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# Sign convention for passive components and sources

A standard practice for labeling current and voltage on resistors, capacitors, and inductors. Labeling voltage and current sources. Written by Willy McAllister.

## Sign convention for passive components

We need a simple, widely understood way to refer to voltages and currents in a circuit. The purpose of the sign convention developed here is to define what we mean by positive and negative voltages and currents.
Why do we need a sign convention? Passive components (resistors, capacitors, inductors) have a defining equation (Ohm's Law and others). These equations establish a relationship between voltage and current. We can't just assign voltage polarity and current direction any which way. Voltage polarity and current direction have to be consistent with each other. The universal convention for voltage polarity and current direction for two-terminal components is shown below:
This is called the sign convention for passive components.
Voltage polarity: The illustration above shows voltage polarity with two notations in orange: plus and minus signs, and an arrow. The voltage arrow points from minus towards plus. The signs and the arrow are redundant, they mean exactly the same thing. You can use either, or both in your schematics. Anything that enhances clarity is always a good idea. The voltage arrow is drawn with a slight curve. This helps identify it as a voltage arrow, and not mistaken for a straight current arrow.
Current direction: The blue arrow shows the direction assigned to positive current flow. Current arrows should be drawn so current flows into the plus voltage terminal and flows out of the minus voltage terminal.
All three of the current arrows in the next image mean the same thing.
The reason for this convention is so the signs of current and voltage come out right when we apply the defining equations for each component, like Ohm's Law for a resistor.

### Example 1

This 250, \Omega resistor has been labeled using the sign convention for passive components. The voltage polarity (orange signs and arrow) have been assigned with plus at the top of the resistor. This direction was an arbitrary choice. The blue current arrow points into the positive terminal. This was not an arbitrary choice. Positive current has to flow into the plus sign.
Something (not shown, a voltage source or surrounding circuit) has caused 2, start text, v, o, l, t, s, end text to appear across the resistor.
What is i?
To find the current, apply Ohm's Law:
i, equals, start fraction, v, divided by, start text, R, end text, end fraction
i, equals, start fraction, plus, 2, start text, V, end text, divided by, 250, \Omega, end fraction
i, equals, plus, 8, start text, m, A, end text
The voltage arrow tells us the top of the resistor is 2, start text, V, end text above the bottom of the resistor. Ohm's Law tells us the current is plus, 8, start text, m, A, end text. The plus sign on current means it is flowing in the direction of the arrow, from top to bottom.

### Example 1X - the wrong way

What would happen if we labeled the resistor with the wrong sign convention? The diagram below shows the same resistor with the same voltage polarity, but the current arrow points out of the positive terminal, so the sign convention for passives is not being used.
Apply Ohm's Law, exactly the same as Example 1,
i, equals, start fraction, plus, 2, start text, V, end text, divided by, 250, \Omega, end fraction
i, equals, plus, 8, start text, m, A, end text
Ohm's Law is telling us the current is plus, 8, start text, m, A, end text. The plus sign on current means it is flowing in the direction of the arrow, from bottom? to top?. What? In a real resistor the current would be flowing the other way. We got the wrong answer. Lesson: use the sign convention for passives.

### Example 2

This 10, start text, k, end text, \Omega resistor has been labeled with the same sign convention as the first example: The orange voltage polarity has plus at the top and the blue current arrow points down. This time, the current is specified instead of the voltage. The value of the current is minus, 20, mu, start text, A, end text. This may look a little odd, to show minus, 20, mu, start text, A, end text current flowing in the direction of the arrow. If you like, think of it as a plus, 20, mu, start text, A, end text current going in the opposite direction (flowing from bottom to top in the resistor).
What is v?
We use Ohm's Law to solve for the unknown voltage. Since we've been careful to use the sign convention, all we have to do is plug in the actual values shown on the schematic. (Avoid the temptation to flip signs around in your head as you write these equations. This often leads to errors.)
v, equals, i, start text, R, end text
v, equals, minus, 20, mu, start text, A, end text, dot, 10, start text, k, end text, \Omega
v, equals, left parenthesis, minus, 20, times, 10, start superscript, minus, 6, end superscript, right parenthesis, dot, left parenthesis, 10, times, 10, start superscript, plus, 3, end superscript, right parenthesis
v, equals, minus, 0, point, 2, start text, V, end text
The answer came out with a negative sign, meaning the resistor terminal with the plus voltage polarity (the top terminal) is 0, point, 2, start text, V, end text below the terminal with the minus sign (the bottom of the resistor). Using the labeling convention lets the math produce the correct sign, even with the quirky-looking negative current.
This labeling convention for passives is not just a good idea, it is the only way to get the right answer when analyzing a circuit.

