# 2: Resistor Circuits - Imperial College London

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2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)2: Resistor CircuitsResistor Circuits: 2 – 1 / 13

Kirchoff’s Voltage Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksThe five nodes are labelledA, B, C, D, E where E is thereference node.Each component that links a pair ofnodes is called a branch of thenetwork. Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 2 / 13

Kirchoff’s Voltage Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryThe five nodes are labelledA, B, C, D, E where E is thereference node.Each component that links a pair ofnodes is called a branch of thenetwork.Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the workdone in moving a charge from one node to another does not depend on theroute you take; in particular the work done in going from one node back tothe same node by any route is zero.E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 2 / 13

Kirchoff’s Voltage Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryThe five nodes are labelledA, B, C, D, E where E is thereference node.Each component that links a pair ofnodes is called a branch of thenetwork.Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the workdone in moving a charge from one node to another does not depend on theroute you take; in particular the work done in going from one node back tothe same node by any route is zero.KVL: the sum of the voltage changes around any closed loop is zero.E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 2 / 13

Kirchoff’s Voltage Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryThe five nodes are labelledA, B, C, D, E where E is thereference node.Each component that links a pair ofnodes is called a branch of thenetwork.Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the workdone in moving a charge from one node to another does not depend on theroute you take; in particular the work done in going from one node back tothe same node by any route is zero.KVL: the sum of the voltage changes around any closed loop is zero.Example: VDE VBD VAB VEA 0E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 2 / 13

Kirchoff’s Voltage Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryThe five nodes are labelledA, B, C, D, E where E is thereference node.Each component that links a pair ofnodes is called a branch of thenetwork.Kirchoff’s Voltage Law (KVL) is a consequence of the fact that the workdone in moving a charge from one node to another does not depend on theroute you take; in particular the work done in going from one node back tothe same node by any route is zero.KVL: the sum of the voltage changes around any closed loop is zero.Example: VDE VBD VAB VEA 0Equivalent formulation:VXY VXE VY E VX VY for any nodes X and Y .E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 2 / 13

Kirchoff’s Current Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Wherever charges are free to move around, they will move to ensurecharge neutrality everywhere at all times.Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 3 / 13

Kirchoff’s Current Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Wherever charges are free to move around, they will move to ensurecharge neutrality everywhere at all times.A consequence is Kirchoff’s Current Law (KCL) which says that the currentgoing into any closed region of a circuit must equal the current coming out.Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 3 / 13

Kirchoff’s Current Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksWherever charges are free to move around, they will move to ensurecharge neutrality everywhere at all times.A consequence is Kirchoff’s Current Law (KCL) which says that the currentgoing into any closed region of a circuit must equal the current coming out.KCL: The currents flowing out of any closed region of a circuit sum to zero. Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 3 / 13

Kirchoff’s Current Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksWherever charges are free to move around, they will move to ensurecharge neutrality everywhere at all times.A consequence is Kirchoff’s Current Law (KCL) which says that the currentgoing into any closed region of a circuit must equal the current coming out.KCL: The currents flowing out of any closed region of a circuit sum to zero.Green: I1 I7 Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 3 / 13

Kirchoff’s Current Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryWherever charges are free to move around, they will move to ensurecharge neutrality everywhere at all times.A consequence is Kirchoff’s Current Law (KCL) which says that the currentgoing into any closed region of a circuit must equal the current coming out.KCL: The currents flowing out of any closed region of a circuit sum to zero.Green: I1 I7Blue: I1 I2 I5 0E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 3 / 13

Kirchoff’s Current Law2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryWherever charges are free to move around, they will move to ensurecharge neutrality everywhere at all times.A consequence is Kirchoff’s Current Law (KCL) which says that the currentgoing into any closed region of a circuit must equal the current coming out.KCL: The currents flowing out of any closed region of a circuit sum to zero.Green: I1 I7Blue: I1 I2 I5 0Gray: I2 I4 I6 I7 0E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 3 / 13

KCL Example2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:The currents and voltages in any linear circuit can be determined by usingKCL, KVL and Ohm’s law.Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 4 / 13

KCL Example2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksThe currents and voltages in any linear circuit can be determined by usingKCL, KVL and Ohm’s law.Sometimes KCL allows you to determine currents very easily withouthaving to solve any simultaneous equations:How do we calculate I ? Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 4 / 13

KCL Example2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryThe currents and voltages in any linear circuit can be determined by usingKCL, KVL and Ohm’s law.Sometimes KCL allows you to determine currents very easily withouthaving to solve any simultaneous equations:How do we calculate I ?KCL: 1 I 3 0E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 4 / 13

KCL Example2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryThe currents and voltages in any linear circuit can be determined by usingKCL, KVL and Ohm’s law.Sometimes KCL allows you to determine currents very easily withouthaving to solve any simultaneous equations:How do we calculate I ?KCL: 1 I 3 0 I 2 AE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 4 / 13

