Circuit elements

An electric circuit is made of elements. Elements include at least one source. The source is connected to a bunch of components. We are going to describe sources and components with ideal mathematical abstractions. By the end of this article, we will have a nice collection of equations, which can be combined to generate lots of useful electronic functions. The next article describes real-world components that come close to the ideal abstractions we define here.

Elements are either sources or components.

Sources provide energy to a circuit. There are two basic types.

• Voltage source
• Current source

Components come in three basic types, each characterized by a different voltage-current relationship.

• Resistor
• Capacitor
• Inductor

These sources and components have two terminals or connection points. Not surprisingly, they are referred to as $2$-terminal elements.

Ideal sources

Constant voltage source

An ideal constant voltage source has a fixed output voltage, independent of the current drawn by the components connected to its terminals, as shown in this current versus voltage plot:

[[☃ image 11]]

The equation for a constant voltage source is,

$v = \text V$

where $\text V$ is some constant output voltage, like $v=3,\text V$.

You frequently see the variable name $e$ associated with voltage, derived from the term "electromotive force" or emf. This term is sometimes used when talking about the voltage from a source (battery or generator).

[[☃ explanation 2]]

The two common symbols for constant voltage sources:

[[☃ image 15]]

The symbol on the left is used for a battery. The longer horizontal line on the battery symbol represents the positive terminal of the battery, and the shorter horizontal line represents the negative terminal. The circle symbol represents some other source of voltage, often a power supply. It is a good practice to draw the $+$ and $-$ signs inside the circle.

Variable voltage source

An ideal variable voltage source generates a known voltage as a function of time, independent of the current drawn by the components connected to its terminals, as shown in this $voltage$ versus $time$ plot:

[[☃ image 7]]

The equation for a variable voltage source is,

$v = v(t)$

$v(t)$ can be a sine wave or any other time-varying voltage, for example, a single voltage step, or a repeating square wave.
[[☃ explanation 1]]

The symbol for a variable voltage source:

[[☃ image 10]]

The squiggle inside the circle suggests this particular symbol represents a sine wave generator. You will come across variations of this symbol for different waveform shapes.

These ideal mathematical abstractions of voltage sources can produce arbitrarily huge output current if the components they are connected to demand it. That doesn't happen in real life, of course. One place gigantic currents pop up is when you simulate a circuit. The computer doesn't mind a current of a zillion amperes, but it's probably not what you intended.

Constant current source

An ideal constant current source has a fixed output current, independent of the voltage connected to its terminals, as shown in this $current$ versus $voltage$ plot:

[[☃ image 7]]

The equation for a constant current source is,

$i = \text I$

where $\text I$ is a constant output current, like $i=2,\text{mA}$.

The symbol for a constant current source:

[[☃ image 10]]

The arrow indicates the direction of positive current flow.

The voltage at the terminals of an ideal current source becomes whatever is required to push out the constant current, even if that voltage is gigantic. When we build real current sources, of course, the range of operation is significantly restricted compared to the ideal current source abstraction.

Resistor

The voltage across a resistor is directly proportional to the current flowing through it.

$\large v = \text R , i \qquad \normalsize \text{Ohm's Law}$

This relationship is known as Ohm's law. You'll use this equation a lot in your work with circuits.

$\text R$ is a constant of proportionality, representing the resistance. Resistance has units of ohms, denoted by the Greek capital Omega symbol, $\Omega$.

The $i$-$v$ graph for a resistor is shown below. The equation plotted is $i=v/\text R$, so the slope of the line is $1/\text R$.

[[☃ interaction 5]]

[[☃ explanation 3]]

The symbols for a resistor:

[[☃ image 15]]

In the US and Japan the resistor symbol is a zig-zag. In the UK, Europe and other parts of the world, the resistor is often drawn as a box.

Ohm's Law can be written a number of ways, all of them useful,

$v = i,\text R \qquad\qquad i = \dfrac{v}{\text R} \qquad\qquad \text R = \dfrac{v}{i}$

Ohm's Law is worth committing to memory.

[[☃ explanation 2]]

Power in a resistor

Power is dissipated by a resistor when current flows through it.
[[☃ explanation 1]]

Energy in flowing electrons becomes bulk heat as electrons collide with atoms in the resistor material. Power can be expressed a few ways using Ohm's Law. These are all equivalent,

$p = v,i$

$p = (\text i,\text R), i = i^2 ,\text R$

$p = v\left (\dfrac{v}{\text R}\right ) = \dfrac{v^2}{\text R}$

The last two expressions reveal that power in a resistor goes up (or down) proportional to the square of voltage or current.

