1 ECE 3336 Introduction to Circuits & Electronics Note Set #2 Circuit Elements, Ohm’s Law, Kirchhoff’s Laws Spring 2015, TUE&TH 5:30-7:00 pm Dr. Wanda Wosik

2 Basic Elements of the Circuits Voltage sources Current sources Resistors Ohm’s Law Current flows through the resistor Voltage builds-up on the resistor terminals. Independent and Dependent Independent sources are real Dependent sources represent behavior of other systems/elements (later)

3 The 5 Basic Circuit Elements There are 5 basic circuit elements: 1.Voltage sources 2.Current sources 3.Resistors 4.Inductors (later) 5.Capacitors (later)

4 Voltage Sources A voltage source is a two-terminal circuit element that maintains a voltage V across its terminals. The value of the voltage is the defining characteristic of a voltage source. Location in the circuit and polarity of this source is important. Any value of the current can go through the voltage source, in any direction. The current can also be zero. The voltage source does not “care about” current. It “cares” only about voltage. Real voltage source has a resistance in series

Figure 2.32, 2.33 Voltage Sources - Battery

6 Voltage Source and Current Direction A battery supplies voltage, which allows for current (see conventions for the current vs. positive charges or electrons) to flow through a diode D to a lamp. When the battery is discharging the light becomes deemed and then goes off. i DD i

7 Voltage Sources Voltage sources: 1.Independent voltage sources (DC and AC) 2.Dependent voltage sources, of which there are 2 forms: i.Voltage-dependent voltage sources ii.Current-dependent voltage sources +-+-

8 Voltage Sources Symbol for Independent Sources The schematic symbol can be labeled either with a variable, like v S, or a value and units. Battery Independent AC voltage sources

Ideal Voltage Sources Various Representations of an Electrical System

Practical Voltage Source

Using the Voltmeter to Measure Voltage. Nigel P. Cook Electronics: A Complete Course, 2e Copyright ©2004 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved. 11

Measurement of Voltage

13 Dependent Voltage Sources Voltage-dependent voltage sources; µ[V/V] dimensionless Current-dependent voltage sources; r[V/A] has a dimension

14 Current Sources A current source is a two-terminal circuit element that maintains a current through its terminals, independently on the circuit connection. The value and sign of the current source is given. Any voltage can be across the current source, in either polarity. It can also be zero. The current source does not “care about” voltage. It “cares” only about current. NEVER assume that you know current in the voltage source! Calculate the voltage value and find its direction! Ideal current sources Real current sources Just add R in parallel

15 Current Sources Current sources: 1.Independent current sources 2.Dependent current sources: i.Voltage-dependent current sources ii.Current-dependent current sources

Practical Current Source 16

Measurement of Current

18 Dependent Current Sources i.Voltage-dependent current sources; coefficient g has dimension [A/V] ii.Current-dependent current sources; coefficient  is dimensionless [A/A] The values for these coefficients are always shown without units.

19 Voltage and Current Polarities Importance of reference polarities of currents and voltages. Notice that the schematic symbols for the voltage sources and current sources indicate these polarities. The voltage sources have a “+” and a “–” to show the voltage reference polarity. The current sources have an arrow to show the current reference polarity.

20 Units We will show units for the values of the coefficients  and g in a problem description not on the schematic. Variables should not have units. Values of variables must have the units. Examples: v X = 120 V i Q = 35 A p abs = 24.5 kW p del = v Q (13 A) p abs = v X i X Examples of missing units: v X = 1.5 p del = 25 i Q i X = 15

21 Resistors A resistor is a two terminal element that has a constant ratio of the voltage to the current through its terminals. It’s unit is Ohm or Ω=V/A The resistor does not have polarity. IMPORTANT: use Ohm’s Law only on resistors. It does not hold for sources iRiR - + vRvR R=v R /i R

Microscopic View of Ohm’s Law 22 Drift velocity Conductivity Ohm’s Law

23 Resistor Polarities There is no corresponding polarity to a resistor. The resistor is symmetrical (like a pipe of specified x-section). However, direction REALLY matters. The voltage and current is in the passive sign convention – then power (vi) dissipated by the resistor is + R v iRiR - Current is treated as a flow of positive charges so it flows from higher to lower potential i.e. from “+” to “-”. In the active sign convention we would have i x <0 and power delivered <0 (=dissipation) RESISTANCE CANNOT BE <0

24 Resistors, Resistance, and Ohm’s Law Resistance of RESISTORS is always positive. It can have some nonlinear behavior (ex. for larger currents) Resistance, as a ratio of voltage to current, can be negative in some devices/conditions: Devices with negative resistance provide positive power. This can be seen in dependent sources (various elements). I(A) V(V) I(A) V(V) R=V/I

2-8 The Resistance and Resistor

Ohmmeter and Measurement of Resistance

27 Extremely useful laws for currents and voltages in circuits: Kirchhoff’s Current Law (KCL) Kirchhoff’s Voltage Law (KVL) Here: connections of elements by wires will be very IMPORTANT. Wires have ≈ ∞ (high) conductivity (  ) so R≈0[Ω] Kirchhoff’s Laws

28 Nodes Important for KCL A node is defined as a point where two or more components are connected. We connect components with wires, which do not cause voltage drop (R wire =0 Ω). branches hmm, really?

