EGR 2201 Unit 2 Basic Laws  Read Alexander & Sadiku, Chapter 2.  Homework #2 and Lab #2 due next week.  Quiz next week.

Ohm’s Law  Ohm’s law says that the voltage v across a resistor is equal to the current i through the resistor times the resistor’s resistance R. In symbols: v = i  R  The voltage’s polarity and current’s direction must obey the passive sign convention, as shown in the diagram at right. Otherwise, you need a negative sign in this equation: v = i  R.

Ohm’s Law Rearranged  The equation v = i  R is useful if we know i and R, and we’re trying to find v.  Often we’ll need to rearrange the equation to one of the following forms: i = v  R or R = v  i

Short Circuit  An element with R=0 (or with an extremely small resistance) is called a short circuit.  Since a short circuit’s resistance is zero, Ohm’s law tells us that the voltage across a short circuit must also be zero: v = i  R = i  0 = 0  But we can’t use Ohm’s law to compute a short circuit’s current: i = v  R = 0  0 = ???

Open Circuit  An element with R= (or with an extremely large resistance) is called an open circuit.  Since an open circuit’s resistance is infinite, Ohm’s law tells us that the current through an open circuit must be zero: i = v  R = v   = 0  But we can’t use Ohm’s law to compute an open circuit’s voltage: v = i  R = 0   = ???

Power Dissipated by a Resistor  When current flows through a resistor, electric energy is converted to heat, at a rate given by the power law: p = v  i  Once the energy has been given off as heat, we can’t easily reverse this process and convert the heat back to electric energy. We therefore say that resistors dissipate energy. In contrast, we’ll see later that capacitors and inductors store energy, which can easily be recovered.

Other Power Formulas for Resistors  By combining the power law (p = v  i) with Ohm’s law (v = i  R or i = v  R), we can easily derive two other useful formulas for the power dissipated by a resistor: p = i 2  R p = v 2  R  Of course, each of these equations can in turn be rearranged, resulting in a number of useful equations that are summarized in the “power wheel”….

The “Power Wheel”  This is a useful aid for people who aren’t comfortable with basic algebra, but you shouldn’t need it.  The important point is that if you know any two of the quantities in the wheel’s center (P, V, I, and R), you can compute the other two, as long as you remember p=vi and v=iR.

Conductance  It’s sometimes useful to work with the reciprocal of resistance, which we call conductance.  The symbol for conductance is G: G = 1  R  Its unit of measure is the siemens (S).  Example: If a resistor’s resistance is 20 , its conductance is 50 mS.

Review: Some Quantities and Their Units QuantitySymbolSI UnitSymbol for the Unit CurrentI or iampereA VoltageV or vvoltV ResistanceRohm  ChargeQ or qcoulombC Timetseconds EnergyW or wjouleJ PowerP or pwattW ConductanceGsiemensS

Ohm’s Law and the Power Formulas Using Conductance  As the book discusses, Ohm’s law and our power formulas can be rewritten using conductance G instead of resistance R.  Example: Instead of writing v = i  R we can write v = i  G  But I advise you to ignore this, and always use R instead of G.

Circuit Topology: Branches  When describing a circuit’s layout (or “topology”), it’s often useful to identify the circuit’s branches, nodes, and loops.  A branch represents a single circuit element such as a voltage source or a resistor. Example: This circuit (from Figure 2.10) has five branches.

Dots or No Dots?  In schematic diagrams, our textbook sometimes draws dots at the points where two or more branches meet, as in this diagram.  But most of the time it omits these dots, as in this diagram.  There’s no difference in meaning.

Circuit Topology: Nodes  A node is the point of connection between two or more branches. Example: This circuit has three nodes, labeled a, b, and c.

Circuit Topology: Loops  A loop is any closed path in a circuit. Example: This circuit has six loops.  We won’t need to worry about the book’s distinction between loops and independent loops.

Elements in Series  Two elements are connected in series if they are connected to each other at exactly one node and there are no other elements connected to that node. Example: In this circuit, the voltage source and the 5- resistor are connected in series.

 If two elements are connected in series, they must carry the same current. Example: In this circuit, the current through the voltage source must equal the current through the 5- resistor.  But usually their voltages are different. Example: In the circuit above, we wouldn’t expect the voltage across the 5- resistor to be 10 V. Current Through Series- Connected Elements

Elements in Parallel  Two or more elements are connected in parallel if they are connected to the same two nodes. Example: In this circuit, the 2- resistor, the 3- resistor, and the current source are connected in parallel.

 If two elements are connected in parallel, they must have the same voltage across them. Example: In this circuit, the voltage across the 2- resistor, the 3- resistor, and the current source must be the same.  But usually their currents are different. Example: In the circuit above, we wouldn’t expect the current through the 3- resistor to be 2 A. Voltage Across Parallel-Connected Elements

 Sometimes elements are connected to each other, but they are neither connected in series nor connected in parallel. Example: In this circuit, the 5- resistor and the 2- resistor are connected, but they’re not connected in series or in parallel. In such a case, we wouldn’t expect the two elements to have the same current or the same voltage. This type of connection doesn’t have a special name. Some Connections are Neither Series Nor Parallel

Series Circuits  Two simple circuit layouts are series circuits and parallel circuits.  In a series circuit, each connection between elements is a series connection. Therefore, current is the same for every element. (But usually voltage is different for every element.)  A series circuit is a “one-loop” circuit. Another series circuit A series circuit: Same circuit on the breadboard:

