# Diode as a circuit element

Diodes conduct current in one direction but not the other. We solve a diode circuit graphically by plotting a diode i-v curve and resistor to find the intersection. Written by Willy McAllister.
These small glass packages have silicon diodes inside. The black band at one end indicates the side where current flows out of the diode.
The diode is our first semiconductor device. The diode's distinctive feature is that it conducts current in one direction, but not the other. We won't go into the details of how a diode does this, or how it's made. Fortunately, you don't have to know how to make a diode before using it in a circuit.
Semiconductor materials fall between insulators and conductors. They usually act like insulators, but we can control how much they conduct by changing the way they are made and by applying voltages.
The most well-known and well-understood semiconductor material is Silicon (Si, atomic number $14$). Silicon is by far the most common material used for creating semiconductor devices. More is known about silicon than perhaps any other material on Earth.
Other semiconductor materials include germanium (Ge, atomic number $32$, right below Silicon on the periodic table), and gallium-arsenide, a $1:1$ ratio of Gallium and Arsenic (atomic numbers $31$ and $33$, on either side of germanium).
A portion of the periodic table showing silicon (Si) and other nearby semiconductor materials. B - boron, C - carbon, N - nitrogen, Al - aluminum, Si - silicon, P - phosphorus, Ga - gallium, Ge - germanium, As - arsenic.
Our ability to finely control the conducting properties of silicon allows us to create modern marvels like computers and mobile phones and every other complex electronic device. The details of how a semiconductor works are governed by our understanding of quantum mechanics.

The $i$-$v$ curve of a diode is modeled by this non-linear equation:
$\text I_{\text S}$ is the reverse saturation current. For silicon, typically $10^{-12}\,\text A$.
$q$ is the charge on an electron, $1.602 \times 10^{-19} \,\text C$.
$k$ is Boltzmann's constant, $1.380\times 10^{-23} \,\text{J/K}$
$\text T$ is the temperature in kelvin.
• We will define terms like forward bias, reverse bias, and saturation current.
• You will learn some tips for identifying the terminals of a real-world diode.
• We will solve a diode circuit using a graphical method.

## Diode symbol

The schematic symbol for a diode looks like this:
The black arrow ▶ in the symbol points in the direction of the diode's forward current, $\blueD i$, the direction where current flow happens. The diode's voltage, $\goldD v$, is oriented with the $+$ sign on the end where forward current comes into the diode. We use the sign convention for passive components. The optional curved orange arrow also indicates the voltage polarity.

## Diode $i$-$v$ curve

This is a typical $i$-$v$ curve for a silicon diode. A diode is a non-linear device:
Diode $i$-$v$ curve of a silicon diode. A positive voltage means the diode is forward biased. A negative voltage means the diode is operating with reverse bias.

## Forward and reverse current

### Forward current

Let's say we place a very small positive voltage, like $+0.2$ volts, across a silicon diode. That puts us on the right side of the $i$-$v$ curve. With this small positive voltage, almost no forward current flows. When the voltage increases up to around $0.6\,\text V$ measurable current starts to flow through the diode in the forward direction. As the voltage moves a little above $0.6\,\text V$, the current through the diode rises rapidly. The $i$-$v$ curve is nearly vertical at this point (it tips a little to the right).
With a positive voltage on its terminals, we say the diode is forward biased. A diode is forward biased when its voltage is anywhere on the $+$voltage side of the origin. In normal operation, the voltage across a forward biased silicon diode is somewhere between $0.60 -0.75\,\text V$. If you externally force the voltage higher than $0.75$ volts, the diode current gets very large and it may overheat.