• Increase either voltage or current by a factor of $2$, the power consumed goes up by a factor of $4$.

• Reduce either voltage or current by half, and you reduce the power by [[☃ explanation 2]]

• Aaron finds a way to cut the voltage across a resistor by a factor of two. When Beth looks at Aaron's new design, she figures out how to reduce the current in the resistor by a factor of two.
  [[☃ explanation 3]]

Capacitor

The basic equation describing a capacitor relates charge on the capacitor to the voltage across the capacitor. [[☃ explanation 1]]

$\text Q = \text C,\text V$

The constant of proportionality $\text C$ is the capacitance. Capacitance has units of farads, symbolized by the capital letter $\text F$. The unit of capacitance is the farad, and from the equation above we see that, $1 ,\text{farad} = 1 ,\text{coulomb}/\text{volt}$

If the charge can move, we have a term for this; moving charge is called current. Current is the time rate of change of charge,

$i = \dfrac{dq}{dt}$

Using this idea, let's take the derivative of both sides of $\text Q = \text C,\text V$ with respect to time and see what we get,

$\dfrac{dq}{dt} = \text C , \dfrac{dv}{dt}$

and we end up with an equation saying the current in a capacitor is directly proportional to the time rate of change of the voltage across the capacitor,

$i = \text C , \dfrac{dv}{dt}$

This capacitor equation captures the $i$-$v$ relationship for capacitors. It also tells us that electric circuits can be affected by time.

The symbols for a capacitor:

[[☃ image 9]]

The version with the curved line is used for capacitors manufactured in a way that requires one terminal to have a positive voltage with respect to the other terminal. The curved line indicates the terminal that needs to be kept at the more negative voltage.

We can flip the capacitor equation around to solve for $v$ in terms of $i$ by integrating both sides, resulting in the integral form of the capacitor equation,

$\displaystyle v = \dfrac1{\text C}, \int_{-\infty}^{,T} i,dt$

The $-\infty$ lower limit on the integral suggests that the capacitor's voltage at time $T$ depends not just on the capacitor current right now, but on the entire past history of the current. That's a long time ago, so we often write this integral starting at some known voltage $v_0$ at some known time like $t=0$, and then keeping track of the changes from there.

$\displaystyle v = \dfrac1{\text C}, \int_{,0}^{,T} i,dt + v_0$

[[☃ explanation 2]]

Power and energy in a capacitor

The instantaneous power in watts associated with a capacitor is,

$p = v,i$

$p = v,\text C ,\dfrac{dv}{dt} $

The energy $(U)$ stored in a capacitor is power integrated over time,

$\displaystyle U = \int p ,dt = \int v,\text C ,\dfrac{dv}{dt},dt = \text C\int v ,dv$

If we assume the capacitor voltage was $0,\text V$ at the beginning of the integration, then the integral evaluates to:

$U = \dfrac 12 ,\text C ,v^2$

Unlike a resistor, where the energy is lost to heat, the energy in an ideal capacitor does not dissipate. Instead, energy in the capacitor, in the form of stored charge, is recovered when the charge flows back out of the capacitor.

Inductor

The voltage across an inductor is directly proportional to the time rate of change of current through the inductor, [[☃ explanation 5]]

$\large v = \text L , \dfrac{di}{dt}$

This property arises from the inductor's ability to store energy in a surrounding magnetic field. The stored magnetic energy can return to the circuit by generating an electric current.

The constant of proportionality $\text L$ is the called the inductance. The unit of inductance is the henry, denoted by the capital letter H. [[☃ explanation 3]]

The reason this property of inductance arises in coils of wire is a complex topic involving the intimate relationship between electricity and magnetism, which is beyond the scope of this article. For now, please trust that the voltage across an inductor is proportional to the rate of change of current.

The symbol for an inductor:

[[☃ image 10]]

It looks like a wire wrapped in a coil, since that is the usual way to make an inductor.

Similar to the capacitor equation, we can write the inductor equation in integral form to get $i$ in terms of $v$. Notice the kinship between the capacitor and inductor equations.

