Electrical Circuit Analysis (3130906) Darshan Unit-1,5

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1. Network Theorems 1.1. Define following terms (a)

Charge Electrons in the outer orbit of an atom can easily evacuated by application of some external force. Electrons that are forced out of their orbits can result in a lack of electrons, hence called positive charge i.e. more protons than electrons. Electrons where they come to rest can result in excess of electrons, hence called negative charge i.e. more electrons than protons. A positive or negative charge is an effect of absence or excess of electrons. The number of protons remains constant. Charge is measured in coulombs.

(b)

Potential Work done against the force of repulsion to bring a charge closer to the one another is called potential. Potential is measured in volt.

(c)

Potential difference The potential difference between two points is “One volt” when one joule of work is done to displace a unit charge of one coulomb from the point of lower potential to point of higher potential. Potential difference is measured in volt

(d)

Electro motive force (EMF) Emf is the potential difference that moves the electrons to flow in any conductor. Emf is measured in volt.

(e)

Current An amount of charge passing through the conductor in unit time is called current. It is measured in ampere.

(f)

Current density It is the amount of current flowing per unit cross section area of a conductor. Current density is measured in A/mm2.

(g)

Power Rate of change of energy with respect to time is called power. It is measured in watt.

(h)

Electrical energy Electrical power consumed in unit time is called electric energy. It is measured in Kwh.

(i)

Linear element and Nonlinear element An elements such as resistor, inductor and capacitor whose voltage vs current characteristics is linear and their resistance, inductance and capacitance do not vary with the change in applied voltage or circuit current are called linear elements. An elements such as semiconductor devices whose voltage vs current characteristics is nonlinear and their resistance, inductance and capacitance may vary with the change in applied voltage or circuit current are called nonlinear elements.

(j)

Active element and Passive element An element such as vacuum tube, transistor, Opams with the capacity of boosting the energy level of signal passing through it are called active elements. An element such as resistor, inductor, capacitor, thermistor that do not have capacity of

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Electrical Circuits Analysis (3130906)

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1. Network Theorems boosting the energy level of signal passing through it are called passive elements. (k)

Unilateral element and Bilateral element When the amount of current passing through element is affected by the change in polarity of applied voltage then it is called unilateral element. This element offers varying impedance with the variation in current. Diode, transistors etc. are the examples of unilateral elements. When the amount of current passing through element is not affected by the change in polarity of applied voltage then it is called bilateral element. This element offers same impedance irrespective of variation in current. Resistance, inductance and capacitance are the examples of bilateral elements.

(l)

Lumped network and Distributed network A network in which circuit elements like resistance, inductance and capacitance are physically separable for analysis purposes, is called lumped network. Most of the electric networks are lumped in nature. A network in which circuit elements like resistance, inductance and capacitance cannot be physically separated for analysis purposes, is called distributed network. A transmission line where resistance, inductance and capacitance of a transmission line are distributed all along its length and cannot separated anywhere in the circuit.

(m)

Linear network and Non-linear network A network whose parameters remain constant irrespective of the change in time, voltage, temperature etc. is known as linear circuit. Ohm’s law is applicable to such network. This type of circuit can be solved using super position law. A network whose parameters change their values with the change in time, voltage, temperature etc. is known as non-linear circuit. Ohm’s law is not applicable to such network. This type of circuit does not follow super position law.

(n)

Unilateral network and Bilateral network A network whose characteristic dependents on the direction of current i.e. characteristics changes if direction of current is changed. Network with diode, transistors etc. that has diverse characteristics in different direction of current. A network whose characteristic is independent of the direction of current i.e. characteristics remains same if direction of current is changed. Network with only resistance has similar characteristics in different direction of current.

(o)

Active network and Passive network A network that contains one or more energy source such as voltage or current is called active network. A network that does not contain any energy source such as voltage or current is called passive network.

(p)

Ideal energy source and Particle energy source Energy sources are the devices that converts any source of energy into electrical energy. Types of sources available in the electrical network are voltage source and current

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1. Network Theorems

Figure 1. 1 Ideal voltage source



Figure 1. 2 Practical voltage source

Current Source

Current Source

Voltage Source

Voltage Source

sources. A voltage source has a driving role of emf whereas the current source has a driving job of current.

