Circuit Theory I Basic Laws Assistant Professor Suna BOLAT Eastern Mediterranean University Electric and electronic department ef2: Anant Agarwaland Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare(http://ocw.mit.edu/), Massachusetts Institute of Technology
Basic laws Physics laws makes our lifes easier. Ohm s Law 2. Kirchhoff Laws form the foundation for electric circuit analysis 2
esistance All materials resist to the flow of current esistance of an element denotes its ability to resist the flow of electric current, which is measured in ohms (Ω). A cylindrical material of length l and cross-sectional area A has the following resistance: = ρ l A 3
Conductors (e.g. Wires) have very low resistance (<0. Ω), which is usually be neglected (i.e. We will assume that wires have zero resistance). Insulators (e.g. air) have very large resistance (>50 MΩ) that can be usually ignored ( omitted from circuit for analysis). esistors have a medium range of resistance and must be accounted for the circuit analysis. 4
Ohm s Law There is a linear relationship between the voltage and the current ν = i ν = voltage in volts (V), i = current in (A), = resistance in (Ω). During this course, we will assume (naively) that Ohm s law holds! 5
Don t trust Ohm s law! esistivity is a strong function of temperature. If the temperature is increased all the electrons in the material (i.e. Copper conductor) gets faster, resistivity goes up! If resistivity goes up, resistance goes up as well. Higher temperature, higher resistance ν = i (T) For example, for a light bulb, which is a resistor, when current runs through it, the bulb heats up, resistance of the bulb gets higher. 6
Passive sign convention (revisit) Note that the relationship between current and voltage are sign sensitive. PSC is satisfied if the current enters the positive terminal of an element: if PSC is satisfied : ν = i if PSC is not satisfied : ν = i 7
Equations derived from Ohm s law: Ohm s law ν = i = v i i = v Ω = V/A ecall p = v i = i 2 = v2 esistors cannot produce power, so the power absorbed by a resistors will always be positive 8
Short circuit Short circuit is zero resistance An element (or wire) with = 0 is called a short circuit. Short circuit is just drawn as a wire (line). 9
Short circuit as voltage source An ideal voltage source with Vs = 0 V is equivalent to a short circuit. Since ν = i and = 0, ν = 0 regardless of i. You could draw a source with Vs = 0 V, but it is not done in practice. You cannot connect a voltage source to a short circuit. If connected, usually wire wins and the voltage source melts (smoke comes out) if not protected. 0
Open circuit Opposite of short circuit An element (or wire) with = is called an open circuit. Such an element is just omitted.
Open circuit as a current source An ideal current source with I = 0 A is equivalent to a open circuit. SinceSince ν = i and I = 0 then =. You could draw a source with I=0 A, but it is not done in practice. You cannot connect a current source to an open circuit. If connected, usually you blow the current source (smoke comes out) if not protected. The insulator (air) wins. Else, sparks fly. 2
Conductance Conductance (G) is the ability of an element to conduct electric current. Conductance is the inverse of resistance. G = = i v Units: siemens (S) or mho ( ) ν = i & p = v i = i 2 = v2 i = G v & p = v i = v 2 G = i2 G 3
Circuit building blocks Nodes, Branches and Loops circuits are modelled to be the same as networks. Networks are composed of nodes, braches and loops 4
Branch A branch represents a single element such as voltage source, resistor or current source. Any two terminal element is represented by a branch (Examples: voltage source/current source/resistors) Q: How many branches? A: How many elements? Wire segments are not counted as branches. 5
Node A node is a point of connection between two or more branches Node: a connection point between two or more branches. May include a portion of circuit (more than a single point). Essential Node: the point of connection between three or more branches. Q: How many nodes? There are 3 nodes (a, b and c) 2 essential nodes (b and c) 6
Loop A loop is a closed path in a circuit. Loop: a closed path in a circuit. Independent Loop: A loop is independent if it contains at least one branch which is not a part of any other independent loop. Q: How many loops? There are 6 loops 3 independent loops. 7
Kirchhoff s Laws To define equations for circuit elements: use Ohm s law Defining equations (from Ohm s law) tell us how the voltage and current within a circuit element are related. To define relaship between the element? Kirchhoff s Current Law (KCL) Kirchhoff s Voltage Law (KVL) Kirchhoff s laws tell us how the voltages and currents in different branches are related. 8
Kirchhoff s current law (KCL) Kirchhoff s current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero. The sum of currents entering a node is equal to the sum of the currents leaving the node. 9
Kirchhoff s current law (KCL) Applying KCL to node a: Equivalent circuit can be generated as follows: 20
KCL for closed boundaries KCL also applies to a closed boundary 2
Example: Apply KCL to the each essential node in the circuit. Essential node : Essential node 2: Essential node 3: 22
Connecting ideal current sources Ideal current sources cannot be connected in series. ecall: ideal current sources guarantee the current flowing through source is at specified value. ecall: the current entering a circuit element must be equal to the current leaving the circuit element: I in = I out. Ideal current sources do not exist. 23
Kirchhoff s Voltage Law (KVL) Kirchhoff s voltage law (KVL) states that the algebraic sum of voltages around a closed path (or loop) is zero. 24
KVL Apply KVL to each loop in the following circuit: Loop : Loop 2: Loop 3: Loop 4: Loop 5: Loop 6: 25
Example 26
Circuit analysis Goal: Find all element v s and i s write element v-i relationships (Ohm s law) write KCL for all nodes write KVL for all loops.2.3 lots of unknowns lots of equations lots of fun (?) solve 27
Element relationships For ν = i V 0 For voltage source ν = V 0 I 0 For current source i = I 0 28
Apply KVL KCL What do we do?! Apply element combination rules 29
Element combination rules 30
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Series connection 32 4 3 2 4 3 2 4 3 2 4 3 2 ) ( G G G G G I I I I I I V eq eq eq s s s s s s s Applying KVL
Series connection (Applying KVL) (voltage divider circuit) voltage accross each resistor, is proportional to its resistance. Larger the resistance, larger the voltage drop on that resistor:
Parallel connection 4 3 2 4 3 2 4 3 2 ) ( V V I I I I I eq eq eq s s s Applying KCL
Parallel connection (Applying KCL at node a) (current divider circuit) Given the total current i entering to node a the current is shared by the resistors by inverse proportion to their resistance:
Equivalent resistance 36
Examples Calculate the eq for the following circuit. 37
Calculate the ab for the following circuit. 38
Calculate the G eq for the following circuit. 39
Sometimes connections are complicated There are cases where the resistors are neither in parallel nor in series 40
Wye-Delta Transformations Wye (Y) Networks Delta(Δ) Networks 4
Wye-Delta transformation 42
Converting a D network to a Y network 43
Examples Convert the following Ynetwork to a D network. 44
Examples Calculate ab and and use it to calculate i. 45
Examples Calculate eq and Power delivered by the source. 46
Examples Calculate ab and and use it to calculate i. 47