ECE 101 Exploring Electrical Engineering

Circuits 2

• Resistor

• Series and parallel connection of resistors

• Voltage Dividers

• Voltage & Current Sources

Many of the slides are modified from course notes by P.K. Wong and M. Holtzman

1

Resistor

A resistor is a passive electronic component that obeys Ohm’s Law. It has resistance R.

SI Unit: ohm ()

Symbol:

I-V relationship:

R

i(t)v(t)

)()( tiRtv

)(1

)( tvR

ti

2

3

Resistors Connected in Series (end-to-end)

If N resistors are connected in series, with thei-th resistor having a resistance Ri , then the equivalent resistance Req is:

N

i

iRR1

eq

R1 R2 Req = R1 + R2

Req = R1 + R2 + R3R1 R2 R3

4

Properties of Series Resistances (DC):

The amount of current Iin entering one end of a series circuit is equal to the amount of current Iout

leaving the other end.

The current is the same through each resistor in the series and is equal to Iin.

Iin = Iout = I1 = I2 = I3

→I1

Iin →→I2

→I3

→ Iout

R1 R2 R3

5

The amount of voltage drop across each resistor in a series circuit is given by Ohm’s Law.

V1 = IR1 , V2 = IR2 , V3= IR3

By convention, the resistor terminal that the current enters is labeled “+”, and the terminal the current exits is labeled “–”.

KVL: total voltage drop = sum of individual drops (watch out for sign!)

+ –

V1

I → → I+ –

V2

+ –

V3

R1 R2 R3

6

Example:2 6 3

A BI

VAB = 3 V

a) Calculate the current I that flows through the resistors.

b) Find the voltage drop across the 6 resistor.

Solution:

a) Approach – Use Ohm’s law:

11362eqRCalculate the equivalent resistance:

eq

AB

R

VI

Calculate the current: A273.011

V3

I

b) Use Ohm’s law again: V64.16A273.066 IRV

7

Example:

100 V Source, 5 and 20 ohm resistors in series; find Rs, I, P, V1, V2

100 V Source, 5 and 20 ohm resistors in parallel; find Rp, I, P, I1, I2

8

Resistors Connected in Parallel (side-by-side)

If N resistors are in parallel, with the i-th resistor having a resistance Ri , then the equivalent resistance is:

21

21

1

21

eq

11

RR

RR

RRR

R1 R2

1

321

eq

111

RRRRR3R1 R2

1

1

eq

1

N

i iRR

9

Properties of Parallel Resistances (DC):

The amount of current Iin entering one end of a parallel circuit is equal to the amount of current Iout

leaving the other end: Iin = Iout

For parallel resistors, the voltage drop across each resistor is the same: V1 = V2 = V3

KCL: Σ (sum) of currents entering a node = Σ (sum) of currents leaving a node

Iin → → Iout

+ V1 –

R3

R1

R2

+ V2 –

+ V3 –

10

Example:2

6

3

A B

I = 4 A

VAB

a) Find the current Ix through the 2 resistor.

b) What is the power dissipated by the 3 resistor?

Assume 3 significant figures.

Solution:

a) Approach – Use Ohm’s law:

13

1

6

1

2

11

eqRCalculate the equivalent resistance:

eqAB IRV

Calculate the voltage drop: V41A4 ABV

b) Use power equation: W33.53

2

ABVP

Find the current: A00.22

x

ABVI

Ix

11

Voltage Divider

0

21

22

0

21

11

VRR

RV

VRR

RV

Voltage Divider

R2

I

R1

+ –

+

–

+

–

V0

V1

V2

12

Example:

What is the voltage drop across Ra?

What is the voltage drop across Rb?

S

ba

bb V

RR

RV

Vb

Ra = 1 Ω Va

+–Vs = 8 V

S

ba

aa V

RR

RV

Rb = 3 Ω

V2V831

1

aV

V6V831

3

aV

13

DC Voltage & Current Sources

+–Vs

I

+

–Vs

I

An ideal DC current source

outputs a constant current

regardless of the amount

of voltage across it.

An ideal DC voltage source

outputs a constant voltage

regardless of the amount of

current through it.

Is V

Prefixes – common engineering style

10-9 nano n

10-6 micro m

10-3 milli m

103 kilo k

106 mega M

109 giga G

14

More circuit examples

Find currents i1 and i2

Find voltage v

Further questions:

1. Find the power in each resistor in the series and parallel examples.

2. Are household appliances connected in series or parallel? Why?