Sign convention
The sign (positive or negative) of a variable is used to represent an orientation of its physical quantity. For instance, \(-3\) m/s could mean \(3\) m/s to the right or left. No one can say which is better (right is better). Deciding how the physical quantity corresponds to the sign of the variable is called sign assignment. When we use a sign convention, we make the assignment in a conventional manner. For instance, the sign convention for normal stress is that compression is negative and tension is positive.
Why use a sign convention? If we follow a convention when constructing a problem, we can use the convention’s interpretation of the result. For complicated systems, this helps us keep things straight. Furthermore, if someone else attempts to understand our work, it is much easier to simply say “using the standard sign convention, …” than explaining our own snowflake sign assignment. However, it is nonetheless true that we can assign signs arbitrarily.
Arbitrary? Vive la revolution! But wait. If a source is present, we must observe some caution. A source typically comes with its own convention. For instance, if we hook up a power supply to the circuit with the \(+\) and \(-\) leads a certain way, unless we want to get very confused, we should probably accommodate that sign.
A sign convention for each of the energy domains we’ve considered follows.
Electronic systems
We use the passive sign convention of electrical engineering, defined below.
Definition
Power flowing in to a component is considered to be positive and power flowing out of a component is considered negative.
Passive sign convention for electronic systems in terms of voltage v and current i. Passive elements are on the left, active on the right.
Because power \(\mathcal{P} = v i\), this implies the current and voltage signs are prescribed by the convention. For passive elements, the electrical potential must drop in the direction of positive current flow. This means the assumed direction of voltage drop across a passive element must be the same as that of the current flow. For active elements, which supply power to the circuit, the converse is true: the voltage drop and current flow must be in opposite directions. figure 2.4 illustrates the possible configurations.
When drawing a linear graph of a circuit, for each passive element’s edge, draw the arrow beside it pointing in the direction of assumed current flow and voltage drop.
The purpose of a sign convention is to help us interpret the signs of our results. For instance, if, at a given instant, a capacitor has voltage \(v_C = 3\) V and current \(i_C = -2\) A, we compute \(\mathcal{P}_C = -6\) W and we know \(6\) W of power is flowing from the capacitor into the circuit.
For passive elements, there is no preferred direction of “assumed” voltage drop and current flow. If a voltage or current value discovered by performing a circuit analysis is positive, this means the “assumed” and “actual” directions are the same. For a negative value, the directions are opposite.
For active elements, choose the sign in accordance with the physical situation. For instance, if a positive terminal of a battery is connected to a certain terminal in a circuit, it ill behooves one to simply say “but Darling, I’m going to call that negative.” It’s positive whether you like it or not, Nancy.
Translational mechanical systems
The following steps can be applied to any translational mechanical system. We introduce the convention with an inline example. Consider the simple mechanical system shown at right.
Assign the sign by drawing a coordinate arrow, as shown at right. The direction of the arrow is arbitrary, however, if possible, assign the positive direction to match the sources. If the problem allows, it is best practice to have all sources and the coordinate arrow in the same direction.
There are two nodes with distinct velocities: ground and the mass, as shown at right. The mass node is always drawn to ground. The spring connects between the mass and ground. Finally, the force source connects to the mass, where it is applied, and also connects to ground, which is impervious to it.
On each spring and damper element, define the positive velocity drop and edge arrow to be in the direction of the coordinate arrow.
On each mass element, define the positive velocity drop and edge arrow to be toward ground. Sometimes we dash the latter half of the mass edge in to signify that it is “virtually” connected to ground.
On each force source element, define the positive direction as
follows.
(ideal) If the force source has the same definition of positive
as your coordinate arrows, draw it toward the node of
application.
(if needed) If the force source has the opposite definition of
positive as your coordinate arrow, draw it away from the node of
application.
On each velocity source element, define the positive direction as follows. (ideal) If the velocity source has the same definition of positive as your coordinate arrows, draw it away from the node of application. (if needed) If the velocity source has the opposite definition of positive as your coordinate arrow, draw it toward the node of application.
This convention yields the interpretations of the following table.
For the system shown, draw a linear graph and assign signs according to the sign convention.
linear graph
Rotational mechanical systems
The following steps can be applied to any rotational mechanical system. We introduce the convention with an inline example. Consider the simple system shown at right.
Assign the sign by drawing a coordinate arrow, as shown at right. The direction of the arrow is arbitrary, however, if possible, assign the positive direction to match the sources. If the problem allows, it is best practice to have all sources and the coordinate arrow in the same direction. The right-hand rule is always implied.
There are two nodes with distinct velocities: ground and the inertia, as shown at right. The inertia node is always drawn to ground. The spring connects between the inertia and ground. Finally, the torque source connects to the mass, where it is applied, and also connects to ground, which is impervious to it.
On each inline spring and damper element, define the positive velocity drop and edge arrow to be in the direction of the coordinate arrow. Springs and dampers that aren’t inline typically connect to ground, toward which edge arrows should point.
On each inertia element, define the positive angular velocity drop and edge arrow to be toward ground. Sometimes we dash the latter half of the inertia edge to signify that it is “virtually” connected to ground.
On each torque source element, define the positive direction as
follows. (ideal) If the torque source has the same definition
of positive as your coordinate arrows, draw it toward the node of
application.
(if needed) If the torque source has the opposite definition of
positive as your coordinate arrow, draw it away from the node of
application.
On each angular velocity source element, define the positive
direction as follows. (ideal) If the source has the same
definition of positive as your coordinate arrows, draw it away from
the node of application.
(if needed) If the source has the opposite definition of
positive as your coordinate arrow, draw it toward the node of
application.
This convention yields the interpretations of the following table.
For the system shown, draw a linear graph and assign signs according to the sign convention.
linear graph
uh huh
Online Resources for Section 2.2
No online resources.