CIRCUIT CONCEPTS AND LAWS

ELECTRIC CIRCUITS

1.1 CIRCUIT CONCEPTS AND LAWS

Network is a combination of elements and it may or may not consist of a closed path. Circuit is a combination of elements and should consist of closed path.

Now, under what conditions is the network or circuit theory valid?

Consider the following network elements (Fig. 1.1),

l

R

≅

Figure 1.1

|

Conditions for network theory to be valid.

Here, l <<<l, where l =3×10⁸

f

m/s

If the distributed element length is very much less than the wavelength (l) corresponding to the maximum fre- quency of excitation, then the distributed parameters are approximated into lumped parameters. The inter- connection of such lumped parameters (elements) is called electrical network. At higher frequencies we cannot construct the lumped electric circuit, and hence the network theory is not applicable. The network theory is valid only at low frequencies, that is, upto 1 MHz frequencies only. For above 1 MHz frequencies,

we use field theory. The field theory approach of solv- ing the electrical network is valid for all frequencies starting from zero onwards.

The flow chart for use of field theory and circuit theory for solving problems related to complicated elec- trical networks is depicted in Fig. 1.2.

Complicated electrical

network

Field theory approach (Ohms’s law and Maxwell’s

equations)

1. Exact 2. Complicated 3. Too many variables (E, H, D, J) 4. Distributed circuit

Circuit theory approach (Ohm’s law, Kirchoff’s law)

1. Approximate 2. Simple 3. Less number of variables (V, I)

4. Lumped circuit

Figure 1.2

|

Application of field theory and circuit theory on analysis of electrical network.

1.1.1 Electrical Quantities and Units

1.1.1.1 Ohm’s Law

Before, the discussion on Ohm’s law, let us recapitulate the mechanism of energy flow through the conductor.

The conductivity (s) of conductors is attributed to the presence of free electrons and their mobility through the conductor. In the absence of electric field (E), there is no net momentum of electrons in a conductor so there is no net current, that is i = 0. In the presence of axial electric field (E), force is exerted on the free electrons, that is

F =E⋅e

Here, e = —1.602 × 10^—19 C and net charge Q = ne, where n is the number of free electrons. Figure 1.3 shows the flow of electrons under the influence of an electric field through the conductor with length l and cross-sectional area A.

E A

l

⊕ s

⊕⊕

⊕⊕

⊕

⊕⊕

⊕

⊕⊕

⊕

Figure 1.3

|

Conductivity under the influence of an electric field.

In the presence of electric field, different free elec- trons will move with different velocities. But only one

velocity called drift velocity is defined. It is an aver- age velocity of all the free electrons present with in a conductor given by v

d = mE m/s where m is the mobility of free electron (m²/V-s) in a conductor and E is the applied electrical field. The direction of force will be opposite to the field direction because the free electrons have a negative charge. The net momentum of charge will exist and will be opposite to the direction of the field.

In the presence of an electric field, all the free electrons will have the drift velocity. So that the kinetic energy (K.E.) associated with each electron is given by

K.E. = 1 2

2

m ve d Joules The mass of electron m = 9.1 × 10⁻³¹ kg and m

e = effective mass of the electrons where m

e> m.

In the presence of electric field, each electron will appear to have reduced mass. So we will consider the effective mass of the mobile electrons in the above equation (m

e).

In the absence electric field (E), drift velocity is zero and hence the kinetic energy is zero. So, the total energy (W) of electron is equal to the potential energy.

T.E. = P.E. + K.E.

W = P.E. Joules

Since the conductor is an open circuit at room tem- perature (27°C or 300°K), the free electrons will acquire the thermal energy and they begin to move in different direction in a random manner. Hence the net momentum or net flow of free electrons in any direction is zero. So the net flow of charge is zero and the current zero

dq dt

= 0 ⇒ i = 0 also J = 0 that is, when E = 0 ⇒ J = 0.

The time rate of flow of electric charges is defined as an electrical current (i), that is

i d Q

dt

= ( )

Ampere

Since Q is negative, so the current will flow in the oppo- site direction to that of electron motion, that is, in the direction of the applied electric field (E).

Current density (J) is defined as the current per unit cross-section area and is given by

J

i A

= A/m²

Since A is a scalar, the direction of J is in the direction of the current, that is, in the direction of E.