### Reverse current

If you put a negative voltage to a diode, so the $-$ terminal is at a higher voltage than the $+$ terminal, this puts us over on the left side of the $i$-$v$ curve. We say the diode is reverse biased. In the reverse direction, the current is very close to zero, just ever so slightly negative, below the voltage axis.
There is a tiny bit of current that flows when a diode is reverse biased. We call it the reverse saturation current. A well-made diode has a typical reverse current of $\text I_{\text S} = 10^{-9}$ to $10^{-12}\,\text A$. In most situations, this is close enough to zero to be ignored.
In some cases (an integrated circuit with millions of diodes, or you want to detect really tiny currents), the reverse saturation current becomes important and you give it a bad-sounding name: leakage current.
A reverse biased diode can't hold out forever. When the voltage reaches a high negative value known as the breakdown voltage, $\text{V}_\text{BR}$, the diode starts to conduct in the reverse direction. At breakdown, the current sharply increases and becomes very high in the negative direction. A breakdown voltage $\text{V}_\text{BR}$ of $-50\,\text V$ is typical of ordinary diodes. Most of the time you don't allow the diode voltage to get near $\text{V}_\text{BR}$.
The Zener diode is intentionally designed to operate in its breakdown region. Parts catalogs give you a choice of many different breakdown voltages to choose from. The symbol for a Zener diode has little "z" strokes on the bar.
This diode makes an appearance in circuits where the sharply vertical breakdown region of the $i$-$v$ curve makes it look like a constant voltage source.
The word bias is an older term. It does not have a single precise definition.
In everyday use, bias can refer to a person's attitude, usually implying unfairness or favoritism. You also might use it in a favorable way, "They show a bias for taking action." Or you might describe a tendency, "The goalkeeper has a bias for jumping to the left on penalty kicks."
In electronics, we use bias in the sense of pulling towards or favoring one side. It only comes up in a few situations. One of them is when talking about diodes as we are doing here. Forward bias means an externally applied voltage is tugging the diode towards the forward conducting side of its $i$-$v$ curve. Reverse bias is the opposite, a negative external voltage pulls the diode into its reverse bias region.
The other place you come across this term: You apply a bias voltage to the terminals of a device (usually a transistor) to position it within a voltage range where it works best. For example, if a transistor works best between $1$ and $3$ volts, you design a bias voltage centered at $2$ volts, right in the middle of its happy range.

## Diode terminals

When you draw diodes, the schematic symbol clearly indicates the direction of forward current flow. You don't really need names for the two terminals. But, if you are handling real diodes to build a circuit, you have to figure out which way to point the diode. Real diodes are so small there isn't room to paint a little diode symbol on them, so you need to identify the terminals some other way.
The two terminals of a diode are the anode and cathode.

### How can I remember the anode and cathode?

For the longest time I could not remember which end of the diode was called the anode and which was the cathode, I looked it up every time. I finally came up with this memory aid. The German word for cathode is Kathode. The big K kind of looks like a diode symbol.
Flip the diode symbol around until it reads like a K. The Kathode is the terminal on the left.

### Identifying the terminals of a real-life diode

Diodes are made on small chips of silicon. They are delivered to you in all sorts of tiny packages. There are a few different ways to indicate which diode terminal is which.
Diode packages like the glass and black plastic cylinders shown above usually have a painted bar near one end. The bar on the package is the bar of the diode symbol, so it indicates the cathode.
The stripe (any contrasting color) corresponds to the diode's cathode.
This red LED (light emitting diode) has wire leads of different length. The forward current goes into the longer lead (anode). The package may have a bump or tab sticking out on the forward current side.
The longer lead corresponds to the diode's anode.

### Identify the anode and cathode with a meter

A good way to verify the identity of the terminals is using an ohm-meter to figure out the forward current direction. On the resistance setting, $\Omega$, the meter puts a small voltage on its test leads (this is why an ohm meter needs a battery). You use that small voltage to see which way current flows.
The diode is flipped in each image. If the ohm-meter reads a finite resistance, that means the diode is conducting a small current in the forward direction, and the red $+$ lead from the meter is touching the anode. If the resistance reads O.L (for overload), the diode is not conducting current. That means the red $+$ test lead is touching the cathode.
Your meter might have a diode setting, a little diode symbol. If it does, it will display the forward voltage when the red lead is touching the forward current terminal (the anode) as shown below.

## Diode $i$-$v$ equation

The diode $i$-$v$ relationship can be modeled with an equation. This equation is based on the physics underlying the diode action, along with careful measurements on real diodes.
The $i$-$v$ curve for a typical silicon diode.
The plot above doesn't look very much like an exponential curve, and the current for negative voltages appears to be $0$. If we expand the current scale a whole bunch $(\text{milliamperes}$ $\rightarrow$ $\text{picoamperes})$ the exponential shape becomes apparent (the voltage scale is expanded, too). You can see a tiny negative $\text I_{\text S}$ flows when the diode is reverse biased:
$\text I_{\text S}$ is the saturation current. This current flows backwards when the diode is reverse biased. A typical value for $\text I_{\text S}$ in silicon is $10^{-12}\,\text A$, ($1$ picoampere). For germanium diodes, a typical value for $\text I_{\text S}$ is $10^{-6}\,\text A$, ($1$ microampere).
It is best to think of this diode equation as a model of a diode, rather than as a law. The equation represents an abstract ideal diode. The actual behavior depends on how it is made, its temperature, and how much you care about the fine details.

