Fundamentals of Electrical EngineeringElectrical Engineering

Electronic & Communication EngineeringDanang University of Technology

Lecture 6L, C, Mutual InductanceL, C, Mutual Inductance

(chapter 6)

Preview � To use the equations for voltage, current, power, and

energy in an inductor, capacitor

� To understand how an inductor behaves in the presence of constant current

� To understand how a capacitor behaves in the � To understand how a capacitor behaves in the presence of constant voltage

� To combine inductors/capacitors with initial conditions in series and in parallel to form an equivalent inductor/capacitor

� To understand the basic concept of mutual inductance. To write mesh-current equations for a circuit containing magnetically coupled coils using the dot convention

Inductors and Capacitors

� Two more circuit elements (besides resistor)� Inductors and capacitors are passive circuit elements: they

cannot generate energy (just like resistors).� But unlike resistors, both inductors and capacitors can store

energy, and deliver energy that was previously storedenergy, and deliver energy that was previously stored� Inductor: electromagnetic field, created by moving

charges (current)� Capacitor: electrostatic field, created by displaced

charges (voltage).� Do not discuss electromagnetic or electrostatic fields. Neither will we discuss the

internal behavior or the physical foundations of inductors or capacitors.� Black box approach: interested only in terminal behavior: voltages and currents

as function of time.

The Inductor

� A circuit element described by inductance L• Symbolized by coil• Measured in Henry [H]

� The inductor v-i equation: (ideal inductor)di

Lv =

- Voltage across is “time rate of current change”- Note passive sign convention: the current reference is in the direction of the voltage drop across the inductor- Called inductor branch relationship (Ohm’s law equivalent)

dt

diLv =

Constant Current

� If i=const then the voltage across the ideal inductor v=0� The inductor behaves as short circuit in this case

� Current can not change instantaneously in an inductor

∞→= vdt

diL

� Inductor opposes any change in current

� How to manipulate inductor:The water flow analogy for an inductor involves a water wheel. When pressure (voltage) is applied across the wheel (inductor), it starts to accelerate, moving more and more water (current). The momentum of the wheel represents the energy stored in the magnetic field by the flowing current.

dt

Current in terms of Voltage

� Current in terms of voltage

∫∫ −==

=ti

ti

ttitiLdiLvd

Ldivdt

0

)(

)(0

11

))()((0

τ

- where i(t) is the current at t, i(t0) is the value of current at t0(when we initiate the integration). In some apps., t0 is zero

- the reference current direction is in the direction of the voltage drop (passive sign convention)

∫∫ +=→+=tt

tivd

Ltitivd

Lti

00 )0(1

)()(1

)(0

ττ

Power and Energy

� Power

+=== ∫t

ttivd

Lv

dt

diLivip

0

)(1

0τ

� Energy

2

00 2

1LiwydyLdx

Lididwdt

diLi

dt

dwp

iw

∫∫ =→=

=→==

Example

� Given a circuit with the voltage source:

≥

<=

− 0 20

0 0)(

10 tte

ttv

t

Find the current i(t) ?

Example

1

0for 2220

02010

)0(1

)(

1010

0

10

0

≥+−−=

+=

+=

−−

−∫

∫

tete

de

ivdL

ti

tt

t

t

ττ

τ

τ

Example

� Given a circuit with the voltage source:

a. Find the current i(t) with i(0)=0

≥<

=0 V5

0 0)(

t

ttv

a. Find the current i(t) with i(0)=0

Conclusion ?b. What will happen if we switch off source v at t=10s ?