1.1 CIRCUIT CONCEpTS aND LaWS 5

According to the Ohm’s law in the field theory form or point form, there exists a linear relation between the current density (J) and the electric field (E), that is

J ∝E (under constant temperature)

J =sE (1.1)

where s = conductivity of material = constant (Ωm)⁻¹. Therefore, power dissipation

J⋅E E ⋅E E 

 



= ( ) =

W m

s s ²

3

The characteristic curve for J-E is shown in Fig.1.4.

−E

−J

E J

0 s

s = Slope

Figure 1.4

|

J-E Characteristic curve.

The limitations in the description of conductivity and Ohms’ law based on network (field) theory are:

  1. The Ohm’s law is valid when the proportionality constant (s) is kept constant, that is, the tem- pearture is kept constant otherwise temperature increases as s decreases.

  2. At a constant applied field, E, as the temperature increases the free electrons will acquire extra ther- mal energy, which leads to the increase in collisions and hence the mobility (m) falls, so conductivity (s) decreases.

  3. At a constant temperatue, as E increases, the colli- sions between the free electrons and the positive ions (larger in size) increases, which leads to the fall in the drift velocity and hence loss in K.E. This lost energy is dissipated in the form of heat, which results in a voltage drop (V) across the conductor. The amount of power dissipated within the conductor is

P =JE = E

A m

V

2 m

 2

 



 

 s W/m³

To overcome these, we define Ohm’s law in circuit theory, which states that the voltage across many types of conducting materials is directly proportional to the current flowing through the element (material).

V ∝ ⇒i V =RiVolts (1.2) where the constant of proportionality R (resistance) is constant, that is, temperature is constant.

We have from Eq. (1.1) that J = sE i A

V l

=s. V

l A

i

= s⋅



 

 (1.3)

Substituting for V from Eq. (1.3) in Eq. (1.2), we get Resistance R

l A

=s⋅ W

The resistivity of the material is defined as r=s

1 W-m

If temperature, length of the conductor and surface area increases, l/A is almost constant.

When the conductivity decreases, resitance increases, hence for a conductor

RT = R

0 (1 + at) Hence, temperature coefficents (a) is positive.

Another form of Ohms law is

i=G⋅V Ampere (1.4) where G is the conductance, expressed in Siemens or mho =

1 R

;s is conductivity, expressed in ( m) m W- ⁻¹= or Siemens m.

As i

dq dt

= Therefore,

V R

dq dt

= ⋅ (1.5)

1.1.1.2 Power and Energy

The rate of change of energy is called power P

dW dt

dW dq

dq dt

Vi

= = . = Watts

Using Eq. (1.2), we have P = i²R P

V R

=

2

W Energy is given by

dW = P .dt W =

∫

P⋅dt ^Joules The energy associated with resistor is

E i R dt

R =

∫

² ⋅ ^Joules

V d

dt L

di dt

L = y = ⋅ Volts (1.7) The current across the inductor is

i L

V dt

t L =

1 ⋅

∫

∞

−

Amperes Power is given by

P = Vi From Eq. (1.6)

P =L =

di dt

i Li di dt

⋅ ⋅ Watts (1.7)

Energy is given by

W =

∫

P⋅dt ^(1.8)

From Eq. (1.7)

W Li

di dt

dt

=



 

⋅

∫

^{Joules (1.9)}

Comparing Eqs. (1.8) and (1.9), we have P Li

di dt

d dt

Li

= ⋅ = 

 



1 2

2

Energy of inductor

W P dt

d dt

L i dt

L = ⋅ =  ⋅

 

⋅

∫

∫

¹₂ ²

W L i

L = 1 2

⋅ 2 Joules

Figure 1.7 shows the characteristic current and flux curve for the inductor. Thus, the energy stored in an inductor at any instant depends on the current through the induc- tor at that instant. From the characteristic curve, we can see that inductor is a linear, passive, bilateral and time invariant element.

−i

−y 0 i

L = Slope y

Figure 1.7

|

Current-flux characteristic curve for inductor.

Capacitor (C )

Figure 1.8 shows the circuit for a capacitor.

V_C C i_C +

−

Figure 1.8

|

^Capacitor.

1.1.1.3 Electric Parameters

Resistor (R)

We have from Ohm’s law that V = iR where R is the resistance.

The convention for representing current voltage and resistor is shown in Fig. 1.5(a) and the voltage-current characteristics are depicted in Fig. 1.5(b).

