introduction to ac circuits (capacitors and inductors) 2 days ago · ac circuits dr. mohamed refky...

Introduction to AC Circuits

(Capacitors and Inductors)

Dr. Mohamed Refky Amin

Electronics and Electrical Communications Engineering Department (EECE)

Cairo University

[email protected]

http://scholar.cu.edu.eg/refky/

OUTLINE

• Previously on ELCN102

• AC Circuits

• Capacitors

• Inductors

• Transient Analysis

RC Circuits

RL Circuits

Dr. Mohamed Refky 2

Previously on ELCN102

Dr. Mohamed Refky

DefinitionElectric circuit theorems are always beneficial to help find

voltage and currents in multi loop circuits.

The network theorems include:

• Superposition Theorem

• Thevenin’s Theorem

• Norton’s Theorem

• Maximum Power Transfer Theorem3


Dr. Mohamed Refky

Norton’s TheoremA linear two-terminal circuit can be replaced by equivalent circuit

consisting of a current source 𝐼𝑁 in parallel with a resistor 𝑅𝑁

4


Dr. Mohamed Refky

Steps of Norton’s Theorem1) Identify the load resistance and introduce two nodes 𝑎 and 𝑏

2) Remove the load resistance between node 𝑎 and 𝑏 and set allthe independent sources to zero (voltage sources are SC and

current sources are OC) and calculate the resistance seen

between nodes 𝑎 and 𝑏. This resistance is 𝑅𝑁 of the Nortonequivalent circuit.

3) Replace the load resistance with a short circuit and calculate

the short circuit current between nodes 𝑎 and 𝑏. This currentis 𝐼𝑁 of the Norton equivalent circuit.

5


Dr. Mohamed Refky

Thevenin and Norton equivalent circuits

Thevenin equivalent circuit must be equivalent to Norton

equivalent circuit

𝑅𝑁 = 𝑅𝑡ℎ, 𝑉𝑡ℎ = 𝐼𝑁𝑅𝑁, 𝐼𝑁 =𝑉𝑡ℎ𝑅𝑡ℎ

→ 𝑅𝑡ℎ =𝑉𝑡ℎ𝐼𝑁

6


Dr. Mohamed Refky

Maximum Power Transfer TheoremThe maximum amount of power will be dissipated by a load

resistance (𝑅𝐿 ) when that load resistance is equal to theThevenin/Norton resistance of the network supplying the power.

→ 𝑅𝐿 = 𝑅𝑡ℎ = 𝑅𝑁For maximum power 𝑃𝑅𝐿

7

AC Circuits

Dr. Mohamed Refky

DefinitionAn AC circuit is a combination of active elements (Voltage and

current sources) and passive elements (resistors, capacitors and

coils).

Unlike resistance, capacitors and coils can store energy and do

not dissipate it. Thus, capacitors and coils are called storage

elements.8

AC Circuits

Dr. Mohamed Refky

DefinitionAn AC circuit is a combination of active elements (Voltage and

current sources) and passive elements (resistors, capacitors and

coils).

The sources are usually sinusoidal voltage or current sources

9

Capacitors

Dr. Mohamed Refky

Definition and StructureA capacitor is a passive element designed to store energy in its

electric field.

Capacitors are the most common component beside resistors. It

used in electronics, communication, and computer systems.

A capacitor is an electrical device

constructed of two parallel plates

separated by an insulating material

called the dielectric.

10

Capacitors

Dr. Mohamed Refky

Definition and StructureWhen a voltage source is connected to a capacitor, an electric

field is generated in the dielectric and charges are accumulated on

the plates.

𝑄 = 𝐶 × 𝑉

𝐶 =𝑄

𝑉

The amount of charge (𝑄) that a capacitor can store per voltacross the plates, is its capacitance (𝐶).

Coulomb Farad

Volt

11

Capacitors

Dr. Mohamed Refky

Definition and Structure

Most capacitors in electronics have capacitance values of micro-

Farad (𝜇𝐹 = 10−6 𝐹) to pico-Farad (𝑝𝐹 = 10−12 𝐹) .

