The physical quantities that characterize many types of systems are defined at all points in space and time. For example, Maxwell's equations describe the relationship between the electric field, magnetic field, and charge at every point in space and time. The equations govern the operation of electric circuits. Cmputing the field and charge at every point is a very thorough approach to analyzing an electrical system. However, such a detailed model includes far too much information to be practical.
Analysis of many systems can be dramatically simplified by using circuit models. Circuit models characterize a system only at a small number of well-defined points by collapsing regions of space into lumped elements. A circuit model is simplified representation of a system that consists of a network of lumped elements connected together by wires. Regions of space that do not significantly impact system behavior are ignored.
Resistor symbol with terminal parameters defined.
A lumped element is an idealized abstraction of a physical component, such as a resistor. Elements are bracketed by terminals. Every element obeys a constitutive equation that describes the relationship between the state variables at its terminals. In elements with two terminals, one is typically designated (+) and the other (-). Each type of element has an associated graphic symbol. Symbols can be arranged in schematic diagrams to depict a system of interconnected elements. The symbol for a resistor is shown at right.
The state variables for an electric circuit are voltage (v) and current (i). Current measures the flow rate of electrons moving through the element. Voltage quantifies the electrical potential difference between two points. Higher voltages attract electrons more intensely and thus result in larger current flows. Every element has a single voltage across it and current flow through it. The constitutive equation for an ideal resistor is given by Ohm's law, v=iR.
Review of electrical units and equations
- Electric charge is measured in Coulombs, usually abbreviated with a capital C. Electrons have a charge of about 1.610×10−19 C. (One mole of electrons has a charge of about 96500 Coulombs.)
- Electric potential difference has the units of energy per charge and is measured in Volts. 1 Volt = 1 Joule/Coulomb = 1 kg m2 / sec 2/Coulomb. The variable v is most often used for voltage.
- Current flow has the units of charge per time and is measured in Amps. 1 Amp = 1 Coulomb / sec.  The variable i is normally used for current.
- Resistance has units of Ohms. 1 Ohm = 1 Volt divided by 1 Amp. Resistance is usually denoted by R
- Electric power is equal to voltage times current, P = v i. Power has units of energy per time called Watts. 1 Watt = 1 Volt x 1 Amp = 1 Joule / second.
Networks of lumped elements
Table 1 lists the name, symbol, constitutive relation and its graphical representation for four ideal elements: a voltage source, a current source, a resistor, and a conductor. Each constitutive equation and associated plot relate voltage (v) across an element and current (i) through that element.
|Table 1: Constitutive relations for ideal elements.
An ideal voltage source maintains a constant potential difference across its terminals regardless of the amount of current flowing through it. Thus, its voltage versus current graph is a horizontal line. The current flowing through an ideal current source is constant no matter what potential difference is across its terminals. The ideal resistor obeys Ohm’s law, v = iR — a sloped line on the voltage versus current graph. Ideal conductors have zero resistance, meaning that the voltage at all points on the conductor is the same for any value of current flow.
To form a network of elements, terminals are connected together by ideal wires. Elements interact only through their terminals. Terminals may be shown as filled or open circles to indicate whether or not that terminal connects to another electrical element. (Open circles represent no connection.) Pairs of terminals are sometimes called ports, whether for a single element or across multiple elements. A port approach is particularly useful for abstracting away (black-boxing) multiple circuit elements and treating them as an equivalent element with defined inputs and/or outputs. We'll return to this idea in sections 5.4 and 5.6.
A voltage v and current i can be defined at any terminal. For terminals A and B in Figure 1, the element voltage v = vAB = vA − vB and the element current i = iA = iB. Here, vA and vB are the absolute potentials (defined relative to a reference), and v is the potential difference between the two terminals.
Two limiting cases of the resistor element are useful for solving circuit problems: the open circuit and the short circuit (Fig. 2). Infinite resistance means that no current passes through that element; this is also called an open circuit across said element and is equivalent to an unconnected port. Zero resistance means no potential difference appears across the element; this is also called a short circuit across that element and is equivalent to an ideal wire.
Figure 2: Open and short circuits.