+ V

V = Ri Volts

V = R (−i) Volts

−

+ V − i i

R

(a)

−i

−V i V

0 1

−11

−1

−2

−2 2 2

R=V i

R=−V

−i

(b)

Figure 1.5

|

(a) Voltage-current relation and (b) Voltage-current characteristics.

From the current-voltage characteristics we can see that the resistor is a passive, linear, bilateral and time invari- ant element.

Inductor (L)

When conductor is bound in the form of coil (Fig. 1.6), it will exhibit an opposition, called inductance.

L i V_L

+

−

Figure 1.6

|

^Inductor.

When a time varying current flows through an inductor, it produces a time varying magnetic flux. The total flux (y) produced by it is given by

y =Nf Webers

where N is the number of turns of coil and f is flux per turn.

The amount of flux produced is proportional to the current through the coil

y ∝ ⇒i y =Li Webers (1.6) The voltage across the inductor is

1.1 CIRCUIT CONCEpTS aND LaWS 7

So the voltage-current relation in an inductor is linear.

Hence, the relation

V L

di dt

= ⋅ (1.10)

is another form of Ohm’s law.

For a capacitor:

i C dV

dt

= Here, i V

1← 1 and i V

1← 2

Therefore, i

1 + i

2← V

1 + V

2

So the voltage-current relationship is linear in a capacitor. Hence, the relation

i C dV

dt

= ⋅ (1.11)

is yet another form of Ohm’s law.

Comparison of different electrical paramters in circuit and field theory is given in Table 1.1.

Table 1.1

|

Electric parameters in circuit and field theory

Parameter Circuit   Theory

Field Theory

R P =i R² Watts P =sE² W/m³

L E = Li

1 2

2

Joules W H

H= J m

1 2

2 3

m ( / )

C E= CV

1 2

2

Joules W E

E = J m

1 2

2 3

e ( / )

Then current and voltage is given by i H dt

i

=

∫

⋅ ^{and V E dt}

i

=

∫

⋅ ^(1.12)

From the relation given in Eq. (1.12), we can conclude that:

  1. The inductor will store the energy in the mag- netic field.

  2. The capacitor will store the energy in the elec- tric field.

1.1.2 Types of Circuits

  1. Passive circuit: A network will be called passive if it cannot generate or amplify energy, that is, if the energy which it has supplied since the begin- ning of time cannot exceed the energy which was fed into it.

Current across the capacitor is i

dq dt

C= q=C⋅V

i C

dV dt

C= . Amperes Voltage across the capacitor

V C

i dt

t C =

1

−∞

∫

^Volts

Power in the capacitor P Vi C

dV dt

V CV dV

dt

= = ⋅ ⋅ = Watts

Energy of the capacitor is given by

W C V

dV dt

dt CV

= ⋅  =

 



∫

¹₂ ^{2 Joules}

The energy stored in the capacitor at any instant depends on the voltage across the capacitor at that instant. The charge-voltage characteristic curve is shown in Fig. 1.9.

From the characteristic curve, we can see that capacitor is a linear, passive bilateral and time invariant element.

−q

V V

C

C q

Figure 1.9

|

Charge-voltage chacateristic curve for a capacitor.

Voltage and Curent Relations in Inductors  and Capacitors

For an inductor:

V L

di dt

= ⋅ i V

1 1

i V

2 ® 2

V L

di dt

1

= ⋅ 1

V L

di dt

2

= ⋅ 2

V L

d i i dt

=

⋅ ( + )

1 2

=L +

di dt

L di

dt

⋅ ¹ ⋅ ²

⇒ V = V

1 + V

2

R S

s @

l

V i

+ −

(a) (b)

Figure 1.11

|

(a) Lumped and (b) distributed elements.

  8.  Bilateral and unilateral: An element is said to be bilateral, if the voltage-current relationship is the same for current flowing in either direction.

Resistor, capacitor and inductor are the examples of bilateral elements. An element is said to be uni- lateral, if the voltage-current relationship is dif- ferent for two directions of current flow. Diode is a unilateral element.

Note: In the linear-time invariant case, resistors, capac- itor and inductors are passive if and only if R ≥ 0, C ≥ 0, L ≥ 0; otherwise active, that is, if R < 0, C < 0 and L < 0. If the ratio of voltage to current at any point on characteristic curve is negative, then the element is active; otherwise it is passive. Every linear element is bidirectional. If the characteristic curve is similar in opposite quadrants, then the element is bidirectional;

otherwise, it is unidirectional.

1.1.2.1 V-I Characterictics of Elements   1. If the V-I characteristics of an element of Fig. 1.12(a)

are as shown in Fig. 1.12(b), then the element is linear, passive and bilateral.

−I

−V I I

0 V

V Z_L +

−

(a) (b) Figure 1.12

  2. If the V-I characteristics of an element are as shown in Fig. 1.13, then the element is non-linear, passive and unilateral. For example, diode.

I V

0

Figure 1.13

E t V i d

i t n

( ) [ ( ) ( )]

i i

= t t t ≥

∞

∑

=

∫

⁰

− 1

where E(t) is the difference between the energies entering and leaving the circuit, that is, the net energy supplied to the network. If E(t) equals the energy stored in circuit then the circuit is neither generating, nor losing energy. Such a passive circuit is called a loss-less network. Otherwise, the passive circuit is lossy. For example, resistor, capacitor, inductor, diode, bulb, transformer.

  2.  Active  circuit: If E(t) < 0, the circuit is active, such a circuit can supply excess energy. In other words, when the elements are capable of deliver- ing energy independently for a long time (approxi- mately infinite time) or when element is having property of internal amplification then element is called as active element. For example, voltage sources, current sources, transistors, op-amps.

  3.  Linear  circuit: A network is called linear if the amplitude of the response is always directly propor- tional to the amplitude of the excitation (Fig. 1.10).

−I

−V 0 I V t₃

t₃

t₂

t₂

t₁ t₁

Figure 1.10

|

Linear element.

  4.  Non-linear  circuit: If the response is not pro- portional to the amplitude of the excitation, the circuit is called non-linear.

  5.  Time-invariant  and  time-variant  circuits:

A circuit is time-invariant if the relation between its response and its excitation is applied; otherwise, the network is time-varying.

  6.  Lumped  circuit: A circuit can be considered lumped if the physical dimensions of all its compo- nents are negligible compared with the wavelength of the electromagnetic signal inside the compo- nent. In such a network, since Kirchhoff’s volt- age and current relations hold; the current within any branch is the same at all points of the branch between its terminating nodes [Fig. 1.11(a)].

  7.  Distributed  circuit: If the physical dimensions of the elements are comparable to the signal wave- length, the spatial variations of voltages and cur- rent along the wires and in the components must be taken into consideration. In such circuits, called distributed network, Kirchhoff’s laws are no longer valid, and the more general laws of Maxwell must be applied [Fig. 1.11(b)].

1.1 CIRCUIT CONCEpTS aND LaWS 9

From the above results,

  1. All passive elements need not be linear elements.

  2. All linear elements need not be passive elements.

  3. All linear elements are always bilateral, but con- verse need not be true.

1.1.3 Current and Voltage Sources

There are two types of energy sources:

  1. Independent sources

• V (ideal, practical)

• I (ideal, practical)   2. Dependent sources

• Voltage-controlled voltage source (VCVS)

• Voltage-controlled current source (VCCS)

• Current-controlled voltage source (CCVS)

• Current-controlled current source (CCCS)

1.1.3.1 Independent Sources

Ideal Voltage Source

Ideal voltage source delivers energy at specified volt- age (V

S), which is independent of load current. Internal resistance of ideal voltage source is zero [Fig. 1.18(a)]. In an ideal voltage source, the terminal voltage is indepen- dent of the terminal current so the source will be there like a non-linear element [Fig. 1.18(b)].

V = V_S V_S

V V

(a) (b)

I

I Ideal V_S +

− Z_L

Figure 1.18

|

Ideal voltage source place. (a) Circuit.

(b) V—I characteristics.

Practical Voltage Source

Practical voltage source delivers energy at specified volt- age (V), which depends on current delivered by source [Fig. 1.19(a)], therefore,

V =V IR

S− S

If I increases, V decreases as shown in Fig. 1.19(b). Note that the terminal voltage is a function of the terminal current. Now, when I = 0, V = V

S. So, the current through a passive element can be zero so that two voltages are equal and vice versa [Fig. 1.19(c)].

  3. If the V-I characteristics of an element are as shown in Fig. 1.14, then the element is non-linear, passive and bilateral.

I

−I

V

−V 0

Figure 1.14

  4. If the V-I characteristics of an element are as shown in Fig. 1.15, then the element is non-linear, active and unilateral.