12

Capacitors

Dr. Mohamed Refky

Instantaneous Current

𝑄 = 𝐶 × 𝑉

𝑖𝑐 𝑡 =𝑑𝑞 𝑡

𝑑𝑡=𝑑 𝑐 𝑡 × 𝑣𝑐 𝑡

𝑑𝑡

For constant capacitance (𝑐 𝑡 = 𝐶)

𝑞 𝑡 = 𝐶 × 𝑣𝑐 𝑡

𝑖𝑐(𝑡) = 𝐶𝑑𝑣𝑐(𝑡)

𝑑𝑡

13

Capacitors

Dr. Mohamed Refky

Instantaneous Power and Energy storedInstantaneous power is given by

𝑞(𝑡) = 𝑣 𝑡 × 𝑖 𝑡

= 𝑣 𝑡 × 𝐶𝑑𝑣(𝑡)

𝑑𝑡

Energy stored in the capacitor is given by

𝑤 = න𝑞 𝑡 𝑑𝑡 = න 𝑣 𝑡 × 𝐶𝑑𝑣(𝑡)

𝑑𝑡𝑑𝑡

= 𝐶න𝑣 𝑡 𝑑𝑣(𝑡) =1

2𝐶𝑣2 𝑡

14

Capacitors

Dr. Mohamed Refky

Series and Parallel Combinations

𝑣𝑒𝑞𝑢 = 𝑣1 + 𝑣2 +⋯+ 𝑣𝑁

Series Capacitors

𝑄

𝐶𝑒𝑞=𝑄

𝐶1+𝑄

𝐶2+⋯+

𝑄

𝐶𝑁

𝑄 = 𝐶 × 𝑉

𝑄 −𝑄 𝑄 −𝑄 𝑄 −𝑄

𝑄 −𝑄

15

Capacitors

Dr. Mohamed Refky


Series Capacitors

1

𝐶𝑒𝑞=

1

𝐶1+

1

𝐶2+⋯+

1

𝐶𝑁

𝑄 = 𝐶 × 𝑉

16

Capacitors

Dr. Mohamed Refky


Parallel Capacitors 𝑄 = 𝐶 × 𝑉

𝑄𝑒𝑞 = 𝑄1 + 𝑄2 +⋯+ 𝑄𝑁

𝐶𝑒𝑞𝑣 = 𝐶1𝑣 + 𝐶2𝑣 + ⋯+ 𝐶𝑁𝑣

17

Capacitors

Dr. Mohamed Refky


Parallel Capacitors 𝑄 = 𝐶 × 𝑉

𝐶𝑒𝑞 = 𝐶1 + 𝐶2 +⋯+ 𝐶𝑁

18

Capacitors

Dr. Mohamed Refky

Example (1)Find the equivalent capacitance seen between terminals 𝑎 and 𝑏of the circuit shown.

19

Inductors

Dr. Mohamed Refky

Definition and StructureAn inductor is a passive element designed to store energy in its

magnetic field.

Inductors are used in power supplies, transformers, radios, TVs,

radars, and electric motors.

An inductor is constructed of a wire

wounded into a coil.

20

Inductors

Dr. Mohamed Refky

Definition and StructureWhen the current flowing through an

inductor changes, the magnetic field induces

a voltage in the conductor, according to

Faraday’s law of electromagnetic induction,

to resist this change in the current.

𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡

𝑑𝑡

𝐿 is the inductance in Henri (𝐻)

21

Inductors

Dr. Mohamed Refky

Definition and Structure

Most inductors in electronics have inductance values of mille

Henri (𝑚𝐻 = 10−3 𝐻) to micro- Henri (𝜇𝐻 = 10−6 𝐻) .