In the open circuit (left), the maximum potential difference exists across the resistor, and VAB = V
. In the short circuit (right), the potential is the same everywhere because a wire connects the two resistor terminals, and VAB = 0
Illustrative circuit analogies
We are now in a position to appreciate some analogies to electronic circuits. We’ll only describe in detail the most physically intuitive one, while the rest are listed in Table 2.
Table 2: Circuit variables in other engineering disciplines.
In a hydraulic system (Fig. 3), pressure differences between two points are analogous to voltages in electronic circuits. A water reservoir is like a voltage source, and water flow is analogous to current flow. The higher the pressure differential (greater amount/height of water in the reservoir), the faster the volumetric flow, all else being equal. Resistance can be thought of as the narrowness of the conduit connecting two points. The narrower the pipe, the slower the flow, all else being equal. Or it can be thought of as a pipe packed with beads. The smaller the beads, or the greater in number, the slower the flow, all else being equal.
Figure 3: Fluid dynamics analogy for circuits.
At left, the (pipe) resistance is high, slowing water flow from the reservoir, but the voltage equivalent is also high, quickening water flow compared to a less full reservoir. At right, the parameters are just the opposite.
The idealized resistor model is an excellent approximation of a real resistor's behavior under most circumstances.
Connecting circuit elements
Multiple elements may be joined at their terminals to build a circuit (Fig. 4). The joined terminals are now called nodes. Any path between two (independent -- see below) nodes is called a branch. Any path from one node back around to itself is called a loop. We’ll revisit these terms when we solve circuit problems in section 5.
Figure 4: Circuit illustrating definitions for node, branch, and loop.
Note that two (or more) apparent nodes that are connected by an ideal conductor are considered equivalent, because the conductor by definition cannot maintain a potential difference. Thus, A and the (A)'s above represent only one independent node, not three, as do B and the (B)'s. The two Steve's are of course independent nodes :)
Now is a good time to emphasize the difference between an element variable and a node variable. The element voltage vAB is the difference between the two node voltages, vA and vB. In other words, an element voltage equals the potential difference across two terminals, while a node voltage equals the absolute potential at a terminal. Of course, this "absolute" potential must still be with respect to a common reference point.
Series and parallel
When two elements are joined end to end, they are said to be in series. In this case, the (-) terminal of one element is joined to the (+) terminal of the second element. If instead the (+) terminals of the two elements are connected, as are their (-) terminals, then the elements are said to be in parallel.
Figure 5: Resistors in series and parallel (with respect to one voltage source).
Resistors in series are additive: R = R1 + R2.
Resistors in parallel are inversely additive: , and thus
These relationships are consistent with resistance being proportional to length and inversely proportional to area for a given material. The current that flows through each resistor in series must be identical, while for resistors in parallel, the voltage across each one must be identical. We will learn why when we discuss Kirchoff’s laws in section 5.1.
Similarly, voltage sources connected in series are additive, with current unaffected. Voltage sources linked in parallel maintain the potential difference of a single voltage source (so all sources must be identical or things get tricky ), with increased current flow.
Series and parallel relationships can be used to reduce complex circuits to more simple ones as an intermediate step in problem-solving.
Note that a voltage, or potential difference, is always a relative description. For example, for a voltage source the nominal (supposed) voltage -- say, 12 V -- is the potential difference between the (+) and (-) terminals. It is usually really convenient when solving circuit problems to define the (-) terminal as zero. (The relative nature of voltage allows this arbitrary choice.) This potential level is called ground. Ground can be considered a reservoir of electrons. By definition, any terminal connected to ground by a wire must also be at 0V. Ground is also the usual reference point for defining terminal/node voltages.
Figure 6: Multi-component circuit with all (dependent) grounded nodes marked.
However, there is nothing stopping us from defining the (+) and (-) terminals as +6V and -6V, and this may be convenient in certain problems.
Understanding equivalent diagram topologies and notations, part 1
A common group of elements comprises a voltage source and two resistors in series. For reasons we shall see shortly, this grouping or "primitive" is called a voltage divider. Two common ways to depict this circuit are shown below.
Figure 7B: Voltage divider circuit, separate grounds.
Figure 7A: Voltage divider circuit, complete connections.