I

−I

V

−V 0

Figure 1.15

  5. If the V-I relation across a resistor is I = 2V ² as shown in Fig. 1.16, then the element is non-linear, active, unilateral. For example, bipolar junction transistor.

I I

V

−V

Figure 1.16

  6. If the V-I characteristics of an element are as shown in Fig. 1.17, then the element is linear, active and bilateral.

I

−I

V

−V 0

Figure 1.17

0 I

V I_S

(b)

− −

+ +

V = 0 R_S

0V 0A

I

I_S

(c)

Figure 1.21

|

Practical current source. (a) Circuit.

(b) V—I characteristics. (c) Current flow- ing through minimum resistance path.

1.1.3.2 Dependent Sources

All the controlled sources shown in Fig. 1.22 are the linear control sources since the control variables are the linear variables. The controlled sources are most of the time active elements and non-bilateral in nature. For exam- ple, (a) BJT ® CCCS, VCVS; (b) Op-amp ® VCVS;

(c) JFET ® VCCS.

K₁V₁ K₂V₂ K₃I₁ K₃I₂

− +

− +

(a) VCVS (b) VCCS (c) CCVS (d) CCCS Figure 1.22

|

Controlled sources.

Inherently, the dependent sources are also non-linear in nature; since the voltage and relation is non-linear. But the linearity of the controlled sources is defined with respect to the controlled variables if they vary in a linear manner, their magnitude varies in a linear manner, so these called linear controlled variables. All the linear controlled sources are always bilateral (non-linear con- trolled sources are unilateral). So, the dependent sources are linear, active and bilateral elements. The presence of these elements in the network makes it an active, linear and bilateral network.

+− V R_S

I

Load V_S

0 V

I V_S

(a) (b)

+- V ≅ I R

V_S

V_S (c)

Figure 1.19

|

Ideal voltage source. (a) Circuit. (b) V-I characteristics.(c) Current through passive element.

Ideal Current Source

Ideal current source delivers energy at specified current (IS), which is independent of voltage across the source [Fig. 1.20(a)]. Internal resistance of ideal current source is infinite. The terminal current is independent of termi- nal voltage [Fig. 1.20(b)].

V I

I = I_S Load I_S

0 I

Ideal

V I_S

Figure 1.20

|

Ideal current source. (a) Circuit.

(b) V—I characteristics.

Practical Current Source

Practical current source delivers energy at specified cur- rent (I), which depends on voltage across the source [Fig. 1.21(a)].

−I V R

S I

S

+ + = 0

So, I I V R

= S S

−

As the terminal voltage increases, I will decrease [Fig. 1.21(b)]. So, the terminal current will be a function of the terminal voltage. When V = 0 ⇒ I = I

S. So, the current always chooses a minimum resistance path [Fig. 1.21(c)].

R_S V I I

Load I_S

(a)

Problem 1.1: The V-I characteristics of a network of Fig. (a) is shown in Fig. (b). If a 10 Ω resistor is con- nected across the terminals a, b of the network, then the terms V and I are, respectively,

1.1 CIRCUIT CONCEpTS aND LaWS 11

0 5 3

3 4 6 7 t(µs) i(t)

(a) 12.5 μs (b) 13 μs (c) 14.5 μs (d) 15 μs Solution: We know that

i t

dq t dt ( )

( )

=

q t i t dt i t dt

i t dt i t dt i t dt ( ) ( ). ( )

( ) ( ) ( )

=

= + +

∫ ∫

∫

∫

∫

=

0 5

1 2 3

4 5

3 4

0 3

= + + + +

1 2

3 5 1 3 1 2

1 2 1 3 1 2

1 1

× ×  × × × × × ×

 

 

 



= + +

15 2

4 7 2

= 15 μs

Ans. (d)

Problem 1.3: The current through 10 Ω resistor is shown in the following figure. Determine the power dissipated in the resistor.