## Detailed look inside the diode $i$-$v$ equation

[This next part takes apart the diode equation in some detail. You don't need this to use a diode in a circuit. It is okay to jump down to the example.]
There are many new parameters in the diode equation. Let's go through them carefully.
$v$ is the voltage across the diode. We find it up top in the exponential term, which explains why current $i$ has an exponential dependence on voltage $v$.
Now lets look at all that other stuff up in the exponent of .
We know exponents have no dimensions, so the other terms in the exponent have to end up with units of $1/v$.
$q$ is the charge on an electron, in coulombs:
$q = 1.602 \times 10^{-19} \,\text C$.
$k$ is Boltzmann's constant, a very important number in physics. The energy of a particle increases with temperature. If you know the temperature of a particle, $k$ tells you how much kinetic energy the particle has just by virtue of being warm. The units of Boltzmann's constant are energy per kelvin.
When we measure the temperature in Celsius or Fahrenheit, we say "degrees celsius" or "degrees fahrenheit". You write temperatures as $23\,^\circ\text C$ or $73\,^\circ\text F$, with the little circle $^\circ$ degree symbol. The units of absolute temperature named after Lord Kelvin are defined to already be degrees. So we say just "kelvin" instead of "degrees kelvin", since that would be redundant. The temperature in kelvin is written without the little degree circle, like this: $-273\,^{\circ}\text C\ = 0\,\text K$.
Also, make a note to not confuse big $\text K$ for kelvin with little $k$ for Boltzmann's constant.
$k = 1.380\times 10^{-23} \,\text{J/K}\,\text{(joules per kelvin)}$
$\text T$ is the temperature measured from absolute zero in $\text{kelvin}$ or $\text K$. A temperature of absolute zero, or $0\,\text K$ is $-273\,^{\circ}\text C\,\text {(celsius)}$.
Since $\text T$ appears in the diode $i$-$v$ equation, we know the diode curve changes at different temperatures.
Increasing temperature moves the $i$-$v$ curve to the right.
If a particle is at $\text T = 300\,\text K$, (room temperature), then it has energy:
$k\text T = 1.380\times 10^{-23} \,\text{J/K}\cdot 300\,\text K = 4.14\times 10^{-21}\,\text J$
Engineers like to round to easy-to-remember numbers, so room temperature is $300\,\text K$. This is close enough for circuit design.
If our particle is an electron, it has a known charge, and we can talk about its energy per charge. Energy per charge might sound familiar. Its other name is voltage.
$\dfrac{k\text T}{q} = \dfrac {4.14 \times 10^{-21}\,\text J} {1.602\times 10^{-19}\,\text C} = 25.8 \approx 26 \,\text{mV}$
At room temperature (around $300 \,\text K$), $k\text T/q$ is $26$ millivolts. That's the energy of a normal everyday electron. The exponent of the diode equation, $v/26\,\text{mV}$, is comparing the diode voltage to the energy of an ordinary electron.
My physics professor called this fraction "kT on q". I always thought that sounded nice. The reciprocal is, of course, "q on kT".
If you feel like it, you can write the diode equation for room temperature as:
This non-linear diode $i$-$v$ equation is harder to deal with than the linear $i$-$v$ equations for $\text R$, $\text L$, and $\text C$. There are very few cases where you will be asked to use this equation to find an analytical solution. The usual approach to diode circuits is to perform a graphical solution or to use a circuit simulation program to get an approximate answer.

## Diode circuit example

Let's build a circuit with a diode. This circuit has a green light-emitting diode (LED).
We do this example with the $i$-$v$ curve of a Silicon diode, with a forward voltage of about $0.7\, \text V$. A normal LED is made of different materials from the periodic table and has a forward voltage somewhere between $2$ and $4\,\text V$ depending on the color. If you have a specific LED in mind, consult its datasheet to find its forward voltage and light output vs. current.
The resistor and diode share the same current, $\blueD i$. We want to find $\blueD i$ and the voltage that appears across the diode, $\goldD{v_\text{D}}$.
All elements share the same current, so we'll focus on equations for current.
For the diode, we get the current $i$ in terms of $v_{\text D}$ from the diode equation at room temperature:
For the resistor, if we can come up with an equation for $i$ in terms of $v_{\text D}$, we'll be able to plot that equation on the same $i$-$v$ graph as the diode. Ohm's Law for the resistor is:
$i = \dfrac{v_\text{R}}{330\,\Omega}$
We know $v_{\text R} = 3\,\text V - v_{\text{D}}$, so the resistor current becomes:
$i = \dfrac{3\,\text V - v_{\text{D}}}{330\,\Omega}$
I'll rearrange the equation so it looks like the slope-intercept form of a line:
$i = -\dfrac{1}{330\,\Omega}\,v_{\text D} + \dfrac{3\,\text V}{330\,\Omega}$
$i = -\dfrac{1}{330\,\Omega}\,v_{\text D} + 9\,\text{mA}\qquad$
Our resistor has its upper terminal connected to the battery. It produces this equation with the distinctive negative slope. This line has the nickname load line. Load lines come up again when we study transistors.
The slope of the resistor load line is $-\dfrac{1}{330}$.
The $i$-intercept is $9\,\text{mA}$.

