Kirchhoff's Laws and Circuit Analysis (EC ) Circuit analysis: solving for I and V at each element Linear circuits: involve resistors, capacitors, inductors Initial analysis uses only resistors Power sources, constant voltage and current Solved using Kirchhoff's Laws (Current and Voltage)
Circuit odes and Loops ode: a point where several wires electrically connect Symbolized by a dot or circle at the wire crossing If wires cross without dot, then not connected odes also called junctions Typically give notes a number or letter Branches: lines with devices connecting two nodes Loop: an independent closed path in a circuit There may be several possible closed paths
Kirchhoff's Current Law (KCL) Kirchhoff's Current Law (KCL) The algebraic sum of currents entering any node (junction) is zero. j I j 0 where number of lines entering the node OTE: the sign convention, Currents are positive when they entering the node Currents negative when leaving Or the reverse of this. KCL is called a continuity equation: It says current is not created or destroyed at any node.
Example of Kirchhoff's Current Law (KCL) Consider the simple parallel resistances below At node define current positive into resistors Since V on 6V the current is V 5 I 5 ma 000 Same V on 6V the current is Thus by KCL at node Thus the total current is V 5 I ma 5000 I + I + I3 0.005 + 0.00+ I3 I I I 6 ma 3 ode has the negatives of these values 0
Kirchhoff's Voltage Law (KVL) Kirchhoff's Voltage Law (KVL) Algebraic sum of the voltage drops around any loop or circuit 0 j where number of voltage drops V j 0 OTE: the sign convention Voltage drops are positive in the direction of the set loop current. Voltage drops negative when opposite loop current. Voltage sources positive if current flows out of + side Voltage sources negative if current flows into + side A loop is an independent closed path in the circuit. Define a "loop current" along that path eal currents may be made up of several loop currents I I I
Example Kirchhoff's Voltage Law (KVL) Consider a simple one loop circuit Voltages are number by the element name eg. V or V : voltage across Going around loop in the loop direction ecall by the rules: Voltage drops negative when opposite loop current. Voltage sources positive if current flows out of + side V 0 V s
Example Kirchhoff's Voltage Law (KVL) Continued Thus voltage directions are easily defined here: V 0 V s Why negative V? Opposes current flow I Since V I V s I Thus this reduces to the Ohms law expression I V s 0
KVL Example esistor Voltage Divider Consider a series of resistors and a voltage source Then using KVL Since by Ohm s law Then V I V V V V 0 I V I ( + ) 0 I V I Thus V 5 I ma + 000 + 3000 i.e. get the resistors in series formula total + 5 KΩ
KVL Example esistor Voltage Divider Continued What is the voltage across each resistor ow we can relate V and V to the applied V With the substitution I V + Thus V V 5(000) V I V + 000 + 3000 Similarly for the V V 5(3000) V I 3V + 000 + 3000
KVL and KCL for Different Circuits With multiple voltage sources best to use KVL Can write KVL equation for each loop With multiple current sources best to use KCL Can write KCL equations at each node. In practice can solve whole circuit with either method
esistors in Series (EC3) esistors in series add to give the total resistance total j j Example: total of,, and 3 Kohm resistors in series Thus total is total + + 000 + 000 + 3000 6 KΩ 3 esistors in series law comes directly from KVL
esistors in Parallel esistors in parallel: Inverse of the total equals the sum of the inverses total j j This comes directly from KCL at the node I total V total OTE: inverse of resistance called conductance (G) Unites are mhos (ohms spelled backwards) j I G total G j Thus when work in conductance change parallel to series j j j V j
Example Parallel esistors Example K and K resistors in parallel total j j total 000 + 000 666 Ω 3000 000000
Example Parallel esistors Example: two Kohm resistors in parallel total j j 000 + 000 000 K ork total 500 Ω Thus adding same resistors cuts get total Good way to get lower values