$\displaystyle i = \dfrac1{\text L}\int_{-\infty}^{,T} v,dt$

$\grayF{\displaystyle v = \dfrac1{\text C}, \int_{-\infty}^{,T} i,dt} $

The $-\infty$ lower limit on the integral means the inductor's current at time $T$ depends on the entire past history of the inductor voltage. We usually write this integral starting from some known current $i_0$ at some known time like $t=0$, and then keeping track of the changes from there.

$\displaystyle i = \dfrac1{\text L}, \int_{,0}^{,T} v,dt + i_0$

Power and energy in an inductor

The instantaneous power in watts associated with an inductor is

$p = i,v$

$p = i,\text L , \dfrac{di}{dt}$

The energy $(U)$ stored in the magnetic field of an inductor is power integrated over time,

$\displaystyle U = \int p ,dt = \int i,\text L , \dfrac{di}{dt},dt = \text L\int i ,di$

$U = \dfrac 12 ,\text L ,i^2$

Unlike a resistor, where the energy is lost to heat, the energy in an ideal inductor does not dissipate. Instead, the energy stored in the inductor's magnetic field can be fully recovered when the energy in the magnetic field gets converted back into an electric current in the wire.

Summary of ideal component equations

Here are the three important circuit component $i$-$v$ equations,

$\large v = i,\text R\quad\qquad$ Ohm's Law

$\large i = \text C ,\dfrac{dv}{dt}\qquad$ capacitor equation

$\large v = \text L ,\dfrac{di}{dt}\qquad$ inductor equation

These three expression are your tools of the trade for circuit analysis.

In addition, we also developed these expressions for power and energy.

The power in a resistor is

$p =i,v \quad$ or$\quad i^2 ,r \quad$or $\quad v^2/r$

The energy in a capacitor is $\dfrac 12 ,\text C ,v^2$

The energy in an inductor is $\dfrac 12 ,\text L ,i^2$

The next article describes how real-world physical components come close to the mathematical ideal.

[[☃ explanation 1]]

Non-ideal circuit elements

The circuit elements discussed in the previous article are ideal circuit elements. Real-world circuit elements come close to the ideal mathematical models, but inevitably will be imperfect. Being a good engineer means being aware of the limitations of real components compared to their the ideal abstractions.

One simple variation from ideal is that physical devices like resistors, inductors, and capacitors $(\text{R, L, C})$ have some level of tolerance around the ideal value (the tighter the tolerance, the more money you pay). Real components never have exactly their specified values.

Real circuit elements deviate from the ideal equations when voltage or current are taken to extremes. The straight line mathematical abstraction of a resistor does not go out to $\infty$ voltage or $\infty$ current for real resistors. The model breaks down at some point and the component could be destroyed. Abstract models of all the ideal components and sources have a limited range in the real world.

A real component is not just that component. Using a resistor as an example: because the wires connected to the ends of a resistor generate a surrounding magnetic field, it will inevitably display some inductive properties. In addition, resistors are made of conducting materials, and are usually located near other conductors. Together these conductors act like the plates of a capacitor, so resistors also display some capacitive properties.

These parasitic effects can be relevant at high frequencies, or when voltage or current changes sharply.
If parasitics matter, you can model a component as a combination of ideal elements, as shown here for a resistor:

[[☃ image 8]]

The properties of real-world components are sensitive to their environment. Most components show some degree of temperature sensitivity; parameters drift high or low depending on how hot or cold the component is. If your circuit has to work over a wide temperature range, you will want to know the temperature behavior of the components you use.

Note: In the electrical engineering subjects covered at Khan Academy, you won't have to worry about parasitic effects. They are mentioned here so you know they exist. When you simulate electronic circuits, you don't need to complicate matters by modeling all potential parasitic effects, unless you have (or learn of) a reason to think they are important.

Real-world resistors

When making real resistors, the goal is to create a component that comes as close as possible to performing like the ideal resistor equation, Ohm's Law, $v = i,\text R$.

The resistance value of a resistor depends on two things:

• what it is made of
• how it is shaped

The bulk material (what it is made of) affects how difficult it is for electrons to flow through. You could think of it as how often electrons bump into the atoms in the material as they try to flow through. This property of a bulk material is called resistivity. You might also hear the term conductivity, which is just the inverse of resistivity.

After selecting a bulk material with a certain resistivity, the resistance of the resistor is determined by its shape. A longer resistor has higher resistance than a shorter resistor because the electrons suffer more collisions as they pass through the jungle of atoms in the material. A resistor with a greater cross-sectional area has lower resistance than a resistor with smaller cross-section, because electrons have a greater number of available paths to follow.

