load line method examples.

Notice that, when the load subcircuit is a resistor, the operating point is in the upper-right quadrant, so the source subcircuit is delivering power $(P<0)$ and the load subcircuit is absorbing power $(P>0)$. But when the load subcircuit is a current source, the operating point is in the lower-right quadrant, so the source subcircuit is absorbing power $(P>0)$ and the load subcircuit is delivering power $(P<0)$.

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