22

Inductors

Dr. Mohamed Refky


𝑣𝑒𝑞𝑢 = 𝑣1 + 𝑣2 +⋯+ 𝑣𝑁

Series Inductors

𝐿𝑒𝑞𝑑𝑖

𝑑𝑡= 𝐿1

𝑑𝑖

𝑑𝑡+ 𝐿2

𝑑𝑖

𝑑𝑡+ ⋯+ 𝐿𝑁

𝑑𝑖

𝑑𝑡 23

Inductors

Dr. Mohamed Refky


Series Inductors

𝐿𝑒𝑞 = 𝐿1 + 𝐿2 +⋯+ 𝐿𝑁

24

Inductors

Dr. Mohamed Refky


Parallel Inductors

𝑖 = 𝑖1 + 𝑖2 +⋯+ 𝑖𝑁

1

𝐿𝑒𝑞න𝑣 𝑑𝑡 =

1

𝐿1න𝑣 𝑑𝑡 +

1

𝐿2න𝑣 𝑑𝑡 + ⋯+

1

𝐿𝑁න𝑣 𝑑𝑡

𝑣 = 𝐿𝑑𝑖

𝑑𝑡

25

Inductors

Dr. Mohamed Refky


Parallel Inductors

1

𝐿𝑒𝑞=

1

𝐿1+

1

𝐿2+⋯+

1

𝐿𝑁

26

Inductors

Dr. Mohamed Refky

Example (2)Find the equivalent inductance seen between terminals 𝑎 and 𝑏 ofthe circuit shown.

27

Transient Analysis

Dr. Mohamed Refky

DefinitionThe transient response of the circuit is the response when the input

is changed suddenly or a switches status is changed.

𝑣 𝑡 =

𝑣1 𝑡 , 𝑡0 < 𝑡 < 𝑡1𝑣2 𝑡 , 𝑡1 < 𝑡 < 𝑡2

⋮𝑣𝑛 𝑡 , 𝑡𝑛−1 < 𝑡 < 𝑡𝑛

𝑣 𝑡 the same

28

Transient Analysis

Dr. Mohamed Refky

DefinitionThe transient response of the circuit is the response when the input

is changed suddenly or a switches status is changed.

𝑣 𝑡 the same

29

Transient Analysis

Dr. Mohamed Refky

Time Domain Analysis 1st Order Systems

At 𝑡 = 0−, capacitor is initiallycharged to 𝑉0.

𝑣𝑐 0 = 𝑉0

At 𝑡 = 0, the switch is closed.

Source Free RC Circuits

From the resistor 𝑖 𝑡 =𝑣𝑐 𝑡

𝑅

From the capacitor 𝑖 𝑡 = −𝐶𝑑𝑣𝑐 𝑡

𝑑𝑡

30

Transient Analysis

Dr. Mohamed Refky


−𝐶𝑑𝑣𝑐 𝑡

𝑑𝑡=𝑣𝑐 𝑡

𝑅

𝑑𝑣𝑐 𝑡

𝑑𝑡= −

𝑣𝑐 𝑡

𝐶𝑅


𝑣𝑐 0 = 𝑉0

This is a first order differential

equation with an initial condition

𝑑

𝑑𝑡𝑎𝑒−𝑏𝑡 = −𝑎𝑏 𝑒−𝑏𝑡

Hint:

31

Transient Analysis

Dr. Mohamed Refky


𝑑𝑣𝑐 𝑡

𝑑𝑡= −

𝑣𝑐 𝑡

𝐶𝑅

𝑣𝑐 𝑡 = 𝛼𝑒−𝑡𝑅𝐶


𝑣𝑐 0 = 𝛼𝑒0𝑅𝐶 = 𝑉0

Using the initial condition

𝛼 = 𝑉032

Transient Analysis

Dr. Mohamed Refky


𝑣𝑐 𝑡 = 𝑉0𝑒−𝑡𝑅𝐶

𝑅𝐶 is called the time constantof the circuit (𝜏)


The time constant (𝜏) of a circuit isthe time required for the response

to decay to 1/𝑒 or 36.8% of itsinitial value (a change of 63%).

33

Transient Analysis

Dr. Mohamed Refky


𝑣𝑐 𝑡 = 𝑉0𝑒−𝑡𝑅𝐶


It takes about 5𝜏 to changethe voltage by 99%. This

period of 5𝜏 is calledtransient time.

𝑣𝑐 𝑡 Change

𝜏 0.3678𝑉0 63%

2𝜏 0.1353𝑉0 86%

3𝜏 0.0497𝑉0 95%

4𝜏 0.0183𝑉0 98%

5𝜏 0.0067𝑉0 99%

34

Transient Analysis

Dr. Mohamed Refky


𝑣𝑅 𝑡 = 𝑣𝑐 𝑡 = 𝑉0𝑒−𝑡𝑅𝐶

𝑖 𝑡 =𝑉0𝑅𝑒−𝑡𝑅𝐶

Power dissipated in the resistor


𝑝 𝑡 = 𝑣𝑅 𝑡 × 𝑖 𝑡 =𝑉02

𝑅𝑒−2𝑡𝑅𝐶

35

Transient Analysis

Dr. Mohamed Refky


Energy dissipated by the resistor


𝑝 𝑡 = 𝑣𝑅 𝑡 × 𝑖 𝑡 =𝑉02

𝑅𝑒−2𝑡𝑅𝐶

𝑤 𝑡 = න0

𝑡

𝑝 𝑡 𝑑𝑡

=𝑉02

𝑅න0

𝑡

𝑒−2𝑡𝑅𝐶 𝑑𝑡 =

𝑉02

𝑅

−𝑅𝐶

2𝑒−2𝑡𝑅𝐶

0

𝑡

=𝑉02𝐶

21 − 𝑒

−2𝑡𝑅𝐶

36

Transient Analysis

Dr. Mohamed Refky

Example (3)For the circuit shown, 𝑣𝐶(0) = 15 𝑉. 𝑅1 = 5𝑘Ω, 𝑅2 = 8𝑘Ω,𝑅3 = 12𝑘Ω, 𝐶 = 100𝜇𝐹. Find 𝑣𝐶 𝑡 , 𝑣𝑥 𝑡 , and 𝑖𝑥 𝑡 for 𝑡> 0.

37

Transient Analysis

Dr. Mohamed Refky


The step response is the

response of the circuit due to

a sudden change of voltage

or current source.

𝑖𝑅 𝑡 =𝐸 − 𝑣𝑐 𝑡

𝑅

𝑖𝐶 𝑡 = 𝐶𝑑𝑣𝑐 𝑡

𝑑𝑡

Step response of an RC circuits

38

Transient Analysis

Dr. Mohamed Refky


𝑖𝐶 𝑡 = 𝑖𝑅 𝑡

𝐶𝑑𝑣𝑐 𝑡

𝑑𝑡=𝐸 − 𝑣𝑐 𝑡

𝑅

−𝐶𝑑 𝐸 − 𝑣𝑐 𝑡


𝑅

𝑑 𝐸 − 𝑣𝑐 𝑡

𝑑𝑡= −

𝐸 − 𝑣𝑐 𝑡

𝑅𝐶


39

Transient Analysis

Dr. Mohamed Refky


𝑑 𝐸 − 𝑣𝑐 𝑡

𝑑𝑡= −

𝐸 − 𝑣𝑐 𝑡

𝑅𝐶

𝑣𝑐 0 = 𝑉0

𝐸 − 𝑣𝑐 𝑡 = 𝛼𝑒−𝑡𝑅𝐶

𝐸 − 𝑉0 = 𝛼𝑒−0𝑅𝐶


→ 𝛼 = 𝐸 − 𝑉0

40

Transient Analysis

Dr. Mohamed Refky


𝐸 − 𝑣𝑐 𝑡 = 𝐸 − 𝑉0 𝑒−𝑡𝑅𝐶

𝑣𝑐 𝑡 = 𝐸 − 𝐸 − 𝑉0 𝑒−𝑡𝑅𝐶

If 𝑉0 = 0

𝑣𝑐 𝑡 = 𝐸 − 𝐸𝑒−𝑡𝑅𝐶

= 𝐸 1 − 𝐸 𝑒−𝑡𝑅𝐶


41

Transient Analysis

Dr. Mohamed Refky


𝐸 − 𝑣𝑐 𝑡 = 𝐸 − 𝑉0 𝑒−𝑡𝑅𝐶


If 𝑉0 = 0

𝑣𝑐 𝑡 = 𝐸 − 𝐸𝑒−𝑡𝑅𝐶

= 𝐸 1 − 𝐸 𝑒−𝑡𝑅𝐶


42

Transient Analysis

Dr. Mohamed Refky



𝑖 𝑡 = 𝐶𝑑𝑣𝑐 𝑡


𝑅

=𝐸 − 𝑉0𝑅

𝑒−𝑡𝑅𝐶


43

Transient Analysis

Dr. Mohamed Refky


Steady state response of a circuit is the behavior of the circuit a

long time after an external excitation is applied.

Theoretically, the steady state period starts at 𝑡 = ∞. However,practically, it starts at 𝑡 = 5𝜏.

In steady state, we deal with DC analysis.

Steady State Response

In DC, the capacitor is consider

open circuit (𝑖𝑐 = 0) and the coil isconsider short circuit (𝑣𝐿 = 0). 𝑣𝐿 𝑡 = 𝐿

𝑑𝑖𝐿 𝑡

𝑑𝑡

𝑖𝑐(𝑡) = 𝐶𝑑𝑣𝑐(𝑡)

𝑑𝑡

44

Transient Analysis

Dr. Mohamed Refky


At steady state,

𝑣𝑐 ∞ = 𝐸

𝑖 ∞ = 0



𝑡=∞𝑣𝑐 ∞ = 𝐸

𝑖 𝑡 =𝐸 − 𝑉0𝑅

𝑒−𝑡𝑅𝐶

𝑡=∞𝑖 ∞ = 0

45

Transient Analysis

Dr. Mohamed Refky


Transient response of a circuit is the circuit’s temporary response

that will die out with time. The transient period starts at 𝑡 = 0and ends at 𝑡 = 5𝜏

Generally, For a first order RC circuit

𝑉𝑖 and 𝑉𝑓 are the initial and final capacitor voltages, respectively.

𝜏 = 𝑅𝑒𝑞𝐶, 𝑅𝑒𝑞 is the resistance seen between the capacitor nodes

while all sources are switched off.

Transient response

𝑣𝑐 𝑡 = 𝑉𝑓 − 𝑉𝑓 − 𝑉𝑖 𝑒−𝑡𝜏

46

Transient Analysis

Dr. Mohamed Refky



𝑉𝑖 = 𝑣𝑐 0 = 𝑉0

𝑉𝑓 = 𝑣𝑐 ∞ = 0

𝑣𝑐 𝑡 = 0 − 0 − 𝑉0 𝑒−𝑡𝑅𝐶

= 𝑉0𝑒−𝑡𝑅𝐶


𝜏 = 𝑅𝐶

47

Transient Analysis

Dr. Mohamed Refky




𝑉𝑖 = 𝑣𝑐 0 = 𝑉0

𝑉𝑓 = 𝑣𝑐 ∞ = 𝐸


𝜏 = 𝑅𝐶

48

Transient Analysis

Dr. Mohamed Refky

Example (4)For the shown circuit, switch has been in position 𝑎 for a longtime. At 𝑡 = 0, the switch moves to 𝑏. Determine 𝑣𝑐(𝑡) for 𝑡 > 0and calculate its value at 𝑡 = 1, 4, and 20𝑚𝑠.

49

Transient Analysis

Dr. Mohamed Refky

Example (5)For the shown circuit, switch has been open for a long time and is

closed at 𝑡 = 0. Calculate 𝑣𝑐(𝑡) at 𝑡 = 0.5 and 4𝑚𝑠.