• A resistor is a circuit element, a physical object.
• Resistivity is a property of a bulk material.
• Resistance is property of a resistor, determined by both the resistivity of the material and the shape of the resistor.

A real resistor breaks down (as in burns out and is destroyed) if the power dissipated by the resistor is greater than what its construction materials can withstand. Resistors come with a power rating that should not be exceeded. If you try to dissipate $1$ watt in a $1/8$ watt resistor, you may end up with a burned chunk of something that is no longer a resistor.

Example of a conventional axial resistor:

[[☃ image 19]]

The color bands indicate the resistor value and tolerance. The bands on this resistor are Orange Orange Brown Gold. From the resistor color code chart, the first two bands corresponds to the digits of the value, $3, 3$. The third band is the multiplier, brown stands for $\times 10^1$. The fourth (last) band indicates the tolerance, gold is $\pm 5%$. The resistor value is $330 ,\Omega,\pm5%$.

[[☃ explanation 4]]

This is a precision resistor with 5 color bands:

[[☃ image 18]]

Read the bands from left to right: Red Red Blue Brown Brown $= 2 ,2 ,6 ,1 ,1$. The first three bands $(2,2,6)$ are the value. The fourth band is the multiplier $(\times 10^1)$, The fifth (last) band indicates the tolerance, brown is $1%$. The resistor value is $2260 ,\Omega,\pm 1%$.

This is a surface mount resistor:

[[☃ image 18]]

The resistance value is encoded in the $3$-digit code: $102$, meaning $10 \times 10^2 = 1000 ,\Omega$. The size specification of this resistor happens to be "0603 Metric", indicating its footprint is $0.6 , \text{mm} \times 0.3 , \text{mm}$.

Example of a resistor in an integrated circuit:

[[☃ image 23]]

The designer selects one of the integrated circuit layers with high resistivity and creates (draws) a serpentine pattern to achieve the desired resistance.

Real-world capacitors

When making real capacitors, the goal is to create a component that comes as close as possible to performing like the ideal capacitor equation, $i = \text C ,\text dv/\text dt.$

A capacitor is constructed from two conducting surfaces placed close to each other. Between the plates there can be air, or any other kind of insulating material. The capacitance value depends on a number of factors, the area of the plates, the distance between the plates (the thickness of the insulator), and on the physical properties of the insulating material.

You can learn more about capacitors and how/why they work in the capacitors and capacitance section in Khan Academy Physics.

Real capacitors:

[[☃ image 10]]

Cylindrical capacitors (black, dark blue, or silver, upper left) are made of two metal foil plates rolled up to maximize the area of the plates to achieve large capacitance values in a compact package.

The circle-shaped capacitors (aqua blue and orange, bottom) are simply two metal disks facing each other, separated by an insulator.

Adjustable capacitors (white, right) have air as the insulator. One set of plates rotates to overlap more or less area to the stationary set of plates. Variable air capacitors are used to tune radios, for example.

The most likely departure from the ideal capacitor equation happens if the voltage across the capacitor becomes so large the insulation between the plates breaks down. When this happens, a spark can burn through the insulation and jump between the plates. No more capacitor. Real capacitors have a voltage rating that should not be exceeded.

Since a capacitor has connection wires, it inevitably has a small parasitic resistance and inductance. The parasitic inductance can be important if the capacitor is expected to provide sudden bursts of current, such as when it is connected to the power pin of a digital chip. Providing a sudden surge of current to the digital chip means the inductance of the capacitor connections should be very low.

The material separating the capacitor plates is supposed to be insulating (allow zero current). But not all insulators are perfect, so tiny currents can seep through. These so-called leakage currents appear to flow straight through the capacitor, even if the voltage is not changing (when $\text dv/\text dt = 0)$. Paths for leakage current also happen if the circuit is not clean, and currents flow around the capacitor, along the surface of the component.

A surface mount capacitor is shown here:

[[☃ image 15]]

Leakage currents might flow between the metal ends through the gunk left behind from the soldering process if the circuit board is not cleaned.

A surface-mount capacitor is built up from many layers of interleaved conducting electrode plates and insulating ceramic layers.