We can see that the voltage at node C is trivially 0, and that at node A is the maximum source voltage, because these are connected to the voltage source by ideal wires. The voltage at node B depends on the relative values of the two resistors.
As an example, let's take V = 12 V, R1 = 200 Ω, and R2 = 600 Ω. Each resistor is described by v = iR, where i must be uniform in this purely series-connected circuit. Ohm's law also applies to the circuit as a whole when using the overall series equivalent resistance Req, namely the sum of the two resistances. Thus, v = 12V = iReq = i(R1 + R2) = i(200 + 600)Ω, which can be solved for i = 15 mA.
The first resistor is described by vAB = iR1 = 15mA * 200Ω. Thus, it has an element voltage of vAB = 3V, and the node joining the two resistors has node voltage vB = 12 − 3 = 9V. Indeed, solving the second resistor equation vBC = iR2 = 15mA * 600Ω yields 9 V (with vC= 0 , of course).
Clearly, a voltage divider "divides" the voltage drop across resistors in series in proportion to their resistances. In sections 5.2-5.3 we will see a more universally applicable way to solve this problem, when we cannot rely on our intuition as much. Then we will imagine connecting measurement devices to the circuit and also further discuss topology.
Some other examples of topogically equivalent circuits are shown below (Fig 8).
Figure 8B: Topological equivalence, 3 resistors.
Figure 8A: Topological equivalence, 2 resistors.
Putting it all together: solving circuit problems
Solving problems: fundamentals/theory
Two key relationships provide the foundation for circuit analysis: Kirchoff’s circuit law and Kirchoff’s voltage law, hereafter KCL and KVL.
KCL is based on charge conservation: at any node, the sum of the current magnitudes flowing in must equal the sum of the current magnitudes flowing out. When solving problems in which current directions are trivially known, this form of the law works well.
And now we see why current must be equal for elements connected in series: at the shared node, current coming out from the first element must have the same magnitude as current coming into the second element. No other branches exist at that node.
Returning to the issue of signs, current always flows from the (+) to the (-) terminal for an element, that is, from higher to lower voltage (except in a voltage source). However, the terminal definitions may not always be obvious/known in advance of solving the circuit problem. To keep signs straight, it will be convenient to have a systematic relationship between i and v signs. This relationship is described in the problem-solving section just below.
KVL is based on conservation of energy: the sum of the element voltages (potential difference across each element) around a loop is zero. Equivalently, the potential difference between the starting and ending node in a loop (that is, the same node) must be zero.
Take, for example, the three node voltage divider from section 4.3. Applying KVL directly in a clockwise direction, vAB + vBC + vCA = 3 + 9 − 12 = 0. KVL is even more straightforwardly applied using node voltages. Again going clockwise, clearly node voltages vA − vB + vB − vC − vA + vC = 0, without even plugging in numbers.
Now we also see why voltage must be equal for elements connected in parallel: the two shared nodes form a loop, and hence the potential differences across the two elements must be identical.
Solving problems: strategy
One rigorous way to solve circuit problems is to write the constitutive relation for each element (B equations for B branches), every independent KCL statement (N-1 equations for N nodes), and every independent KVL statement (B-1+N equations for B branches). Then algbraic substitutions are made until every element voltage and current is known. Often it is not obvious which substitutions will most quickly yield the quantities that are actually of interest in that problem.
A faster universal approach for solving circuit problems indirectly incorporates KVL and requires fewer KCL statements. Rarely can there be confusion about which choice of equations will quickly yield meaningful results. (For very complex circuits, however, linear rather than simple algebraic approaches may be required.)
Recall that while an element voltage is the potential difference across said element, a node voltage is the potential difference between that node and a common reference point – usually ground. (If these definitions doesn't sound familiar, see Figure 1 under Primary definitions and also review Connecting circuit elements.)
Figure 9: Voltage divider with current flow convention shown.
The true flow of the current in this circuit is clock-wise. Hence, i2
would be negative because the assumed toward-node direction is opposite the true flow.
Working with node voltages rather than element voltages makes KVL implicit, as we saw in the voltage divider example above. The basic approach is then to write a single KCL equation per unknown node voltage, while immediately substituting in the element constitutive laws in terms of voltages and resistances.