Analyzing a Series Circuit 1. Find the total resistance by adding the series-connected resistors: R T = R 1 + R 2 + R Use Ohm’s law to find the current produced by the voltage source: i s = v s  R T 3. Recognize that this same current passes through each resistor: i s = i 1 = i 2 = i 3 = Use Ohm’s law to find the voltage across each resistor: v 1 = i 1  R 1 and v 2 = i 2  R 2 and... + v 1  + v 2   v 3 + isis i1i1 i2i2 i3i3

Parallel Circuits  In a parallel circuit, every element is in parallel with every other element. Therefore, voltage is the same for every element. (But usually current is different for every element.) Another parallel circuit A parallel circuit: Same circuit on the breadboard:

Analyzing a Parallel Circuit 1. Recognize that the voltage across each resistor is equal to the source voltage: v s = v 1 = v 2 = v 3 = Use Ohm’s law to find the current through each resistor: i 1 = v 1  R 1 and i 2 = v 2  R 2 and... +v1+v1 +v2+v2 +v3+v3 i1i1 i2i2 i3i3

 We’ve seen that series circuits and parallel circuits are easy to analyze, using little more than Ohm’s law.  But most circuits don’t fall into either of these categories, and are harder to analyze.  Some authors call these series-parallel circuits. Others call them complex circuits. More Complicated Circuits Examples that are neither series circuits nor parallel circuits

Kirchhoff’s Current Law (KCL)  Kirchhoff’s current law: The algebraic sum of currents entering a node is zero. Example: In this figure, i 1  i 2 + i 3 + i 4  i 5 = 0  Alternative Form of KCL: The sum of the currents entering a node is equal to the sum of the currents leaving the node. Example: In the figure, i 1 + i 3 + i 4 = i 2 + i 5

KCL Applied to a Closed Boundary  You can also apply KCL to an entire portion of a circuit surrounded by an imaginary boundary. The sum of the currents entering the boundary is equal to the sum of the currents leaving the boundary. Example: In this figure, i 1 + i 5 = i 2 + i 7 + i 8

Kirchhoff’s Voltage Law (KVL)  Kirchhoff’s voltage law: Around any loop in a circuit, the algebraic sum of the voltages is zero. Example: In this figure, v 1 + v 2 + v 3  v 4 + v 5 = 0  Alternative Form of KVL: Around any loop, the sum of the voltage drops is equal to the sum of the voltage rises. Example: In the figure, v 2 + v 3 + v 5 = v 1 + v 4

Equivalent Resistance  In analyzing circuits we will often combine several resistors together to find their equivalent resistance.  Basic idea: What single resistor would present the same resistance to a source as the combination of resistors that it’s actually connected to?

Resistors in Series  The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances: R eq = R 1 + R R N  We’ve already used this earlier in Step 1 of our analysis of series circuits.

Parallel Resistors

More Than Two Resistors in Parallel

Parallel Resistors: Two Shortcut Rules for Special Cases

Series-Parallel Combinations of Resistors  In many cases, you can find the equivalent resistance of combined resistors by repeatedly applying the previous rules for resistors in series and resistors in parallel.  Hint: Always start farthest from the source or (in a case like the one above from Figure 2.36) farthest from the open terminals.

Voltage Division  For resistors in series, the total voltage across them is divided among the resistors in direct proportion to their resistances. Example: In the circuit shown (Figure 2.29), if R 1 is twice as big as R 2, then v 1 will be twice as big as v 2.  See next slide for a formula that captures this…

The Voltage-Divider Rule

The Voltage-Divider Rule in More Complex Circuits  The voltage-divider rule is easiest to apply in series circuits, but it also holds for series resistors in more complex circuits.

Current Division  For resistors in parallel, the total current through them is shared by the resistors in inverse proportion to their resistances. Example: In the circuit shown (Figure 2.31), if R 1 is twice as big as R 2, then i 1 will be one-half as big as i 2.  See next slide for a formula that captures this…

The Current-Divider Rule

The Current-Divider Rule in More Complex Circuits  The current-divider rule is easiest to apply in parallel circuits, but it also holds for parallel resistors in more complex circuits.

Review of Short Circuits  Recall that an element with R=0 (or with an extremely small resistance) is called a short circuit.  Recall also that, since the resistance of a short circuit is zero, the voltage across it must always be zero.  Sometimes short circuits are introduced intentionally into a circuit, but often they result from a circuit failure.

New Observations about Short Circuits  The equivalent resistance of a short circuit in parallel with anything else is zero.  An element or portion of a circuit is short- circuited, or “shorted out,” when there is a short circuit in parallel with it.  No current flows in a short-circuited element; instead all current is diverted through the short circuit itself.

Series-Aiding Voltage Sources  If two series-connected voltage sources produce current in the same direction, they are said to be series-aiding.  The net effect on the circuit is the same as that of a single source whose voltage equals the sum of the two voltage sources.

Series-Opposing Voltage Sources  If two series-connected sources produce current in opposite directions, they’re said to be series-opposing.  The effect is the same as that of a single source equal in magnitude to the difference between the source voltages and having the same polarity as the larger of the two.

Series-Connected Current Sources?  Current sources are not connected in series.

Potentiometers & Rheostats  A potentiometer is an adjustable voltage divider that is widely used in a variety of electronic circuit applications. It is a three-terminal device.  A rheostat is an adjustable resistance. It has only two terminals.

Parallel Voltage Sources  Different-valued voltage sources are not connected in parallel. Exception: rechargeable batteries.  Two identical voltage sources, having the same voltage and the same internal resistance, can be connected in parallel.

Parallel Current Sources  Equal- or different-valued current sources can be connected in parallel. The parallel-connected sources can be replaced by a single equivalent current source that produces a current equal to the algebraic sum (resultant) of the several sources.