KCL Example2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryThe currents and voltages in any linear circuit can be determined by usingKCL, KVL and Ohm’s law.Sometimes KCL allows you to determine currents very easily withouthaving to solve any simultaneous equations:How do we calculate I ?KCL: 1 I 3 0 I 2 ANote that here I ends up negative which means we chose the wrong arrowdirection to label the circuit. This does not matter. You can choose thedirections arbitrarily and let the algebra take care of reality.E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 4 / 13

Series and Parallel2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series: Components that are connected in a chain so that the same currentflows through each one are said to be in series.Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 5 / 13

Series and Parallel2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksSeries: Components that are connected in a chain so that the same currentflows through each one are said to be in series.R1 , R2 , R3 are in series and the samecurrent always flows through each. Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 5 / 13

Series and Parallel2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksSeries: Components that are connected in a chain so that the same currentflows through each one are said to be in series.R1 , R2 , R3 are in series and the samecurrent always flows through each.Within the chain, each internal nodeconnects to only two branches. Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 5 / 13

Series and Parallel2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummarySeries: Components that are connected in a chain so that the same currentflows through each one are said to be in series.R1 , R2 , R3 are in series and the samecurrent always flows through each.Within the chain, each internal nodeconnects to only two branches.R3 and R4 are not in series and do notnecessarily have the same current.E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 5 / 13

Series and Parallel2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummarySeries: Components that are connected in a chain so that the same currentflows through each one are said to be in series.R1 , R2 , R3 are in series and the samecurrent always flows through each.Within the chain, each internal nodeconnects to only two branches.R3 and R4 are not in series and do notnecessarily have the same current.Parallel: Components that are connected to the same pair of nodes aresaid to be in parallel.E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 5 / 13

Series and Parallel2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummarySeries: Components that are connected in a chain so that the same currentflows through each one are said to be in series.R1 , R2 , R3 are in series and the samecurrent always flows through each.Within the chain, each internal nodeconnects to only two branches.R3 and R4 are not in series and do notnecessarily have the same current.Parallel: Components that are connected to the same pair of nodes aresaid to be in parallel.R1 , R2 , R3 are in parallel and the samevoltage is across each resistor (eventhough R3 is not close to the others).E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 5 / 13

Series and Parallel2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummarySeries: Components that are connected in a chain so that the same currentflows through each one are said to be in series.R1 , R2 , R3 are in series and the samecurrent always flows through each.Within the chain, each internal nodeconnects to only two branches.R3 and R4 are not in series and do notnecessarily have the same current.Parallel: Components that are connected to the same pair of nodes aresaid to be in parallel.R1 , R2 , R3 are in parallel and the samevoltage is across each resistor (eventhough R3 is not close to the others).R4 and R5 are also in parallel.E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 5 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:VX V1 V2 V3Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 6 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:VX V1 V2 V3 IR1 IR2 IR3Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 6 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:VX V1 V2 V3 IR1 IR2 IR3 I(R1 R2 R3 )Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 6 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksVX V1 V2 V3 IR1 IR2 IR3 I(R1 R2 R3 )V1VX IR1I(R1 R2 R3 ) Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 6 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworksVX V1 V2 V3 IR1 IR2 IR3 I(R1 R2 R3 )V1VX IR1I(R1 R2 R3 ) R1R1 R2 R3 Non-ideal Voltage Source SummaryE1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 6 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryVX V1 V2 V3 IR1 IR2 IR3 I(R1 R2 R3 )V1VX IR1I(R1 R2 R3 ) R1R1 R2 R3 R1RTwhere RT R1 R2 R3 is thetotal resistance of the chain.E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 6 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance:Series Equivalent Resistance:Parallel Equivalent Resistance:Parallel Formulae Simplifying ResistorNetworks Non-ideal Voltage Source SummaryVX V1 V2 V3 IR1 IR2 IR3 I(R1 R2 R3 )V1VX IR1I(R1 R2 R3 ) R1R1 R2 R3 R1RTwhere RT R1 R2 R3 is thetotal resistance of the chain.VX is divided into V1 : V2 : V3 in the proportions R1 : R2 : R3 .E1.1 Analysis of Circuits (2017-10110)Resistor Circuits: 2 – 6 / 13

Series Resistors: Voltage Divider2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series a

E1.1 Analysis of Circuits (2017-10110) Resistor Circuits: 2 – 5 / 13 Series: Components that are connected in a chain so that the same current ﬂows through each one are said to be in series. Series and Parallel 2: Resistor Circuits Kirchoff’s Voltage Law Kirchoff’s Current Law KCL Example Series and Parallel Dividers Equivalent Resistance: Series Equivalent .

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