Now we make our alternative statement of KCL: the sum of all currents flowing into a node is zero, where current now includes both magnitude and sign. For i = v / R relationships, always subtract the close node voltage from the far node voltage — Vfar − Vclose — to calculate the element voltage v in the numerator, and the sign of the calculated current should always come out right. It will be positive if flowing in as assumed, and negative if actually flowing out. Using this sign convention, a current pointing toward the node is positive, and one pointing away fom the node is negative. However, a voltage source whose far node is the negative terminal must yield a positive current, because a source does work to push charge against the usual direction. [Last part maybe could be phrased more clearly]
Understanding equivalent diagram topologies and notations, part 2
Let’s return to the two-resistor voltage divider described in section 4.3 above to practice our systematic approach for solving circuits, calling the single non-trivial node of interest B for consistency. We can apply KCL to calculate it:
Now let’s imagine connecting a voltmeter to measure the voltage at node B. To read the correct relative potential, the voltmeter must also be connected at ground. In other words, the second resistor and the voltmeter with associated resistance Rm are in parallel. Before attaching the voltmeter, the access port of interest can be shown as:
Figure 10: Voltage divider with access port for voltmeter highlighted.
After attaching the voltmeter, three equivalent circuit diagrams depicting the system are:
Figure 11: Voltage divider with voltmeter attached.
The voltmeter must have a high enough resistance that it does not draw current and substantially affect vB. If instead it has a comparable resistance, say 300 Ω, let’s calculate the effect on vB using our usual KCL approach:
This value is unacceptably far off from our nominal 9 V. However, if Rm is increased an order of magnitude, to 3 kΩ, vBis a respectable 8.57 V. (Try it!)
Wouldn't it be nice to calculate vB for a given Rm without repeating the full calculation each time? The next section describes just such a method.
Thevenin equivalent circuits
Topological equivalence between circuits implies that wires have been moved around or added in place of shorthand. There is also such a thing as functional equivalence between circuits. In fact, any complex circuit consisting only of voltage sources and resistors can be reduced to a circuit with only a single voltage source and resistor! This equivalent circuit cannot be empirically distinguished from the multi-component one.
Let’s return to the voltage divider example with a port across the second resistor. We want to know how the circuit as a whole responds at this access port when there is no load on it, i.e., no element in which energy can be dissipated. We calculate the equivalent circuit by three steps.
First we calculate the open circuit voltage (Voc) at that port, namely 9 V as above in section 5.3.
Next we calculate the short circuit current. In this hypothetical, the attached voltmeter has no internal resistance and behaves as a wire (Fig 12). In turn, vB is shorted to ground, and current magnitude is dependent only on R1, not both R1 and R2. Hence, the short circuit current Isc = 12V / 200Ω = 0.06A = 60mA.
Finally, the Thevenin resistance . The Thevenin equivalent circuit is shown in Figure 13.
Now how does this two-component circuit reduction help us? The voltmeter (or any element!) can be connected in series with RTh, and vB calculated immediately by the simple voltage divider relation,
An alternative and direct calculation of RTh is the total resistance at the port as viewed looking into the circuit (with voltage sources changed to wires): here, R1 and R2 in parallel.
Figure 12: Short circuit current for voltage divider.
Figure 13: Thevenin reduction of voltage divider.
The Thevenin equivalent of the previously described voltage divider is shown (left) with an open port ready to be connected to a voltmeter with arbitrary resistance (right).
Alternative problem-solving strategies
Circuit problems can be approached in many ways and in the interests of time we will primarily treat the approach that (nearly) universally works, namely the node voltage/KCL method described above. Let’s briefly look at one example of another approach.
Consider the circuit shown at right. The parallel resistors R2 and R3 can be temporarily collapsed into an equivalent single resistor. Its resistance is given by
Then, the current through the purely series-connected circuit is simply
where Rtot = 400Ω and by the voltage divider relation vA is clearly 6 V. That is, half the voltage drop occurs across each 200 Ω resistor, the physically real one and the imaginary one (equivalent to two real ones).
