ECE 221Electric Circuit Analysis I

Chapter 4Circuit Elements

Herbert G. Mayer, PSU & CCUTStatus 1/1/2015

Taken with permission from PSU Prof. Phillip Wong

Syllabus Resistor Inductor Capacitor Voltage and Current Sources Voltage and Current Dividers

3

Resistor

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

SI Unit: ohm () Symbol: Impedance:

I-V relationship:

R

i(t)v(t)

)()( tiRtv

)(1

)( tvR

ti

RZ

4

5

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

iiRR

1eq

R1 R2 Req = R1 + R2

Req = R1 + R2 + R3R1 R2 R3

6

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

7

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 “–”

+ –

V1

I → → I+ –

V2

+ –

V3

R1 R2 R3

8

Example: 2 6 3

A B I

VAB = 3 V

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

b) Find the voltage drop across the 6 resistor.

Assume 3 significant figures.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

9

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

21eq

11

RR

RR

RRR

R1 R2

1

321eq

111

RRRRR3R1 R2

1

1eq

1

N

i iRR

10

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

For parallel resistors, the voltage drop across each resistor is the same

Iin = Iout and V1 = V2 = V3

Iin → → Iout

+ V1 –

R3

R1

R2

+ V2 –

+ V3 –

11

The amount of current through each resistor in a parallel circuit is given by Ohm’s Law.

The sum of the currents through each resistor is equal to the original amount of current entering or leaving the parallel circuit

I1 + I2 + I3 = I

I → → I

+ V –

R3

R1

R2

+ V –

+ V –

I1

I2

I3

I1 = V / R1

I2 = V / R2

I3 = V / R3

12

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: A00.22x

ABVI

Ix

13

Inductor & Capacitor

An inductor is a passive component that stores energy within a magnetic field. It has inductance L

SI unit: henry (H) Impedance:

Symbol:

A capacitor is a passive component that stores energy within an electric field. It has capacitance C

SI unit: farad (F) Impedance:

Symbol: non-polarized polarized+

LjZ

1 CjZ

14

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

15

Voltage & Current Dividers

021

22

021

11

VRR

RV

VRR

RV

Voltage Divider

R2

I

R1

+ –

+

–

+

–

V0

V1

V2

IRR

RII

GG

GI

IRR

RII

GG

GI

21

12

21

22

21

21

21

11

Current Divider

I

+

–

VS G2G1

I1 I2

16

Example:

What is the voltage drop across Ra?

What is the voltage drop across Rb?

Sba

bb V

RR

RV

Vb

Ra = 1 Ω Va

+–Vs = 8 V

Sba

aa V

RR

RV

Rb = 3 Ω

V2V831

1

aV

V6V831

3

aV

17

Example:The voltage Vg from a signal generator must be reduced by a factor of three before entering an audio amplifier. Determine the resistor values of a voltage divider that performs this task

3g

out

VV 21 2RR 2123 RRR gV

RR

R

21

2

R2

R1

Vg VoutSignal

GeneratorAudio

Amplifier

Peak output = 3 V Max input = 1 V