Figure 1. 3 Ideal current source

Figure 1. 4 Practical current source

Voltage source Ideal voltage source is a two-terminal device whose voltage at any instant of time is constant and is independent of the current drawn from it. Internal resistance of ideal voltage source is zero, but practically an ideal voltage source cannot be achieved. Practical voltage source is a two-terminal device whose voltage at any instant of time changes with the current drawn from it. Due to internal resistance of voltage source, when current flows voltage drop takes place and it causes terminal voltage to fall down.



Current source Ideal current source is a two-terminal device that provides constant current to any load from zero to infinity. Internal resistance of ideal current source is infinite, but practically an ideal current source cannot be achieved. Practical current source is a two-terminal device whose current at any instant of time changes. Amount off current depends upon the load.

(q)

Independent energy sources

V

+ -

v(t)

Figure 1. 5 Independent voltage source

I

i(t)

Figure 1. 6 Independent current source

Independent voltage source is the two terminal element that provides a specific voltage across its terminal. The value of this voltage at any instant is independent of value or direction of the current that flow through it. Independent current source is the two-terminal elements that provides a specific current across its terminal. The value and direction of this current at any instant is independent of value or direction of the voltage that appears across the terminal of source. Shital Patel, EE Department

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1. Network Theorems (r)

Dependent energy sources

a

+

+

Vab b

μ Vab

+ -

c

Vcd

-

-

d

a

Vab b

Figure 1. 7 Voltage controlled voltage source (VCVS)

a

+ i ab

+ r iab

b

-

+ -

Figure 1. 9 Current controlled voltage source (CCVS)

+

c

Vcd

g m Vab

-

-

d

Figure 1. 8 Voltage controlled current source (VCCS)

c

a

+

d

icd

i ab

+

c

β iab

Vcd

-

Icd

+

b

-

-

d

Figure 1. 10 Current controlled current source (CCCS)

Voltage controlled voltage source (VCVS) is the four terminal network components that establishes a voltage between two-point c and d. Value of Vcd depends upon the controlled voltage Vab and constant μ. Voltage controlled current source (VCCS) is the four terminal network components that establishes a current icd in the branch of circuit. Value of icd depends on the controlled voltage Vab and constant gm. Current controlled voltage source (CCVS) is the four terminal network components that establishes a voltage Vcd between two-point c and d. Value of Vcd depends upon the controlled current iab and constant r. Current controlled current source (CCCS) is the four terminal network components that establishes a current icd in the branch of circuit. Value of icd depends upon the controlled current iab and constant β. (s)

Single port network An active or passive network with two terminals is treated as single port network.

(t)

Two port network An active or passive network with two pairs of terminals is treated as two port network. Where one pair of terminal is designated as input and other pair of terminal is designated as output.

(u)

Multi-port network An active or passive network with n- number of pairs of terminals is treated as multi-port network. Where some pair of terminals are designated as input and some pair of terminals are designated as output.

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1. Network Theorems 1.2. Relation between energy (E) and power (P) for two terminal resistor element 

Relation between voltage and current in resistor element in terms of charge is,

v  Ri  R 

dq dt

Given voltage v (t) across and current i (t) through a resistor L and then associated energy e (t) is,

If v (0)  0 and v (t ) V v (t )  Ri (t ) v (t ) i (t )  R t e (t )   p (t )dt 0

t

  i (t )v (t )dt 0

t

v (t ) v (t )dt R 0

 

1

T

V R

2

dt

0



V2 T R

If v (0)  0,v (t ) V m sin(t ) and energy dissipated for time period T  i (t ) 

2



v (t ) V m sin(t )  R R t

e (t )   p (t )dt 0

t

  i (t )v (t )dt 0

V m sin(t ) V m sin(t ) dt R 0

t



V m2 t 2  sin (t )dt R 0 V m2 T (1  cos(2t ))  dt R 0 2 V m2  T 2R Shital Patel, EE Department

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1. Network Theorems 1.3. Relation between energy (E) and power (P) for two terminal inductor element 

Relation between voltage and current in inductor element in terms of charge is,

v 

d dt

When there is an initial charge of ψo is stored on inductor and it is increasing linearly with time, then charge on inductor at any instant of time is,

   o  kt d  k dt 

Hence, it can be observed that voltage in the inductive system is independent of initial charge. t

   vdt 

t

0

  vdt  vdt 

0

t

  o  vdt 0



Given voltage v (t) across and current i (t) through a inductor L and then associated energy e (t) is,

If i (0)  0 and i (t )  I v (t )  i (t ) 

d d (Li ) di (t )  L dt dt dt t

1

 v (t )dt L  t

e (t )   p (t )dt 0

t

  i (t )v (t )dt 0

t

  i (t )L 0

di (t ) dt dt

I

 L  idi 0

1.4.

1  LI 2 2

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1. Network Theorems If i (0)  0, i (t )  I m sin(t ) and energy dissipated for time period T  v (t )  L

2



d  I m sin(t )  di (t ) L  LI m cos(t ) dt dt

t

e (t )   p (t )dt 0

t

  i (t )v (t )dt 0

t

  I m sin(t )LI m cos(t ) dt 0

 

LI m2  t 2

 2cos(t )sin(t )dt 0

LI m2  T 2

 sin(2t )dt 0

0

1.5. Relation between energy (E) and power (P) for two terminal capacitor element 

Relation between voltage and current in capacitor element in terms of charge is,

i 

dq dt

When there is an initial charge of qo is stored on capacitor and it is increasing linearly with time, then charge on capacitor at any instant of time is,

q  qo  kt dq  k dt 

Hence, it can be observed that current in the capacitive system is independent of initial charge. t

q   idt 

t

0



 idt   idt



0

t

 qo   idt 0



Given voltage v (t) across and current i (t) through a capacitor C and then associated energy e (t) is,

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1. Network Theorems If v (0)  0 and v (t ) V i (t )  v (t ) 

dq (t ) d Cv (t ) dv (t )  C dt dt dt t

1

C

 i (t )dt



t

e (t )   p (t )dt 0

t

  i (t )v (t )dt 0

t

 C 0

dv (t ) v (t )dt dt

V

 C vdv 0

1  Cv 2 2

If v (0)  0,v (t ) V m sin(t ) and energy dissipated for time period T  i (t )  C

2



d V m sin(t )  dv (t ) C  CV m cos(t ) dt dt

t

e (t )   p (t )dt 0

t

  i (t )v (t )dt 0

t

 CV m cos(t )V m sin(t ) dt 0



CV m2 t



CV m

2 2

2

 2cos(t )sin(t )dt 0

T

 sin(2t )dt 0

0

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1. Network Theorems 1.6. Superposition theorem 

Statement: In a linear circuit having several independent sources, the current or voltage of a circuit element equals the algebraic sum of the component voltages or currents produced by the independent sources acting alone.



To reflect the effect of each sources alone, a voltage source that makes no contribution is replaced by a short circuit. Whereas a current source that makes no contribution is replaced by an open-circuit. The internal resistance of the source is kept as it is.



For better understanding consider below circuit with two voltage sources. R2

R1

I1



I2

I3 R3

V1

I1

Current through resistanceR3 , I 3  I 3'  I 3"

V2

R2



Equivalent resistance, Req  R1  I2

I3

R2R3 R 2  R3

Current through resistanceR1 , I 1' 

R3

V1

S.C.

V1 Req

 R3  ' Current through resistanceR2 , I 2'    I 1  R 2  R3   R2  ' Current through resistanceR3 , I 3'    I 1  R 2  R3 

Consider voltage source V2 only

R2

R1 1

S.C.



Current through resistanceR 2 , I 2  I 2'  I 2"

Consider voltage source V1 only R1

I

Current through resistanceR1 , I 1  I 1'  I 1"

I

3

R3

Equivalent resistance, Req  R 2 

I

R1R3 R1  R3

2

Current through resistanceR2 , I 2' 

V2

V2 Req

 R3  ' Current through resistanceR1 , I 1'    I 2  R1  R3   R1  ' Current through resistanceR3 , I 3'    I 2  R1  R3 

Superposition theorem is applicable to linear networks i.e. time varying or time invariant with independent sources, linear dependent sources, linear passive elements such as resistors, inductors, capacitors and linear transformers.

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1. Network Theorems 1.7. Substitution theorem 

Statement: In any circuit if, current through branch or voltage across that branch is known then this branch can be replace by combination of same set of terminal voltage and current without disturbing voltages and currents in entire circuit.



For better understanding consider below circuit with a branch x between node A and B having impedance Zx and current Ix.



Impedance Zx can be replaced by a compensating voltage source having magnitude Vx = Ix Zx or can be replaced by current source having magnitude Ix = Vx / Zx. A

A

A

Ix

Vx

Zx

=

Ix

B

Vx

OR

B

B



While applying substitution theorem, branch k should not be connected to other element i.e. neither the part of magnetically coupled circuit nor part of controlled source.



This theorem is generally used for the circuits that contains single non-liner or time varying elements.



Connect voltage source of magnitude E = Vx at node B and keep node A and node C at same potential. A

A

A=C

A=C

C

Vx

Zx

B

=

E

Zx

B

E

Zx

B

E

B



As branch x, is in parallel with voltage source and hence it can be removed without affecting the other part of circuit i.e. branch x is replaced by independent voltage source.



Similarly branch x can replaced by a current source. Let, current source of magnitude I = Ix is connected between node A and node C such that addition of current cause the current in short circuit branch zero.

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1. Network Theorems

Ix

Vx

Zx

I = Ix

I Ix

=

B



A

A

A

Zx

Zx

B

B

As branch x, is in series with current source and hence it can be removed without affecting the other part of circuit i.e. branch x is replaced by independent current source.

1.8. Compensation theorem 

Statement: In any linear time invariant network when the resistance of R of an uncoupled branch, carrying a current I is changed by ΔR, then currents in all the branches will change. The change in current ΔI is obtained by assuming that an ideal voltage source VC = I (ΔR) is connected in series with (R+ΔR) when all other sources in the network are replaced by their internal resistances.



As it is known that voltage drop across element is replaced by ideal voltage source and current through element is replaced by ideal current source without affecting rest of circuit.



But, if impedance of an element is changed then redistribution of current and voltage in entire circuit takes places.



This theorem is useful to determine current and voltage change in a circuit element when value of its impedance is changed.



Let suppose, circuit is supplied by Thevenin’s voltage and resistance of circuit is changed to RL+ΔR such that current changes from IL to I’L. Rth

Rth

Rth

D IL

IL

IL

RL

Vth

Vth

RL

RL

DR

DR VC=(IL)DR

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Electrical Circuits Analysis (3130906)

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1. Network Theorems 

Thus, change in branch current

DI  I L'  I L 

Vth Vth  Rth  RL  DR Rth  R L

 R  R  R  R  DR  L th L  Vth  th   Rth  R L  DR  Rth  R L       DR Vth  Rth  RL  DR Rth  RL   Vth  DR       Rth  R L  Rth  R L  DR   DR  I L    Rth  R L  DR  I L  DR  Rth  RL  DR  

  

Vc Rth  RL  DR

The voltage source Vc = (IL) DR is called compensation voltage source.

1.9. Thevenin's theorem 

Statement: Any linear bilateral network with circuit element and active source connected to the load can be replaced by single two terminal networks consisting of a single voltage source (Vth) in series with impedance (Zth).



Single voltage source (Vth) is the voltage across load terminal when load ZL is removed i.e. open circuit voltage across load terminal.



Series impedance Zth is the equivalent impedance of passive network viewed from load terminal when ZL is removed. Passive network means effect of sources are considered zero i.e. voltage sources are short circuited and current sources are open circuit.



Thevenin's theorem is used to find current through any branch of the circuit.



