Resistive circuit analysis Kirchhoff s Laws Fundamentals of DC electric circuits A simple model that we can use as a starting point for discussing electronic circuits is given on Figure 1 i Voltage across sourse Vs Source Load Resistance RL internal to load i Figure 1 Fundamental circuit model This circuit is made up of a source which provides a voltage across its terminals labeled VS and a load connected to the source which presents a resistance RL to the current i flowing as indicated around a closed loop In order to characterize the operation of this circuit we must determine What voltage VS does the source provide as a function of current i What resistance RL does the load present In order to completely define the problem we have to establish the relationship between the voltage VS the resistance RL and the current i Before proceeding let s define the physical significance of these new physical variables and establish ways to represent them Current i The current i results from the flow of electric charge around the closed loop shown on Figure 1 Electrons are electrically negatively charged particles and their flow in conductors such as wires results in electric current The current i is equal to the amount of charge Q passing through a cross section per second and it is expressed as 6 071 22 071 Spring 2006 Chaniotakis and Cory 1 i dQ dt 1 1 The unit of charge is the Coulomb One Coulomb is equivalent to 6 24 x 1018 electrons The unit for current is the ampere A One ampere 1 Coulomb sec 1 Voltage In order to move electrons along a conductor some amount of work is required The work required must be somehow supplied by an electromotive force usually provided by a battery or similar device This electromotive force is referred to as the voltage or potential difference between two points or across an element By representing an element with the block diagram shown on Figure 2 the voltage across the element represents the potential difference between terminals a and b Mathematically the voltage vab is given by vab dW dQ 1 2 where work W is measured in Joules and the charge Q in Coulombs Joule Newton meter 1 The voltage is measured in volts V and 1 volt 1 Coulomb Ampere second element a b vab Figure 2 Voltage across an element The positive and negative signs shown on Figure 2 define the polarity of the voltage vab With this definition vab represents the voltage at point a relative to point b Equivalently we may also say that the voltage at point a is vab volts higher than the voltage at point b 1 The SI system of units is based on the following seven base units Length m Mass kg Time s Thermodynamic temperature K Amount of substance mol Luminous intensity cd Current A The purpose for this small diversion is to remind us of the power of dimensional analysis in engineering 6 071 22 071 Spring 2006 Chaniotakis and Cory 2 i v curves The two dynamical variables of electronic circuits are current and voltage It is useful therefore to explore the characteristic relationship between these for various circuit elements The relationship between voltage and current for an element or for an entire circuit as we will explore shortly is fundamental in circuit design and electronics We will start this exploration by looking at the i v space of the two most fundamental sources the voltage source and the current source Ideal DC voltage sources The most common voltage source is a battery The voltage provided by a battery is constant in time and it is called DC voltage In its ideal implementation the battery provides a specific voltage at all times and for all loads The common symbols for a battery are shown on Figure 3 Vs Vs Figure 3 Battery symbols The i v curve of an ideal battery is i 0 Vs v As the i v curve shows regardless of the current flowing through the battery the voltage across the battery remains constant The actual amount of current that is provided by the battery depends on the circuit that is connected to the battery This is not a realistic model of a battery Real batteries contain small internal resistors resulting in a modification of the i v curve We will look at these non ideal effects in more detail shortly 6 071 22 071 Spring 2006 Chaniotakis and Cory 3 Ideal DC current sources The current source is a device that can provide a certain amount of current to a circuit The symbol for a DC current source and the i v characteristic curve of an ideal current source are shown on Figure 4 i Is Is 0 a v b Figure 4 a Symbol of current source and b i v characteristic curve of ideal current source Ideal resistor An ideal resistor is a passive linear two terminal device whose resistance follows Ohm s law given by v iR 1 3 which states that the voltage across an element is directly proportional to the current flowing through the element The constant of proportionality is the resistance R provided by the element The resistance is measured in Ohms and 1 1 V A 1 4 The symbol for a resistor is R Notice that there is no specific polarity to a physical resistor the two leads terminals are equivalent The circuit shown on Figure 5 consists of a voltage source and a resistor These two elements are connected together with wires which are considered to be ideal The current flowing through the resistor is given by i VS R 6 071 22 071 Spring 2006 Chaniotakis and Cory 1 5 4 i Vs R Figure 5 Simple resistive circuit The i v curve for a resistor is a straight line the current is directly proportional to the voltage The slope of the straight line is R1 see Figure 6 For convenience we define the conductance G of a circuit element as the inverse of the resistance i 1 v Gv R 1 6 The SI unit of conductance is the siemens S S 1 V 1 7 The most important use of i v curves is to characterize a component or an entire circuit as we will see later The i v curve of the resistor shown on Figure 6 describes how that resistor will behave for any voltage or current We can therefore use the i v curve to find the operating points of circuits For our circuit Figure 5 the voltage is set by the battery at VS and thus the operating point may be determined as shown graphically on Figure 6 The power of this method should not be dismissed just because of its apparent simplicity The i v curve is one of the most powerful tools for circuit analysis and we will use it extensively in characterizing circuits and electronic components i slope is 1 R Vs R operating point 0 Vs v Figure 6 i v curve of a resistor 6 071 22 071 Spring 2006 Chaniotakis