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. iSource LoadiVoltageacrosssourseVsResistanceinternal toloadRL Figure 1. Fundamental circuit model This circuit is made up of a source which provides a voltage across its terminals, labeled and a loadSV connected to the source which presents a resistance LR to the current flowing as indicated around a closed loop. i In order to characterize the operation of this circuit we must determine: • What voltage does the source provide as a function of current i? SV• What resistance LR does the load present? In order to completely define the problem we have to establish the relationship between the voltage , the resistance SVLR 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 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. i The current, , is equal to the amount of charge, Q, passing through a cross-section per second and it is expressed as i6.071/22.071 Spring 2006. Chaniotakis and Cory 1dQidt= (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 is given by abv abdWvdQ= (1.2) where work (W) is measured in Joules and the charge (Q) in Coulombs. The voltage is measured in volts (V) and Joule Newton meter1 volt 1 =1Coulomb Ampere second≡ . +-abelementvab Figure 2. Voltage across an element The positive (+) and negative (-) signs shown on Figure 2 define the polarity of the voltage . With this definition, represents the voltage at point a relative to point b. Equivalently we may also say that the voltage at point a is volts higher than the voltage at point b. abvabvabv 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/iv 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 space of the two most fundamental sources: the voltage source and the current source. /iv 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 curve of an ideal battery is: /iv vi0Vs As the 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. /iv This is not a realistic model of a battery. Real batteries contain small internal resistors resulting in a modification of the curve. We will look at these non-ideal effects in more detail shortly. /iv 6.071/22.071 Spring 2006. Chaniotakis and Cory 3Ideal 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 characteristic curve of an ideal current source are shown on Figure 4. /iv Is (a) viIs0 (b) Figure 4. (a) Symbol of current source and (b) characteristic curve of ideal current source. /iv Ideal resistor An ideal resistor is a passive, linear, two-terminal device whose resistance follows Ohm’s law given by, viR= (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, Ω, and V1Ω = 1A (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 SViR= (1.5) 6.071/22.071 Spring 2006. Chaniotakis and Cory 4+-VsRi Figure 5. Simple resistive circuit. The curve for a resistor is a straight line (the current is directly proportional to the voltage). The slope of the straight line is /iv1R (see Figure 6) For convenience we define the conductance (G) of a circuit element as the inverse of the resistance. 1ivGvR== (1.6) The SI unit of conductance is the siemens (S) 1 ΑVS==Ω (1.7) The most important use of curves is to characterize a component or an entire circuit as we will see later. The curve of the resistor shown on Figure 6 describes how that resistor will behave for any voltage or current. We can therefore use the curve to find the operating points of circuits. For our circuit (Figure 5) the voltage is set by the battery at and thus the operating point may be determined as shown graphically on Figure 6. /iv/iv/ivSV The power of this method should not be dismissed just because of its