[[☃ image 17]]

Real-world inductors

When creating an inductor, the goal is to come as close as possible to the ideal inductor equation, $v = \text L ,\text di/\text dt.$

A full analysis of how an inductor actually works is an advanced topic and beyond the scope of this article. To learn more about inductors and magnetic fields, see the magnetic fields section of Khan Academy Physics.

Any conductor carrying a current generates a magnetic field in the surrounding region, as represented by the red lines in these images. The magnetic field around a wire wrapped in a coil shape becomes concentrated on the interior of the coil.

[[☃ image 7]]

A good way to think about inductance is to make the comparison to mass in a mechanical system. Magnetic energy is stored in an inductor in the same way kinetic energy is stored in a moving mass. Think of a rotating flywheel (a wheel with a heavy rim). You can't stop a spinning flywheel in an instant. An inductor is similar. The current in an inductor does not stop instantaneously, just like the flywheel. The magnetic field energy keeps pushing it along.

Making inductors: To achieve higher levels of inductance (higher $\text{L}$), inductors are made by winding wire in a coil. The magnetic field can be intensified even more by placing a suitable magnetic material inside the coil. This is toroidal-shaped inductor wound around a core of iron/ceramic material called ferrite. (You can't see the ferrite core shaped like a donut, it is covered by the copper wire.)

[[☃ image 9]]

The ferrite core concentrates and intensifies the magnetic field, which increases the value of the inductance, $\text{L}$.

Real inductors differ from the ideal equation in a few important ways. Since inductors are made of long wires, they often have a significant parasitic resistance.

The other unavoidable feature of inductors is that they take up a lot of space. The magnetic field exists in the volume of space around and inside the inductor, and the coil of wire has to be large enough to surround a large amount of magnetic field if it is to achieve a significant inductance. This is why it is rare to see an inductor designed into an integrated circuit.

We finish up with this astonishing image of an air-core inductor. This large copper coil (an inductor) was part of a wireless telegraph station built in New Jersey, USA in 1912. It could send a message 4000 miles (6400 km), all the way across the Atlantic Ocean to Germany. Wow. Needless to say, most inductors are much smaller.

[[☃ image 3]]

[[☃ explanation 5]]

[[☃ explanation 1]]

We are developing methods for analyzing a circuit. So far we've defined the most common components (resistor, capacitor, and inductor) and sources (voltage and current). Now we need a crisp vocabulary to talk about circuits. This article is a glossary of terms and concepts we use in circuit analysis and design.

Circuit

Circuit comes from the word circle. A circuit is a collection of real components, power sources, and signal sources, all connected so current can flow in a complete circle.

Closed circuit
– A circuit is closed if the circle is complete, if all currents have a path back to where they came from.

Open circuit
– A circuit is open if the circle is not complete, if there is a gap or opening in the path.

Short circuit
– A short happens when a path of low resistance is connected (usually by mistake) to a component. The resistor shown below is the intended path for current, and the curved wire going around it is the short. Current is diverted away from its intended path, sometimes with damaging results. The wire shorts out the resistor by providing a low-resistance path for current (probably not what the designer intended).

[[☃ image 2]]

Make or Break – You make a circuit by closing the current path, such as when you close a switch. Breaking a circuit is the opposite. Opening a switch breaks the circuit.

[[☃ image 6]]

Schematic

A schematic is a drawing of a circuit. A schematic represents circuit elements with symbols and connections as lines.

Elements
– The term elements means "components and sources."

Symbols
– Elements are represented in schematics by symbols. Symbols for common 2-terminal elements are shown here,

[[☃ image 11]]

Lines
– Connections between elements are drawn as lines, which we often think of as "wires". On a schematic, these lines represent perfect conductors with zero resistance. Every component or source terminal touched by a line is at the same voltage.

Dots
– Connections between lines can be indicated by dots. Dots are an unambiguous indication that lines are connected. If the connection is obvious, you don't have to use a dot.

[[☃ image 7]]

Reference designator
– When you place a component in a schematic you often give it a unique name, known as a reference designator. Examples of reference designators are $\text{R1}$, $\text{C6}$, and $\text{V}_\text{BAT}$. The $1$ in $\text{R1}$ is part of the name, and does not indicate the resistance value. Reference designators are by definition unique for each schematic. They let you identify components by name even if some of them have the same value. It is okay to use reference designators in equations. $\text{R1}$ can be assigned a resistance value, $\text{R1} = 4.7,\text k\Omega$, and it can be used as a variable in expressions, as in $\text{R2} \cdot \text{C6} = 4.7,\text k\Omega \cdot 2,\mu\text F$.