29 Find the Nodes How many nodes are there in this circuit? Remember that wires have R≈0Ω so V=0V no because there are 2 pairs of the same nodes 5? so the answer is 3

30 Closed Loops Important for KVL A closed loop is a completely closed contour in the circuit. It starts and ends at the same point. It can, but does not need, to contain elements! It can also jump across open space i.e. will not follow components i.e. elements do not need to be connected.

31 Find Closed Loops Several closed loops are possible. The total number of simple closed loops in this circuit is 13. Example circuit

32 Closed Loop #1 Loop #1 shown red.

33 Loop #2 shown red. Closed Loop #2

34 Closed Loop #3 Loop #3 shown red. Jump across the voltage labeled v X.

35 Closed Loop #4 Loop #4 shown red. The same jump across the voltage labeled v X. The loop also crossed the current source. Remember that a current source can have a voltage across it. Do not assume that you know this voltage but calculate its value and direction

36 A Not-Closed Loop Not a closed contour i.e. not a closed loop

37 Kirchhoff’s Current Law (KCL) The algebraic (or signed i.e with directions) summation of currents through a closed surface (such as a node) must equal zero. It means that there is no charge build-up there.

38 Demonstration of Kirchhoff’s Current Law

39 Kirchhoff’s Current Law (KCL) It means that charge does not build up at this big node so charge is conserved. We may be not interested what happens with other currents? Hmm? Expand the KCL concept to a part of the circuit i.e. bigger node -> super node Here currents in this KCL: I S1, I 3 and I 5

40 Current Polarities in KCL In KCL we have to assign a sign to each reference current polarity that we will use (and do not change it from assignment to assignment!). We can choose: + sign for all currents entering the node - sign for all currents leaving the node or - sign for all currents entering the node + sign for all currents leaving the node These conventions are exactly opposite but equivalent in KCL.

41 KCL in the Example Circuit We will use a convention (w/o flip-floping): + for currents leaving the node – for those that are entering. For this circuit with our notation in KCL, we have the following equation:

42 Currents entering the node (closed surface) must be equal all currents leaving this node. This is the same as before. KCL More Intuitive Approach It is the consequence of KCL

43 Kirchhoff’s Voltage Law (KVL) The algebraic (or signed i.e. directions defined for voltages) summation of voltages around a closed loop must equal zero. -v 1 +v 2 =0 It results from energy conservation within a closed loop

44 KVL and Voltage Polarities In any closed loop we must designate a sign to each reference voltage polarity. We can (but will not) choose: + if the voltage shows a rise (- to +) or - if the voltage shows a drop (+ to -) as we circulate around the closed loop. Once the convention is set DO NOT change it from assignment to assignment. v1v1 v4v4 v2v2 v3v3 v5v

45 KVL – an Example We will use the following convention: positive sign assigned for a voltage drop negative sign assigned to a voltage rise. v1v1 v4v4 v2v2 v3v3 v5v KVL in each loop will give total three equations Loop 1: v 1 -v 2 +v 3 =0 Loop 2: -v 3 +v 4 + v 5 =0 Loop 3: v 1 -v 2 + v 4 +v 5 =0

46 KVL an Example (from the loops ex.) positive sign assigned for a voltage drop negative sign assigned to a voltage rise. KVL, when starting at the bottom, will give the following equation: Entering Positive Entering Negative See mnemonics

47 KVL More Intuitive Notation Voltage drops must equal the voltage rises Earlier we had These are the same equations. It is the consequence of of KVL You can use either method.

48 To solve a circuit we must determine unknown currents, voltages and, if required, also power. For DC conditions, KVL, KCL and Ohm’s Laws will be enough. We will add techniques to make solutions easy and systematic. Number of equations must be equal # of unknowns. Equations to Describe the Circuit

49 Ohm’s Law, KVL and KCL Find the voltage v X and the current i X. Use the following steps. Polarities givev 4 and i 3 (defined) Solve +  4 - for KVL i 3 - for Ohm’s Law

50 Ohm’s Law, KCL and KVL Results for v x and i x Solve the equations We have substituted into our KVL equation from other equations: