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EE101: Basics KCL, KVL, power, Thevenins theorem

M. B. Patilmbpatil@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay

M. B. Patil, IIT Bombay

Kirchhos laws

A i2 v2 v4

v3 R2 i3 v6 R1 v1 i1

B i6 V0

v4 R3 i4 v5

C i5 I0

E

D

M. B. Patil, IIT Bombay

Kirchhos laws

A i2 v2 v4

v3 R2 i3 v6 R1 v1 i1

B i6 V0

v4 R3 i4 v5

C i5 I0

E

D

* Kirchhos current law (KCL): P ik = 0 at each node.

M. B. Patil, IIT Bombay

Kirchhos laws

A i2 v2 v4

v3 R2 i3 v6 R1 v1 i1

B i6 V0

v4 R3 i4 v5

C i5 I0

E

D

* Kirchhos current law (KCL): P ik = 0 at each node. e.g., at node B, i3 + i6 + i4 = 0.

M. B. Patil, IIT Bombay

Kirchhos laws

A i2 v2 v4

v3 R2 i3 v6 R1 v1 i1

B i6 V0

v4 R3 i4 v5

C i5 I0

E

D

* Kirchhos current law (KCL): P ik = 0 at each node. e.g., at node B, i3 + i6 + i4 = 0. (We have followed the convention that current leaving a node is positive.)

M. B. Patil, IIT Bombay

Kirchhos laws

A i2 v2 v4

v3 R2 i3 v6 R1 v1 i1

B i6 V0

v4 R3 i4 v5

C i5 I0

E

D

* Kirchhos current law (KCL): P ik = 0 at each node. e.g., at node B, i3 + i6 + i4 = 0. (We have followed the convention that current leaving a node is positive.) * Kirchhos voltage law (KVL): P vk = 0 for each loop.

M. B. Patil, IIT Bombay

Kirchhos laws

A i2 v2 v4

v3 R2 i3 v6 R1 v1 i1

B i6 V0

v4 R3 i4 v5

C i5 I0

E

D

* Kirchhos current law (KCL): P ik = 0 at each node. e.g., at node B, i3 + i6 + i4 = 0. (We have followed the convention that current leaving a node is positive.) * Kirchhos voltage law (KVL): P vk = 0 for each loop. e.g., v3 + v6 v1 v2 = 0.

M. B. Patil, IIT Bombay

Kirchhos laws

A i2 v2 v4

v3 R2 i3 v6 R1 v1 i1

B i6 V0

v4 R3 i4 v5

C i5 I0

E

D

* Kirchhos current law (KCL): P ik = 0 at each node. e.g., at node B, i3 + i6 + i4 = 0. (We have followed the convention that current leaving a node is positive.) * Kirchhos voltage law (KVL): P vk = 0 for each loop. e.g., v3 + v6 v1 v2 = 0. (We have followed the convention that voltage drop across a branch is positive.)

M. B. Patil, IIT Bombay

Circuit elements

Element Resistor

Symbolv i

Equation v =Ri v =L di dt dv dt

Inductori

v

Capacitori

v

i =Cv iC

Diode BJT

to be discussed to be discussedE

B

M. B. Patil, IIT Bombay

Sources

Element Independent Voltage source Current source Dependent VCVS VCCS CCVS CCCS

Symbolv i v i v i v i v i v i

Equation v (t ) = vs (t ) i (t ) = is (t ) v (t ) = vc (t ) i (t ) = g vc (t ) v (t ) = r ic (t ) i (t ) = ic (t )

* , : dimensionless, r : , g : 1 or

(mho)

* The subscript c denotes the controlling voltage or current.M. B. Patil, IIT Bombay

Instantaneous power absorbed by an element

i1 V1 VN iN

i2 V2 V3 i3

P (t ) = V 1 (t ) i 1 (t ) + V 2 (t ) i 2 (t ) + + V N (t ) i N (t ) , where V1 , V2 , etc. are node voltages (measured with respect to a reference node).

M. B. Patil, IIT Bombay

Instantaneous power absorbed by an element

i1 V1 VN iN

i2 V2 V3 i3

P (t ) = V 1 (t ) i 1 (t ) + V 2 (t ) i 2 (t ) + + V N (t ) i N (t ) , where V1 , V2 , etc. are node voltages (measured with respect to a reference node).

* two-terminal element:v V1 i1 i2 V2

P = V1 i 1 + V2 i 2 = V 1 i 1 + V 2 ( i 1 ) = [V 1 V 2 ] i 1 = v i 1

M. B. Patil, IIT Bombay

Instantaneous power absorbed by an element

i1 V1 VN iN

i2 V2 V3 i3

P (t ) = V 1 (t ) i 1 (t ) + V 2 (t ) i 2 (t ) + + V N (t ) i N (t ) , where V1 , V2 , etc. are node voltages (measured with respect to a reference node).

* two-terminal element:v V1 i1 i2 V2

P = V1 i 1 + V2 i 2 = V 1 i 1 + V 2 ( i 1 ) = [V 1 V 2 ] i 1 = v i 1 P = V B i B + V C i C + V E ( i E )iC iE

* three-terminal element:VC VB iB VE

= V B i B + V C i C V E (i B + i C ) = (VB VE ) iB + (VC VE ) iC = VBE iB + VCE iE

M. B. Patil, IIT Bombay

Instantaneous power

* A resistor can only absorb power (from the circuit) since v and i have the same sign, making P > 0. The energy absorbed by a resistor goes in heating the resistor and the rest of the world.

M. B. Patil, IIT Bombay

Instantaneous power

* A resistor can only absorb power (from the circuit) since v and i have the same sign, making P > 0. The energy absorbed by a resistor goes in heating the resistor and the rest of the world. * Often, a heat sink is provided to dissipate the thermal energy eectively so that the device temperature does not become too high.

M. B. Patil, IIT Bombay

Instantaneous power

* A resistor can only absorb power (from the circuit) since v and i have the same sign, making P > 0. The energy absorbed by a resistor goes in heating the resistor and the rest of the world. * Often, a heat sink is provided to dissipate the thermal energy eectively so that the device temperature does not become too high. * A source (e.g., a DC voltage source) can absorb or deliver power since the signs of v and i are independent. For example, when a battery is charged, it absorbs energy which gets stored within.

M. B. Patil, IIT Bombay

Instantaneous power

* A resistor can only absorb power (from the circuit) since v and i have the same sign, making P > 0. The energy absorbed by a resistor goes in heating the resistor and the rest of the world. * Often, a heat sink is provided to dissipate the thermal energy eectively so that the device temperature does not become too high. * A source (e.g., a DC voltage source) can absorb or deliver power since the signs of v and i are independent. For example, when a battery is charged, it absorbs energy which gets stored within. * A capacitor can absorb or deliver power. When it is absorbing power, its charge builds up. Similarly, an inductor can store energy (in the form of magnetic ux).

M. B. Patil, IIT Bombay

Resistors in series

v1A

v2 R2

v3B A

v iB

i

R1

R3

R

M. B. Patil, IIT Bombay

Resistors in series

v1A

v2 R2

v3B A

v iB

i

R1

R3

R

v1 = i R1 , v2 = i R2 , v3 = i R3 , v = v1 + v2 + v3 = i (R1 + R2 + R3 )

M. B. Patil, IIT Bombay

Resistors in series

v1A

v2 R2

v3B A

v iB

i

R1

R3

R

v1 = i R1 , v2 = i R2 , v3 = i R3 , v = v1 + v2 + v3 = i (R1 + R2 + R3 ) * The equivalent resistance is Req = R1 + R2 + R3 .

M. B. Patil, IIT Bombay

Resistors in series

v1A

v2 R2

v3B A

v iB

i

R1

R3

R

v1 = i R1 , v2 = i R2 , v3 = i R3 , v = v1 + v2 + v3 = i (R1 + R2 + R3 ) * The equivalent resistance is Req = R1 + R2 + R3 . * The voltage drop across Rk is v Rk . Req

M. B. Patil, IIT Bombay

Resistors in parallel

v i1A

R1B A

v iB

i

i2 i3

R2 R3

R

M. B. Patil, IIT Bombay

Resistors in parallel

v i1A

R1B A

v iB

i

i2 i3

R2 R3

R

i1 = G1 v , i2 = G2 v , i3 = G3 v , where G1 = 1/R1 , etc. i = i1 + i2 + i3 = (G1 + G2 + G3 ) v .

M. B. Patil, IIT Bombay

Resistors in parallel

v i1A

R1B A

v iB

i

i2 i3

R2 R3

R

i1 = G1 v , i2 = G2 v , i3 = G3 v , where G1 = 1/R1 , etc. i = i1 + i2 + i3 = (G1 + G2 + G3 ) v . * The equivalent conductance is Geq = G1 + G2 + G3 , and the equivalent resistance is Req = 1/Geq .

M. B. Patil, IIT Bombay

Resistors in parallel

v i1A

R1B A

v iB

i

i2 i3

R2 R3

R

i1 = G1 v , i2 = G2 v , i3 = G3 v , where G1 = 1/R1 , etc. i = i1 + i2 + i3 = (G1 + G2 + G3 ) v . * The equivalent conductance is Geq = G1 + G2 + G3 , and the equivalent resistance is Req = 1/Geq . * The current through Rk is i Gk . Geq

M. B. Patil, IIT Bombay

Resistors in parallel

v i1A

R1B A

v iB

i

i2 i3

R2 R3

R

i1 = G1 v , i2 = G2 v , i3 = G3 v , where G1 = 1/R1 , etc. i = i1 + i2 + i3 = (G1 + G2 + G3 ) v . * The equivalent conductance is Geq = G1 + G2 + G3 , and the equivalent resistance is Req = 1/Geq . * The current through Rk is i Gk . Geq

* If N = 2, we have R1 R2 R2 R1 Req = , i1 = i , i2 = i . R1 + R2 R1 + R2 R1 + R2

M. B. Patil, IIT Bombay

Resistors in parallel

v i1A

R1B A

v iB

i

i2 i3

R2 R3

R

i1 = G1 v , i2 = G2 v , i3 = G3 v , where G1 = 1/R1 , etc. i = i1 + i2 + i3 = (G1 + G2 + G3 ) v . * The equivalent conductance is Geq = G1 + G2 + G3 , and the equivalent resistance is Req = 1/Geq . * The current through Rk is i Gk . Geq

* If N = 2, we have R1 R2 R2 R1 Req = , i1 = i , i2 = i . R1 + R2 R1 + R2 R1 + R2 * If Rk = 0, all of the current will go through Rk .M. B. Patil, IIT Bombay

Example

i1 4 3 6V i2

2

5(a)

2.5

2.5

3

Example

i1 4 3 6V i2

2

5(a)

2.5

2.5

3

Example

i1 4 3 6V i2

2

i1