[[☃ image 12]]

Node
– A junction where $2$ or more elements connect is called a node. The schematic below shows a single node (the black dot) formed by the junction of five elements (abstractly represented by orange rectangles).

[[☃ image 16]]

Since lines on a schematic represent perfect zero-resistance conductors, there is no rule that says lines from multiple elements are required to meet in a single point junction. We can draw the same node as a distributed node like the one in the schematic below. These two representations of the node mean exactly the same thing.

[[☃ image 18]]

A distributed node might be all spread out, with lots of line segments, elbows, and dots. Don't be distracted, it is all just one single node. Connecting schematic elements with perfect conductors means the voltage everywhere on a distributed node is the same.

Here is a realistic-looking schematic with the distributed nodes labeled:

[[☃ image 22]]

[[☃ graded-group 3]]

[[☃ explanation 5]]

Branch
– Branches are the connections between nodes. A branch is an element (resistor, capacitor, source, etc.). The number of branches in a circuit is equal to the number of elements.

[[☃ graded-group 1]]

Loop
– A loop is any closed path going through circuit elements. To draw a loop, select any node as a starting point and draw a path through elements and nodes until the path comes back to the node where you started. There is only one rule: a loop can visit (pass through) a node only one time. It is ok if loops overlap or contain other loops. Some of the loops in our circuit are shown here. (You can find others, too. If I counted right, there are six.)

[[☃ image 3]]

[[☃ explanation 2]]

Mesh
– A mesh is a loop that has no other loops inside it. You can think of this as one mesh for each "open window" of a circuit.

[[☃ graded-group 1]]

Reference Node
– During circuit analysis we usually pick one of the nodes in the circuit to be the reference node. Voltages at other nodes are measured relative to the reference node. Any node can be the reference, but two common choices that simplify circuit analysis are,

• the negative terminal of the voltage or current source powering the circuit, or
• the node connected to the greatest number of branches.

Ground
– The reference node is often referred to as ground. The concept of ground has three important meanings.

[[☃ image 6]]

Ground is

• the reference point from which voltages are measured.
• the return path for electric current back to its source.
• a direct physical connection to the Earth, which is important for safety. [[☃ explanation 1]]

The ground node gets its name from the third meaning. But the other two are equally important.

You will come across various symbols for ground:

[[☃ image 7]]

Schematic equivalence

We need to take a second to talk about the idea of schematic equivalence. This is important because a circuit can be represented by schematics drawn in different ways.

The following two schematics are drawn differently. The schematic on the left shows a voltage source and three resistors in numerical order. In the schematic the right, resistor $\text R3$ appears to the left of the voltage source.

[[☃ image 14]]

Do both of these schematics properly represent the intended circuit? Or said another way,
Are these two schematics equivalent?

We say a real circuit and a schematic (or two schematics), are equivalent if they have the same nodes and branches.

To be equivalent, two schematics must:

• Represent every component and source
• Have the same number of nodes
• Each node must be connected to the same branches

Let's check to see if our two schematics are equivalent:

• Are all components and sources represented in both schematics?
  Roll call ... $\text V$, here!, $\text R1$, here!, $\text R2$, here! $\text R3$, here!
  All elements accounted for.

• Do both schematics have the same number of nodes?
  Yes. Both schematics have $2$ nodes.

• Is each node connected to the same branches?
  – Yes. Each node connects to the three resistors and a source.

The two nodes are marked with orange lines. The four branches are shown as blue arrows.

[[☃ image 16]]

So the answer is: yes, these schematics are equivalent.

Equivalence means the matching nodes will have the same voltage, and the matching branches will have the same current. These are the things we care about being the same.

You could build a real circuit based on either of these schematics. Lay the physical wires and components right on top of either drawing and solder them together. Both schematics will produce the intended circuit, with identical node voltages and branch currents.

This discussion of equivalence may seem rather overdone; what's the big deal? Schematics have a curious property that often catches beginners.

A schematic puzzle

I'm going to point out something that may seem baffling, (but only for a moment). As we just established, the following two schematics are equivalent. But, not everything is exactly the same. The individual point-to-point connections of the lines between elements are not the same.