0 10A

1s 2s

∞ 3s t i(t)

Solution: Average power is given by Pavg

Energy absorbed in one period period

=

E i t Rdt

T

i t Rdt I R

R

T

= ² = ² = ²

0

1

( )⋅

∫

( ) ⋅

∫

^rms

Therefore,

i t( ) t t t

( )

( )

= =

10 0 1 0

10 1

−

− ⋅ 0≤ ≤ 0 ≤ t ≤ 1

E t dt

R = (10)² 10

0 1

∫

⋅

= 1000 3

3 0 1

t

=1000 = 1 3

1000 3



 

 J +

+

−

− a

b V I

V_S

R_S

I

−2 V

−20 0

Figure (a) Figure (b)

Solution: When V = 0 V, I = −2 A, we have

+−

R_S I

V_S

V = V

S − IR

S

0 = V

S − (−2) R

S

V

S = −2R

S (i)

When I = 0, A V = −20 V, we have

+−

a

b V V_S

R_S

VS = V Substituting in Eq (i),V

S = −20 V (ii)

we have R

S = 10 Ω

Now considering the following circuit, we have

+

+

− −

I a

b 10Ω

10Ω

V_S V

I V

= +

S

10 10

= = =

VS

20 20 20

1

− − A

So V = I × 10 Ω = −10 V

Ans. (−10, −1)

Problem 1.2: Current I(t) is passing through a capac- itor as shown in the following figure. Determining charge across the capacitor, up to the first 5 μs is

E dt

R 2

2 2 4

6 1 36 2 72

=

∫

⋅ = ^× = ^J

E L i

di dt

L = dt

1 ⋅ 

 

⋅

∫

0 2

= 2 3 3 =18 =18 = =

2

18 2 36

2 0 2

0 2

0 2

. t .dt t dt

⋅

∫

⋅ t

∫

^{× J}

E L i

di dt

L = dt

2 ⋅ 

 



∫

2 4

= 2 6 0 =0

2 4

∫

⋅ ⋅ ^{.dt J} Therefore,

E=E +E +E +E = + + + =

R1 R2 L L J

1 2 24 72 36 0 132

Ans. (132) Problem 1.4: When a resistor R is connected to a cur-

rent source it consumes a power of 18 W. When the same resistor R is connected to a voltage source whose magnitude is same as the current source, the power absorbed is 4.5 W. The values of current source and R are____________.

Solution: Consider the circuits shown in Fig. (a) for R connected to current source and Fig. (b) for R con- necte to voltage source.

i_S R V_S + R

−

Figure (a) Figure (b)

Given that i

S = V

S

Also,

i RS 2

18

= W

V R

2

4 5

= . W R²

18 4 5

180 45

4

= = =

.

W R = 2 Ω

i i

S S

2 18 2

9 3

= = ⇒ = A

Ans. (3, 2) Therefore,

Pavg = = 1000

3 1

1000 3

W Peak power = (10)². 10 = 1000 W

Ans. (1000)

Problem 1.5: The given figures show the inductor and the waveform of the current passing through an inductor of resistance 1 ohm and inductance 2 H. The energy absorbed by the inductor up to 4 s is

0 6A

2 4 t (s) i(t)

Solution: Given that R = 1 Ω; 0 ≤ t ≤ 2 seconds

E t dt

R ( )

1

2 0 2

3

=

∫

= 9 =9 = =

3

9 8 3

24

2

3 0 2

0 2

t dt

⋅ t

∫

^{× J}

1.1.4 Kirchoff’s Laws

Kirchhoff’s laws are valid only in lumped circuits, and at constant temperature, Ohm’s law is also valid. Some features of these laws are as follows:

  1. The Ohm’s law is defined across an element, and that element can be lumped (R, L, C) or distrib- uted element (J = sE); whereas Kirchhoff’s laws are valid only for lumped electric circuits.

  2. The Kirchhoff’s laws are independent of the nature of the elements; whereas the Ohm’s law is a func- tion of the nature of the elements.

  3. The Ohm’s law is applicable for the active elements like sources (i.e., generators). Since the voltage and current relation is non-linear, Kirchhoff’s laws are applicable only for the linear and passive elements (R, L, C).

  4. Kirchhoff’s laws and the lumped element models are valid only when the length of distributed ele- ment is much less than the wavelength correspond- ing to the frequency under consideration.

1.1.4.1 Kirchoff’s Current Law

For any lumped electrical network, for any of its node’s and at any time, the algebraic sum of all the branch cur- rents entering (or leaving) the nodes is zero. A simple node is one having an interconnection of only two branches. The principle node is an interconnection of at least three branches.

For the segment of circuit, shown in Fig. 1.23, the Kirchhoff’s current law (KCL) is expressed as

−i

1− i

2 + i

3 + i

4 + i

5 = 0