Conclusion ?

tti 50)( =

500050010for 11

)(10

10

0=+=≥+= ∫∫ tvd

Lvd

Lti

tττ

Inductors In Series

� Equivalent L

dt

diL

dt

diLv

njdt

diLv

eq

n

j j

jj

==

==

∑ =1

,...,1for

Inductors In Parallel

� Equivalent L

)0(1

)0(1

,...,1for )0(1

00

0

11ivd

Livd

Li

njivdL

i

t

teq

n

j j

n

j

t

tj

t

t jj

j

+=+=

=+=

∫∑∑ ∫

∫

==ττ

τ

The Capacitor

� A circuit element described by capacitance C• Symbolized by 2 parallel plates• Measured in Farad [F]

� The inductor v-i equation: (ideal capacitor)dv

Ci =

- Current is proportional to “time rate of voltage across”- Note passive sign convention: the current reference is in the direction of the voltage drop across the capacitor

dt

dvCi =

Constant Voltage

� If v = const then the current through the ideal capacitor i = 0� The capacitor behaves as an open circuit in this case

� Voltage can not change instantaneously in a capacitor

∞→= idt

dvC

� Capacitor opposes any change in voltage

� “Displacement current”: applied voltage displaces charges in a dielectric

dt

Capacitance

� The capacitor stores energy in the electric field. The easy way to make a capacitor is to take two plates of conducting material and separate them with a dielectric (an insulator). The capacitance, C, is then

AC ε=

where ε is the permittivity of the dielectric (basically how good an insulator it is), A is the area and d is the distance between the plates.

dC ε=

Capacitor-Water flow

� We can use a water flow analogy to understand the capacitor. It's like a chamber with a rubber membrane stretched across it. Remember that in the stretched across it. Remember that in the water flow analogy, pressure is analogous to voltage. So when charge (water) flows into the capacitor (current flows in), the same amount of water (charge) flows out the other side (current flows through). But the membrane stretches (the electric field becomes stronger between the plates) and so the pressure (voltage) gets higher.

Capacitor-Water flow

� Thus the more charge we stuff in one end, the higher the voltage gets (positive at that end). For capacitors, this is a linear relationship. This gives us the basic relationship. This gives us the basic relationship

q=Cvwhere C is the capacitance in Farads

Voltage in terms of Current� Voltage in terms of current

∫∫ −==

=tv

tv

t

ttvtvCdxCid

Cdvidt

00

))()(( 0

)(

)(τ

- The reference current direction is in the direction of the voltage drop (passive sign convention)

∫

∫∫

+=t

t

tvt

tvidC

tv0

00

)(1

)( 0

)(

τ

Power and Energy� Power

+=== ∫t

ttvid

Ci

dt

dvCvvip

0

)(1

0τ

� Energy

2

00 2

1CvwydyCdx

Cvdvdwdt

dvCv

dt

dwp

vw

∫∫ =→=

=→==

Capacitors in series

� Equivalent C

)0(1

)0(1

,...,1for )0(1

00

0

11vid

Cvid

Cv

njvidC

v

t

teq

n

j j

n

j

t

tj

t

t jj

j

+=+=

=+=

∫∑∑ ∫

∫

==ττ

τ

Capacitors in parallel

� Equivalent C

dt

dvC

dt

dvCi

njdt

dvCi

eq

n

j j

jj

==

==

∑ =1

,...,1for

Rev

iew

Mutual Inductance� Def: Inductance is the circuit parameter that relates a

voltage to a time-varying current in the same circuit (self-inductance).

� If a magnetic field links two circuits then we obtain mutual inductance (in addition to self-inductance).mutual inductance (in addition to self-inductance).

dt

diMv

dt

diLv

212

111

=

=

dt

diMv

dt

diLv

121

222

=

=

Dot Convention� When the reference direction for a current enters the

dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal.

0

0

12222

21111

=−+

=−++−

dt

diM

dt

diLRi

dt

diM

dt

diLRiv

Exa

mpl

e

Coupling Coefficients

� It can be shown that M2 = k2 L1 L2 for a constant k with 0 ≤ k ≤ 1.

� k depends on the physical arrangement of � k depends on the physical arrangement of the inductors.– k = 0 no coupling– k = 1 ideal coupling

Energy calculation

1 1

2< 22

Energy calculation

Example (steady state)