Objective of Lecture Provide step-by-step instructions for loop analysis,

which is a method to calculate voltage drops and mesh currents that flow around loops in a circuit.

Chapter 3.2

Basic Engineering Circuit Analysis by

J.D. Irwin and R.M. Nelms

Loop Analysis Technique to find voltage drops around a loop using

the currents that flow within the loop, Kirchhoff’s Voltage Law, and Ohm’s Law

First result is the calculation of the loop currents

Which can be used to calculate the current flowing through each component

Second result is a calculation of the voltages across the components

Which can be used to calculate the voltage at the nodes.

Definition of a Loop Mesh – the smallest grouping around a subset of

components in a circuit that are all connected to one another

Multiple loops are defined so that every component in the circuit belongs to one or more loops.

Steps in Loop Analysis

Vin

Step 1 Identify all of the loops in the circuit

Vin

Step 2 Label the currents flowing in each loop

i1

i2

Vin

Step 3 Label the voltage across each component in the circuit

i1

i2

+ V1

_

Vin

+ V3

_

+ V5

_

+ V6

_

+ V2 - + V4 -

Step 4 Use Kirchhoff’s Voltage Law

i1

i2

+ V1

_

Vin

+ V3

_

+ V5

_

+ V6

_

+ V2 - + V4 -

0

0

543

6321

VVV

VVVVVin

Step 5 Use Ohm’s Law to relate the voltage drops across each

component to the sum of the currents flowing through them.

Follow the sign convention on the resistor’s voltage.

RIIV baR

Step 5

i1

i2

+ V1

_

Vin

+ V3

_

+ V5

_

+ V6

_

+ V2 - + V4 -

616

525

424

3213

212

111

RiV

RiV

RiV

RiiV

RiV

RiV

Step 6 Solve for the loop currents, i1 and i2

These currents are related to the currents found during the nodal analysis.

213

542

62171

iiI

IIi

IIIIi

Step 7 Once the lopp currents are known, calculate the

voltage across all of the components.

12V

From Previous Slides

616

525

424

3213

212

111

RiV

RiV

RiV

RiiV

RiV

RiV

0

0

543

6321

VVV

VVVVVin

Substituting in Numbers

kiV

kiV

kiV

kiiV

kiV

kiV

1

3

6

5

8

4

16

25

24

213

12

11

0

012

543

6321

VVV

VVVVV

Substituting the results from Ohm’s Law into the KVL equations

0365

0158412

2221

12111

kikikii

kikiikikiV

Chugging through the Math

One or more of the loop currents may have a negative sign.

Mesh Currents (mA)

i1 740

i2 264

Chugging through the Math Voltage across

resistors (V)

VR1 = -i1R2 -2.96

VR2 = i2 R2 5.92

VR3 =(i1 – i2) R3 2.39

VR4 = i2 R4 1.59

VR5 = (V4 – V5) 0.804

VR6 = (V5 – 0V) 0.740

The magnitude of any voltage across a resistor must be less than the sum of all of the voltage sources in the circuit

In this case, no voltage across a resistor can be greater than 12V.

Chugging through More Math

Currents (mA)

IR1 = i1 740

IR2 = i1 740

IR3 = i1- i2 476

IR4 = i2 264

IR5 = i2 264

IR6 = i1 740

I Vin = i1 740

The currents through each component in the circuit.

Check None of the loop currents should be larger than the

current that flows through the equivalent resistor in series with the 12V supply.

ARVI

kR

kkkkkkR

eqeq

eq

eq

m74012

2.16

136584

Summary Steps in Loop Analysis

1. Identify all of the loops in the circuit 2. Label the currents flowing in each loop 3. Label the voltage across each component in the circuit 4. Write the voltage loop equations using Kirchhoff ’s

Voltage Law. 5. Use Ohm’s Law to relate the voltage drops across each

component to the sum of the currents flowing through them.

6. Solve for the loop currents 7. Once the loop currents are known, calculate the voltage

across all of the components.