[[☃ image 7]]

Look at the blue arrow in the left schematic. That wire carries the current flowing towards $\text R2$ and $\text R3$.

Can you find the equivalent wire in the schematic on the right?

(Find a wire carrying the current going to $\text R2$ and $\text R3$.)

[[☃ explanation 2]]

What is going on? It is a trick question, to highlight something about the nature of schematics.

This puzzle reveals a fundamental difference between a real circuit and a drawn schematic. The lines in a schematic diagram do not necessarily represent the specific point-to-point order of the connections the corresponding real circuit might have. The question about one wire carrying current to $\text R2$ and $\text R3$ assumes a specific wiring order that does not exist in the schematic on the right.

How do you avoid getting trapped by this schematic puzzle? You can always count on identical branch currents in every equivalent schematic or real circuit. So always think about current flowing in a branch (flowing in a component or source), not current flowing in a "wire." Currents in "wires" may or may not exist in an equivalent version of the schematic, or in the real circuit built from either schematic.

Concept check: Equivalence

Here is a brainteaser to help you check your understanding of schematic equivalence.

** Which of these schematics represent the same circuit (are equivalent)?**

Assume all resistors have the same value.
Take your time, this isn't simple.
Hint: There are three answers.

[[☃ image 5]]

[[☃ explanation 4]]
[[☃ explanation 1]]
[[☃ explanation 2]]
[[☃ explanation 3]]

Drawing a good schematic

A good schematic serves a number of noble purposes. A good schematic

• captures the design of a circuit in an unambiguous way.
• allows you to share your design with other people.
• helps you remember how your circuit works, even a month from now.

Both you and your colleagues will appreciate these drawing habits for creating good schematics,

• Place inputs on the left, and outputs on the right.
• Let information flow from left to right across the circuit.
• Use up/down on the page to suggest voltage levels. That is, draw higher voltage wires closer to the top of the page, and lower voltages (like ground) near the bottom of the page.

The following schematics are equivalent, but the one on the left is not as easy to read as the one on the right. The one on the right follows the guidelines for a good schematic.

[[☃ image 3]]

[[☃ video 1]]

Good schematics capture your design intent. You convey your meaning more quickly and reliably if you draw schematics to make it obvious what you are trying to do.

As you are asked to read different schematics, take a moment to notice the drawing style. Mimic the drawing style of schematics you find easy to read. Put your creativity into the circuit design, not into drawing a schematic in a new style.

We now have a full vocabulary for talking about circuits and their sub-parts. We are ready to start analyzing.

[[☃ explanation 1]]

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:

[[☃ image 10]]

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.

[[☃ image 7]]

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

[[☃ image 17]]

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 $+$ 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 $2,\text{volts}$ to appear across the resistor.

** What is $i$? **

To find the current, apply Ohm's Law:

$i = \dfrac{v}{\text R}$

$i = \dfrac{+2,\text V}{250,\Omega}$

$i = +8 ,\text{mA}$

The voltage arrow tells us the top of the resistor is $2,\text V$ above the bottom of the resistor. Ohm's Law tells us the current is $+8 ,\text{mA}$. 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.

[[☃ image 15]]

Apply Ohm's Law, exactly the same as Example 1,

$i = \dfrac{+2,\text V}{250,\Omega}$

$i = +8 ,\text{mA}$

Ohm's Law is telling us the current is $+8 ,\text{mA}$. 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

[[☃ image 13]]

This $10,\text k\Omega$ 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 $-20 ,\mu\text A$. This may look a little odd, to show $-20 ,\mu\text A$ current flowing in the direction of the arrow. If you like, think of it as a $+20 ,\mu\text A$ 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.)

$v = i,\text R$

$v = -20 ,\mu\text A \cdot 10,\text k\Omega$

$v = (-20 \times 10^{-6}) \cdot (10 \times 10^{+3})$

$v = -0.2,\text V$

The answer came out with a negative sign, meaning the resistor terminal with the $+$ voltage polarity (the top terminal) is $0.2 ,\text V$ 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: $v = 1.5,\text V$. 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).
   [[☃ explanation 1]]

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).
   [[☃ explanation 2]]

[[☃ image 5]]

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,

[[☃ image 4]]

For a battery symbol, the longer black line indicates the positive terminal of the battery. A battery with two alternative labels,

[[☃ image 5]]

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: $i = 1 ,\text A$. 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.

[[☃ image 3]]

[[☃ explanation 1]]