## Sign convention for ideal sources

### Voltage sources

The voltage across an ideal voltage source is independent of the current flowing through it. An ideal voltage source can be defined by an equation like this: v, equals, start text, V, end text, for example: v, equals, 1, point, 5, start text, V, end text. The equation does not have a term related to the current i.
If you need to label the current through a voltage source, it can be done a few ways. In general, the options are:
1. No current label. Usually you don't need to label current through a voltage source. The surrounding circuit context determines the direction of the current, (illustration 1).
2. If you are doing power calculations, v, dot, i, you probably want the correct sign for power: plus sign for power dissipation and minus sign for generation. Use the same convention we defined for passive components: Current points into the positive voltage terminal of a voltage source (illustration 2).
3. If it is important (or comforting) for the sign of current in a voltage source to have a positive sign, then use a convention where the current arrow points out of the positive voltage terminal, (illustration 3).
In most cases, the current flows out of the positive terminal of a voltage source. If you apply the passives sign convention to the voltage source, in most cases the current ends up with a negative sign. This current arrow direction may feel "wrong," or you may find it annoying, but it is not technically an error. It just means the current has a minus sign, which isn't a big deal.
My preference for labeling voltage sources is the first option: no convention. Different textbooks teach all versions of this sign convention. Be tolerant of those who learned from a different book. Everyone gets the right answer in the end.

### The label does not have to match the actual voltage

The label on a voltage source is usually oriented with the polarity arrow going in the same direction as the actual voltage generated by the source (1a.), but there is no law that says it has to. The black plus and minus signs inside the symbol circle show the actual orientation of the source voltage. It is acceptable to define the label on a voltage source with the opposite polarity of the source itself (1b.). It may look odd, but it is not broken.
A voltage source with two alternative labels,
1. The same voltage source labeled two ways, both valid:
1a. The usual label. v, start subscript, s, 1, end subscript, equals, start text, V, end text.
1b. The same voltage source, with the voltage label reversed. The current arrow is also reversed. This means v, start subscript, s, 2, end subscript, equals, minus, start text, V, end text and i, start subscript, s, 2, end subscript, equals, minus, i, start subscript, s, 1, end subscript.
For a battery symbol, the longer black line indicates the positive terminal of the battery. A battery with two alternative labels,
2. The same battery labeled two ways, both valid:
2a. The usual battery label. v, start subscript, B, 1, end subscript, equals, 1, point, 5, start text, V, end text
2b. The same battery with the label reversed. The current arrow is also reversed. This means v, start subscript, B, 2, end subscript, equals, minus, 1, point, 5, start text, V, end text and i, start subscript, B, 2, end subscript, equals, minus, i, start subscript, B, 1, end subscript.
When might you want to point the voltage label "backwards"? When we learn about Kirchoff's Voltage Law sometimes it is helpful to point all the voltage arrows in the same direction going around a loop, (to make it easier to get the signs in the equation right). If one of the elements in the loop is a battery or voltage source, the voltage arrow may point opposite the actual voltage polarity.
Remember, the voltage labels are just labels; they are there to establish a reference direction for voltage in the context of the overall circuit. The labels don't determine the internal properties of the voltage source or battery; that's the job of the black symbol.
In some ways, a voltage label is similar to a force vector in mechanics, if you assign a vector going up and then run through the math and find your answer is negative, it means it's actually going down. The direction is so you have a clear idea of which way things are actually moving when all is said and done.

## Current sources

The current through an ideal current source is independent of the voltage across it. The equation describing a current source is: i, equals, start text, I, end text, for example: i, equals, 1, start text, A, end text. Voltage v does not appear in this equation.
Current sources are usually labeled with a current arrow matching the direction of the symbol arrow, and no voltage indication. The actual voltage across the current source will emerge from the analysis of the surrounding circuit. If you need to label the voltage for some reason, it is usually done as shown in option 2, similar to the sign convention for passive components.
Options for labeling a current source:
1. Current arrow only. The voltage polarity is determined by the surrounding components.
2. Current arrow and voltage polarity, using the sign convention for passive components.