### Graphical solution

We now have two simultaneous equations with two unknowns, $i$ and $v_\text D$:
$i = -\dfrac{1}{330\,\Omega}\,v_{\text D} + 9\,\text{mA}$
We can solve these two equations by a graphical method, by plotting them on the same scale, and seeing where they cross. At the point where they intersect, the current in the resistor equals the current in the diode.
Graphical solution to the resistor-LED circuit. The blue line is a plot of the diode equation. The green line is a plot of the resistor "load line" equation. The point where the two lines intersect is where the current in the diode and resistor are the same.
We get a pretty accurate answer by reading the intersection point off the graph:
$v_{\text D} = 0.6\,\text V$ and $i = 7.2\,\text{mA}$
Reading from the graph is often all the accuracy you need.

### Concept check

problem 1
What is the current when the diode voltage $v_\text D$ is $0$?
$i =$
$\,\text{mA}$

problem 2
What is the voltage when the resistor line touches the $v$-axis? $(i = 0)$
$v =$
$\text V$

problem 3
Does the $i$-axis intercept of the resistor line depend on the value of $\text R$?

problem 4
Does the $v$-axis intercept of the resistor line depend on the value of $\text R$?

### Brighter

Suppose you build this circuit and the LED is not bright enough. The brightness goes up if you increase the current. How might you do that?
Try changing something about the circuit to increase brightness. Then sketch a new graphical solution.
One way to get more diode current is to reduce the resistance. Lower resistance makes the resistor line steeper. If we reduce the resistor to $200\,\Omega$ and replot the line, we get a new solution:
Making the resistor $200\,\Omega$ increases the LED current to $12\,\text{mA}$, making it brighter.
You could also see what happens if you increase the voltage. The resistor line moves in a different way when the voltage is adjusted. Go ahead give that a try on your own.
Suppose we go for maximum brightness by leaving out the resistor altogether. Is this a good idea or a bad idea?
Imagine how the resistor line changes as the resistor goes from $200\,\Omega$ to $0\,\Omega$.
If we make the resistor smaller and smaller, the green resistor line becomes steeper and steeper as the intersection point on the $i$ axis goes up and up.
The bottom end of the resistor line is stuck at $v=3\,\text V$. With $0\,\Omega$ resistance, the line points straight up and does not intersect the diode curve until way way up there in current. Two things can happen at this point. The diode burns out from the excess heat, or, if the diode survives, the battery runs down in just a little while. So it turns out to be a bad idea to leave out the resistor.

### If we had used an LED $i$-$v$ curve

If we had done this problem with an $i$-$v$ curve for an LED with a forward voltage of $2\,\text V$, the solution would have come out something like this:
A typical LED gives off a nice light with a current of around $20\,\text{mA}$. To get this current you adjust the slope of the resistor load line until the intersection point is at the current you want. Then figure out the resistor value from the slope. The slope of the resistor load line is $-\dfrac{1}{\text R}$.
A standard-value $47\,\Omega$ resistor comes close to the target $20\,\text{mA}$. The resistor value will come out different if you have a different supply voltage, of course.
Most LED datasheets give you a forward voltage specification, but not an equation for a diode curve as I've shown on the plot above. It is okay to approximate the diode with a straight vertical line at the forward voltage. You will get a pretty good graphical solution.

## Summary

The schematic symbol and terminal names for a diode:
The diode equation is:
$\text I_{\text S}$ is the saturation current. For silicon, $\text I_{\text S}=10^{-12}\,\text A$ is a typical value.
In the exponent of the diode equation, the term $k\text T/q$ is equivalent to $26\,\text{mV}$ if the diode is at room temperature. $k$ is Boltzmann's constant, $\text T$ is the temperature in kelvin, and $q$ is the charge on an electron in coulombs. If the diode is near room temperature, the diode equation can be written as:
We demonstrated a graphical solution for a diode circuit, and it gave a pretty good answer without a lot of work. In general, graphical solutions are a good way to go after any circuit with a non-linear element.

• Silicon diode - Silicon is the most common material used to make diodes. Silicon has a typical forward voltage of $0.6-0.7\,\text V$.
• Germanium diode - Made from a different element. Germanium diodes have a lower forward voltage of $0.25-0.30\,\text V$.