2: Resistor Circuits - Imperial College · PDF file · 2017-09-13E1.1 Analysis of...

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

• Summary

E1.1 Analysis of Circuits (2017-10110) Resistor Circuits: 2 – 1 / 13

Kirchoff’s Voltage Law

• KCL Example

• Summary

The 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.

• KCL Example

• Summary

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.

• KCL Example

• Summary

KVL: the sum of the voltage changes around any closed loop is zero.

• KCL Example

• Summary

Example: VDE + VBD + VAB + VEA = 0

• KCL Example

• Summary

Example: VDE + VBD + VAB + VEA = 0

Equivalent formulation:VXY = VXE − VY E = VX − VY for any nodes X and Y .

Kirchoff’s Current Law

• KCL Example

• Summary

Wherever charges are free to move around, they will move to ensurecharge neutrality everywhere at all times.

• KCL Example

• Summary

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.

• KCL Example

• Summary

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.KCL: The currents flowing out of any closed region of a circuit sum to zero.

• KCL Example

• Summary

Green: I1 = I7

• KCL Example

• Summary

Green: I1 = I7

Blue: −I1 + I2 + I5 = 0

• KCL Example

• Summary

Green: I1 = I7

Blue: −I1 + I2 + I5 = 0

Gray: −I2 + I4 − I6 + I7 = 0

KCL Example

• KCL Example

• Summary

The currents and voltages in any linear circuit can be determined by usingKCL, KVL and Ohm’s law.

KCL Example

• KCL Example

• Summary

Sometimes KCL allows you to determine currents very easily withouthaving to solve any simultaneous equations:

How do we calculate I ?

KCL Example

• KCL Example

• Summary

KCL: −1 + I + 3 = 0

KCL Example

• KCL Example

• Summary

KCL: −1 + I + 3 = 0=⇒ I = −2A

KCL Example

• KCL Example

• Summary

KCL: −1 + I + 3 = 0=⇒ I = −2A

Note 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.

Series and Parallel

• KCL Example

• Summary

Series: Components that are connected in a chain so that the same currentflows through each one are said to be in series.

Series and Parallel

• KCL Example

• Summary

R1, R2, R3 are in series and the samecurrent always flows through each.

Series and Parallel

• KCL Example

• Summary

Within the chain, each internal nodeconnects to only two branches.

Series and Parallel

• KCL Example

• Summary

R3 and R4 are not in series and do notnecessarily have the same current.

Series and Parallel

• KCL Example

• Summary

Parallel: Components that are connected to the same pair of nodes aresaid to be in parallel .

Series and Parallel

• KCL Example

• Summary

R1, R2, R3 are in parallel and the samevoltage is across each resistor (eventhough R3 is not close to the others).

Series and Parallel

• KCL Example

• Summary

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.

Series Resistors: Voltage Divider

• KCL Example

• Summary

VX = V1 + V2 + V3

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

= I(R1 +R2 + R3)

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

= I(R1 +R2 + R3)

VX= IR1

I(R1+R2+R3)

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

= I(R1 +R2 + R3)

VX= IR1

I(R1+R2+R3)

R1+R2+R3

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

= I(R1 +R2 + R3)

VX= IR1

I(R1+R2+R3)

R1+R2+R3

where RT = R1 +R2 +R3 is thetotal resistance of the chain.

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

= I(R1 +R2 + R3)

VX= IR1

I(R1+R2+R3)

R1+R2+R3

VX is divided into V1 : V2 : V3 in the proportions R1 : R2 : R3.

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

= I(R1 +R2 + R3)

VX= IR1

I(R1+R2+R3)

R1+R2+R3

Approximate Voltage Divider:

If IY = 0, then VY = RA

RA+RBVX .

• KCL Example

• Summary

VX = V1 + V2 + V3

= IR1 + IR2 + IR3

= I(R1 +R2 + R3)

VX= IR1

I(R1+R2+R3)

R1+R2+R3

Approximate Voltage Divider:

If IY = 0, then VY = RA

RA+RBVX .

If IY ≪ I , then VY ≈ RA

RA+RBVX .

Parallel Resistors: Current Divider

• KCL Example

• Summary

Parallel resistors all share the same V .

• KCL Example

• Summary

I1 = VR1

= V G1 where G1 = 1R1

is the conductance of R1.

• KCL Example

• Summary

I1 = VR1

IX = I1 + I2 + I3

= V G1 + V G2 + V G3

= V (G1 +G2 +G3)

• KCL Example

• Summary

I1 = VR1

IX = I1 + I2 + I3

= V G1 + V G2 + V G3

= V (G1 +G2 +G3)

= V G1

V (G1+G2+G3)= G1

G1+G2+G3

where GP = G1 +G2 +G3 is the total conductance of the resistors.

• KCL Example

• Summary

I1 = VR1

IX = I1 + I2 + I3

= V G1 + V G2 + V G3

= V (G1 +G2 +G3)

= V G1

V (G1+G2+G3)= G1

G1+G2+G3

IX is divided into I1 : I2 : I3 in the proportions G1 : G2 : G3.