50

Transient Analysis

Dr. Mohamed Refky



𝑑𝑡

𝑣𝑅 𝑡 = 𝑅𝑖𝐿 𝑡

𝑣𝐿 𝑡 = −𝑣𝑅 𝑡

𝐿𝑑𝑖𝐿 𝑡

𝑑𝑡= −𝑅𝑖𝐿 𝑡

𝑑𝑖𝐿 𝑡

𝑑𝑡= −

𝑅

𝐿𝑖𝐿 𝑡

Source Free RL Circuits

51

Transient Analysis

Dr. Mohamed Refky


𝑑𝑖𝐿 𝑡

𝑑𝑡= −

𝑅

𝐿𝑖𝐿 𝑡

𝑖𝐿 0 = 𝐼0

𝑖𝐿 𝑡 = 𝐼0𝑒−𝑡𝜏, 𝜏 =

𝐿

𝑅

Generally,


𝑖𝐿 𝑡 = 𝐼𝑓 − 𝐼𝑓 − 𝐼𝑖 𝑒−𝑡𝜏, 𝜏 =

𝐿

𝑅52

Transient Analysis

Dr. Mohamed Refky


𝑖𝐿 𝑡 = 𝐼𝑓 − 𝐼𝑓 − 𝐼𝑖 𝑒−𝑡𝜏, 𝜏 =

𝐿

𝑅

𝐼𝑖 = 𝑖𝐿 0 = 𝐼0

𝐼𝑓 = 0

Then

𝑖𝐿 𝑡 = 0 − 0 − 𝐼0 𝑒−𝑡𝜏

= 𝐼0𝑒−𝑡𝜏


𝜏 =𝐿

𝑅

53

Transient Analysis

Dr. Mohamed Refky


𝑖𝐿 𝑡 = 𝐼0𝑒−𝑡𝜏

𝑣𝑅 𝑡 = 𝑖𝐿 𝑡 × 𝑅

= 𝐼0𝑅𝑒−𝑡𝜏

Power dissipated in the resistor


𝑝 𝑡 = 𝑣𝑅 𝑡 × 𝑖𝐿 𝑡 = 𝐼02𝑅𝑒

−2𝑡𝜏

54

Transient Analysis

Dr. Mohamed Refky



Energy dissipated by the resistor

𝑝 𝑡 = 𝑣𝑅 𝑡 × 𝑖𝐿 𝑡 = 𝐼02𝑅𝑒

−2𝑡𝜏

𝑤 𝑡 = න0

𝑡

𝑝 𝑡 𝑑𝑡

= 𝐼02𝑅න

0

𝑡

𝑒−2𝑡𝜏 𝑑𝑡 = 𝐼0

2𝑅−𝐿

2𝑅𝑒−2𝑡𝜏

0

𝑡

=𝐼02𝐿

21 − 𝑒

−2𝑡𝜏

55

Transient Analysis

Dr. Mohamed Refky



𝑖𝐿 𝑡 = 𝐼𝑓 − 𝐼𝑓 − 𝐼𝑖 𝑒−𝑡𝜏

𝐼𝑖 = 𝑖𝐿 0 = 𝐼0

𝑖𝐿 𝑡 =𝐸

𝑅−

𝐸

𝑅− 𝐼0 𝑒

−𝑡𝜏

𝜏 =𝐿

𝑅𝐼𝑓 =

𝐸

𝑅

→ 𝑖𝐿 𝑡 =𝐸

𝑅1 − 𝑒−

𝑡𝜏

56

Transient Analysis

Dr. Mohamed Refky



𝑖𝐿 𝑡 = 𝐼𝑓 − 𝐼𝑓 − 𝐼𝑖 𝑒−𝑡𝜏

𝐼𝑖 = 𝑖𝐿 0 = 𝐼0

𝑖𝐿 𝑡 =𝐸

𝑅−

𝐸

𝑅− 𝐼0 𝑒

−𝑡𝜏

𝜏 =𝐿

𝑅𝐼𝑓 =

𝐸

𝑅

→ 𝑖𝐿 𝑡 =𝐸

𝑅1 − 𝑒−

𝑡𝜏

57

Transient Analysis

Dr. Mohamed Refky



𝑖𝐿 𝑡 =𝐸

𝑅1 − 𝑒−

𝑡𝜏

𝑣𝑅 𝑡 = 𝑅 × 𝑖𝐿 𝑡

= 𝐸 1 − 𝑒−𝑡𝜏


𝑑𝑡= 𝐸 − 𝑣𝑅 𝑡

= 𝐸𝑒−𝑡𝜏

58

Transient Analysis

Dr. Mohamed Refky

Example (6)For the shown circuit, switch has been closed for a long time and

is opened at 𝑡 = 0. Calculate 𝑖𝐿(𝑡) for 𝑡 > 0.

59

Transient Analysis

Dr. Mohamed Refky

Example (7)For the shown circuit, switch has been opened for a long time and

is closed at 𝑡 = 0. find 𝑖𝑅, 𝑣𝑜, and 𝑖𝐿 for all time.

60

Transient Analysis

Dr. Mohamed Refky

Example (8)For the shown circuit, switch has been closed for a long time and

is opened at 𝑡 = 0. find 𝑖𝐿(𝑡) for 𝑡 > 0.

61

introduction to ac circuits (capacitors and inductors) 2 days ago · ac circuits dr. mohamed refky...

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