We next expand the circuit to once more include the real, parallel branches. The potential difference across each branch must be the same vA, and thus the individual currents can be calculated from vA = ibranchxRbranchx.
For example, .
As expected, i2 and i3 sum to the current through the first resistor.
Yet another general circuit-solving approach is applying energy conservation equations.
Limits of ideal elements: more realistic model of a battery
Recall the ideal i-v curves for our four elements, referring to Table 1 under Primary definitions .
Let’s model a battery as an ideal voltage source connected to no other elements except ideal wires (Fig. 14).
What is the current flowing through this circuit? The resistance of the wires is zero, implying that the current is infinite! Therefore this model, and the associated horizontal i-v curve, is not realistic.
Figure 14: Model for ideal battery.
Figure 15: More realistic battery model with access port.
Instead, let’s model the battery as having an internal resistance, RBATT (Fig. 15). To investigate the i-v curve, we’ll view the output port formed by nodes A and B.
When the port is not connected to anything, resistance is infinite and current is zero. There is no load on the battery, and thus the output voltage at node A must be the full VBATT. This useful quantity is our by now familiar open circuit voltage, Voc.
Next, let’s attach a voltmeter to the port with an associated resistance Rm.
Figure 16: More realistic battery model with voltmeter attached.
When we assume that Rm = 0, vA is shorted to ground. Thus, the current i is at its maximum, namely the short circuit current:
These two quantities define two points on our new, more realistic, i-v curve! Presumably a line connects them, but let’s prove it by solving this simple circuit. We’ll practice our systematic approach once again.
The only unknown node voltage is vA. The other nodes are either ground or VBATT by definition of a perfect conductor. Hence we want to solve KCL at VBATT, substituting in the relevant v = iR constitutive relations:
and substituting for Rm using
to simplify to vA = 9 − I * RBATT,
indeed the equation of a line.
Figure 17: More realistic battery model: i-v plot.
From the model diagram, we see that a real battery only sources a full 9V when no load is applied. As current increases, supplied voltage decreases linearly, with negative slope equalling internal resistance. Here is just one simple example of how we can learn about a system by perturbing it -- whether theoretically or experimentally. In this case we found the function relating voltage and current by imagining putting a load on the battery. Note that the equation we found is ultimately not dependent on the resistance of this added load Rm.
Finally, note that the symbol for a real battery is different from that of the ideal voltage source shown earlier. A typical battery symbol looks like Figure 18A; an arbitrary number of additional lines may be shown in some cases (since most batteries actually contain multiple electrochemical cells), as in Figure 18B.
Figure 18B: Multi-cell battery symbol from public domain.
Figure 18A: Single-cell battery symbol from public domain.
Thanks for reading! I'll appreciate any constructive feedback about the utility of this guide. Agi Stachowiak 21:11, 11 October 2012 (EDT)
Circuit models are useful for solving problems in many engineering domains — electrical, mechanical, thermal, hydraulic, acoustic, and several others. Anything that is well described by a system of linear ordinary differential equations is amenable to circuit analysis. For simplicity, we’ll start with circuit elements whose behavior is not frequency dependent, namely resistors and sources, and build up to more complex systems involving capacitors and inductors in lecture/lab and perhaps someday a subsequent primer.
- ↑ For more about circuit assumptions and abstraction layers, see Agarwal 1.1-1.4: The power of abstraction, The lumped circuit abstraction, The lumped matter discipline, and Limitations of the lumped circuit abstraction.
- ↑ By convention, current is positive in the direction of positive charge flow, which means the electrons are actually flowing in the negative direction.
- ↑ See Agarwal 1.5.1, Practical two-terminal elements: Batteries, for more information.
- ↑ Slightly more complex approaches must be taken for cases in which a voltage source has a negative terminal that is not connected to ground. See Agarwal 3.3.2, The node method: Floating independent voltage sources.
- ↑ You could alternatively subtract the far node from the close node; the key is to be consistent. May as well memorize what Steve does though!
- ↑ Overall, see chapters 2 (Resistive Networks) and 3 (Network Theorems) of Agarwal. Section 2.4 (Intuitive method of circuit analysis: series and parallel simplification) treats series/parallel reductions, section 3.5 (Superposition) treats superposition, etc.