• KCL Example

• Summary

I1 = VR1

IX = I1 + I2 + I3

= V G1 + V G2 + V G3

= V (G1 +G2 +G3)

= V G1

V (G1+G2+G3)= G1

G1+G2+G3

IX is divided into I1 : I2 : I3 in the proportions G1 : G2 : G3.

Special case for only two resistors:

I1 : I2 = G1 : G2 = R2 : R1 ⇒ I1 = R2

Equivalent Resistance: Series

• KCL Example

• Summary

We know that V = V1 + V2 + V3 = I(R1 +R2 +R3) = IRT

So we can replace the three resistorsby a single equivalent resistor ofvalue RT without affecting therelationship between V and I .

• KCL Example

• Summary

• KCL Example

• Summary

Replacing series resistors by theirequivalent resistor will not affect anyof the voltages or currents in the restof the circuit.

• KCL Example

• Summary

Replacing series resistors by theirequivalent resistor will not affect anyof the voltages or currents in the restof the circuit.

However the individual voltages V1,V2 and V3 are no longer accessible.

Equivalent Resistance: Parallel

• KCL Example

• Summary

Similarly we known that I = I1 + I2 + I3 = V (G1 +G2 +G3) = V GP .

• KCL Example

• Summary

So V = IRP where RP = 1GP

= 1G1+G2+G3

= 11/R1+1/R2+1/R3

• KCL Example

• Summary

= 1G1+G2+G3

= 11/R1+1/R2+1/R3

We can use a singleequivalent resistor ofresistance RP withoutaffecting therelationship betweenV and I .

• KCL Example

• Summary

= 1G1+G2+G3

= 11/R1+1/R2+1/R3

Replacing parallel resistors bytheir equivalent resistor will notaffect any of the voltages orcurrents in the rest of the circuit.

• KCL Example

• Summary

= 1G1+G2+G3

= 11/R1+1/R2+1/R3

• KCL Example

• Summary

= 1G1+G2+G3

= 11/R1+1/R2+1/R3

R4and R5 are also in parallel.

• KCL Example

• Summary

= 1G1+G2+G3

= 11/R1+1/R2+1/R3

R4and R5 are also in parallel.

Much simpler - although none of the original currents I1, · · · , I5 are nowaccessible. Current IS and the three node voltages are identical.

Equivalent Resistance: Parallel Formulae

• KCL Example

• Summary

For parallel resistors GP = G1 +G2 +G3

or equivalently RP = R1||R2||R3 = 11/R1+1/R2+1/R3

These formulae work for any number of resistors.

• KCL Example

• Summary

• For the special case of two parallel resistors

RP = 11/R1+1/R2

= R1R2

(“product over sum”)

• KCL Example

• Summary

RP = 11/R1+1/R2

= R1R2

• If one resistor is a multiple of the otherSuppose R2 = kR1, then

RP = R1R2

(k+1)R1

= kk+1R1 = (1− 1

k+1 )R1

• KCL Example

• Summary

RP = 11/R1+1/R2

= R1R2

RP = R1R2

(k+1)R1

= kk+1R1 = (1− 1

k+1 )R1

Example: 1 kΩ || 99 kΩ = 99100 kΩ =

1− 1100

• KCL Example

• Summary

RP = 11/R1+1/R2

= R1R2

RP = R1R2

(k+1)R1

= kk+1R1 = (1− 1

k+1 )R1

Example: 1 kΩ || 99 kΩ = 99100 kΩ =

1− 1100

Important: The equivalent resistance of parallel resistors is always lessthan any of them.

Simplifying Resistor Networks

• KCL Example

• Summary

Many resistor circuits can besimplified by alternately combiningseries and parallel resistors.

• KCL Example

• Summary

Series: 2 k + 1 k = 3 k

• KCL Example

• Summary

Series: 2 k + 1 k = 3 k

Parallel: 3 k || 7 k = 2.1 kParallel: 2 k || 3 k = 1.2 k

• KCL Example

• Summary

Series: 2 k + 1 k = 3 k

Series: 2.1 k + 1.2 k = 3.3 k

• KCL Example

• Summary

Series: 2 k + 1 k = 3 k

Series: 2.1 k + 1.2 k = 3.3 k

Sadly this method does not alwayswork: there are no series or parallelresistors here.

Non-ideal Voltage Source

• KCL Example

• Summary

An ideal battery has a characteristic that isvertical: battery voltage does not vary withcurrent.Normally a battery is supplying energy so V

and I have opposite signs, so I ≤ 0.

• KCL Example

• Summary

An real battery has a characteristic that hasa slight positive slope: battery voltagedecreases as the (negative) currentincreases.

Model this by including a small resistor inseries. V = VB + IRB .

• KCL Example

• Summary

An real battery has a characteristic that hasa slight positive slope: battery voltagedecreases as the (negative) currentincreases.

Model this by including a small resistor inseries. V = VB + IRB .

The equivalent resistance for a batteryincreases at low temperatures.

Summary

• KCL Example

• Summary

• Kichoff’s Voltage and Current Laws• Series and Parallel components• Voltage and Current Dividers• Simplifying Resistor Networks• Battery Internal Resistance

For further details see Hayt Ch 3 or Irwin Ch 2.

2: Resistor Circuits - Imperial College · PDF file · 2017-09-13E1.1 Analysis of...

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