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### Course: Electrical engineering>Unit 2

Lesson 1: Circuit elements

# 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: $+$ and $-$ signs, and an arrow. The voltage arrow points from $-$ towards $+$. 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 $+$ voltage terminal and flows out of the $-$ 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 $250\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }$ resistor has been labeled using the sign convention for passive components. The voltage polarity (orange signs and arrow) have been assigned with $+$ 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 $+$ sign.
Something (not shown, a voltage source or surrounding circuit) has caused $2\phantom{\rule{0.167em}{0ex}}\text{volts}$ to appear across the resistor.
What is $i$?
To find the current, apply Ohm's Law:
$i=\frac{v}{\text{R}}$
$i=\frac{+2\phantom{\rule{0.167em}{0ex}}\text{V}}{250\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}$
$i=+8\phantom{\rule{0.167em}{0ex}}\text{mA}$
The voltage arrow tells us the top of the resistor is $2\phantom{\rule{0.167em}{0ex}}\text{V}$ above the bottom of the resistor. Ohm's Law tells us the current is $+8\phantom{\rule{0.167em}{0ex}}\text{mA}$. The $+$ 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,
$i=\frac{+2\phantom{\rule{0.167em}{0ex}}\text{V}}{250\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}$
$i=+8\phantom{\rule{0.167em}{0ex}}\text{mA}$
Ohm's Law is telling us the current is $+8\phantom{\rule{0.167em}{0ex}}\text{mA}$. The $+$ 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 $10\phantom{\rule{0.167em}{0ex}}\text{k}\mathrm{\Omega }$ resistor has been labeled with the same sign convention as the first example: The orange voltage polarity has $+$ 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 $-20\phantom{\rule{0.167em}{0ex}}\mu \text{A}$. This may look a little odd, to show $-20\phantom{\rule{0.167em}{0ex}}\mu \text{A}$ current flowing in the direction of the arrow. If you like, think of it as a $+20\phantom{\rule{0.167em}{0ex}}\mu \text{A}$ 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.)
$v=i\phantom{\rule{0.167em}{0ex}}\text{R}$
$v=-20\phantom{\rule{0.167em}{0ex}}\mu \text{A}\cdot 10\phantom{\rule{0.167em}{0ex}}\text{k}\mathrm{\Omega }$
$v=\left(-20×{10}^{-6}\right)\cdot \left(10×{10}^{+3}\right)$
$v=-0.2\phantom{\rule{0.167em}{0ex}}\text{V}$
The answer came out with a negative sign, meaning the resistor terminal with the $+$ voltage polarity (the top terminal) is $0.2\phantom{\rule{0.167em}{0ex}}\text{V}$ below the terminal with the $-$ 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: $v=\text{V}$, for example: $v=1.5\phantom{\rule{0.167em}{0ex}}\text{V}$. 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, $v\cdot i$, you probably want the correct sign for power: $+$ sign for power dissipation and $-$ 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 $-$ 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 $+$ and $-$ 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,
For a battery symbol, the longer black line indicates the positive terminal of the battery. A battery with two alternative labels,
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: $i=\text{I}$, for example: $i=1\phantom{\rule{0.167em}{0ex}}\text{A}$. 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.

## Want to join the conversation?

• Is there a topic i can study here regarding the math on ohms law?
Better yet, can someone walk me through the part where it said:
v = −20μA⋅10kΩ
to
v = (-20 x 10^-6) x (10 x 10 ^+3)
• I don't fully understand the current direction and voltage polarity for two-terminal components. First, don't electrons move from negative to positive, so why does the current enter from the positive terminal? Also, why does the voltage polarity face the opposite way from the current?
• Check out this video on how we define the direction of current: https://www.khanacademy.org/science/electrical-engineering/introduction-to-ee/intro-to-ee/v/current-direction

The definition of current direction used by all electrical engineers is to point the current arrow in the direction positive charge would move. Since electrons have a negative charge, they move in the opposite direction the current arrow is pointing. This quirky definition of current direction was created long before anyone knew that electrons existed, and that they carry the current in wires. There are no plans to change this definition, and, once you get used to it, it starts to feel normal.
• Maybe a stupid question, but why would one need a "voltage arrow"? Voltage is the energy per charge... so how can it have a direction? I'm so confused. Feels like these tutorials assumes that I know stuff that I don't know...
• Hello David,

Instead of direction let's call it polarity. You could also call it a rise or fall in potential. From a practical perspective it tells you if the voltmeter will read a positive or a negative voltage.

Yes, I understand your confusion. At first glance this passive sign convention seems completely useless. However, as you continue on it will start to be more and more important. You will appreciate the concept when you learn about Kirchhoff's Voltage Law (KVL).

Ref: https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws

Regards,

APD
• What is the difference between current source and voltage source? I think they are the same because both of them produce electricity.
• Both sources create voltage and current. A voltage source creates a constant voltage, and the current adjusts to whatever is required by the attached circuit. A current source outputs a constant current, and the voltage adjusts depending on what is connected.
• How did you get from "i=+2V/250Ω" to "i=+8mA"
I got "i=1V/125Ω" = 1A/125
• And if you divide 1 by 125, you will get 0.008 Ampere, which is 8 milli-Ampere (milli = one thousandth)
• Example 1: "What is i?"
i = +8mA
Example 1X: "What would happen if we labeled the resistor with the wrong sign convention?"
i = +8mA
Did we though? Because those two answers look awfully similar to me, despite one using the wrong convention. If one of those examples could be i=-8mA due to being misled somehow, then that ambiguity should be pointed out and explained. However, even if the math does work out to be the same for both examples somehow, comparing the two examples is not useful for justifying the use of the correct convention. Also, we're talking about a resistor in these examples, and AFAIK it doesn't actually matter which way they're oriented, mathematically or practically, so again maybe not the best illustration to justify convention? If what you're trying to say is that +8mA IS the wrong answer for a diagram labeled with current flowing the other way, then that is not clearly stated and it also implies that the actual direction of current should be changed instead of fixing the label, which is where the real problem is here. Either way, something needs to be clarified or corrected.
• The answers look similar, but one of them gives you the wrong current direction.

In both Example 1 and 1X there is +2 volts on the resistor, with the + voltage applied to the resistor's upper terminal. In both cases the resulting current will be 8 mA flowing from top to bottom in the resistor.

The sign convention tells us to draw a current arrow pointing into the positive voltage terminal of the resistor. With this convention, Ohm's Law gives you i = +8mA, which is the correct answer. If you accidentally draw the current arrow pointing the other way (coming out of the positive voltage terminal, or equivalently, pointing into the negative voltage terminal) when you apply Ohm's Law you again compute i = +8mA. BUT, since the arrow is pointing up, that says the current is flowing up, which it is not (it is flowing down).

Part of the challenge here is that current direction is indicated by mingling two signs. First you have the arrow symbol, which can point in two directions. Then you have the sign of i, as in i = +8mA or i = -8mA. The arrow direction and the arithmetic sign combine to give you the overall direction of current.
• Why would the label not match the actual?
• In general the label is like a vector in mechanics, if you assign it going up and then run through the math but 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.
• If the convention is to have current flow into the positive terminal of a battery, as this article states, then why are most circuit schematics modeled with the current flowing out of the positive terminal? To me this seems explicitly contradictory.
• Hello Sean,

Yes is is contradictory. Shrug - It's just the convention we use - positive to negative. It is regrettable that this is the first lesson electronics. It's not just here but in just about every electronics and physics textbook.

I think Randall Munroe said it best in his comic https://xkcd.com/567/

Regards,

APD