For better understanding consider below circuit with two voltage sources. Rth

RL=R2

R1

IL

V1

Shital Patel, EE Department

R3

V2

IL Vth

RL

IL 

Vth Rth  R L

Electrical Circuits Analysis (3130906)

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1. Network Theorems 

Determination of Vth R1

Thevenin's voltage,Vth  IR3 V 2

Vth

 V1     R3 -V 2 R  R 3   1

I R3

V1



V2

Determination of Rth R1

Thevenin's resistance, Rth  R1 R3

Rth

 R3

S.C.

R1R3 R1  R3

S.C.

1.10. Norton's theorem 

Statement: Any linear bilateral network with circuit element and active source connected to the load can be replaced by single two terminal networks consisting of a single current source (In) in parallel with impedance (Zn).



Single current source (In) is the current through load terminal when load ZL is removed and terminals are short circuited i.e. short circuit current across load terminal.



Parallel impedance Zn is the equivalent impedance of passive network viewed from load terminal when ZL is removed. Passive network means effect of sources are considered zero i.e. voltage sources are short circuited and current sources are open circuit.



Norton's theorem is used to find current through any branch of the circuit.



For better understanding consider below circuit with two voltage sources. RL=R2

R1

IL

IL



V1

R3

Shital Patel, EE Department

V2

In

Rn

RL

  I n  R n  RL 

I L  

Rn

Electrical Circuits Analysis (3130906)

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1. Network Theorems 

Determination of In R1 I1

Norton's current, I n  I 1  I 2

Vx

I2 R3

V1



In

V V  V    x 1    x   R1   R3  V V   V    2 1    2   R1   R3 

V2

Determination of Rn R1

Norton's resistance, R n  R1 R3

Rn

 R3

S.C.

R1R3 R1  R3

S.C.

1.11. Reciprocity theorem 

Statement: In any linear, bilateral network, the current due to a single source of voltage in the network is equal to the current through that branch in which the source was originally placed when the source is again put in the branch in which the current was originally obtained.



Limitations of reciprocity theorem are o Applicable to the network with only one source of excitation o Network is initially relaxed i.e. all initial condition are zero o Network must be linear and bilateral o Impedance matrix of a network must be symmetric matrix o Network with dependent or controlled sources are excluded even if it is linear



For better understanding consider below circuit. R2

R1

I1

V1

I3 R3

Shital Patel, EE Department

I2

 RR  Equivalent resistance, Req  R1   2 3   R2  R3  V Current through resistance R1 , I1  1 Req  R3  Current through resistance R2 , I2    I1  R2  R3   R2  Current through resistance R3 , I3    I1  R2  R3  Electrical Circuits Analysis (3130906)

14

1. Network Theorems R2

R1 I1

I3

R3

 RR  Equivalent resistance, Req  R2   1 3   R1  R3  V Current through resistance R2 , I2  1 Req

I2

 R3  Current through resistance R1 , I1    I2  R1  R3   R1  Current through resistance R3 , I3    I2  R1  R3 

V1

1.12. Maximum power transfer theorem

(a)



Maximum power transfer theorem helps to determine value of load impedance that allows maximum power to be transferred from source to load.



This theorem is generally used for telecommunication circuit i.e. small amount of power handling capacity and aim is to transfer maximum power from source to load.



It is never used for power system i.e. large amount of power handling capacity and aim is to achieve maximum power transfer efficiency.



For better understanding consider DC circuit and AC circuit separately. DC circuit

Rth

Load current, I L 

IL Vth

RL

Vth Rth  RL

 Vth Power transferred to load, P  I L R L    Rth  R L 2

For power to be maximum,

2

  R L 

dP 0 dRL

 R R 2 1  R 2 R R   th L     L   th L   dP  Vth2  4   dRL Rth  RL     R R 2 1  R 2 R R   th L     L   th L   0 Vth  4   Rth  RL    2 0   Rth  R L  1   R L  2 Rth  R L  2

0  Rth2  2Rth R L  R L2  2Rth R L  2R L2 0  Rth2  R L2

RL2  Rth2 RL  Rth Shital Patel, EE Department

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1. Network Theorems  This shows that, in DC circuit maximum power can be transferred when load resistance is equal to the internal resistance of network.

Maximum power, Pmax

 Vth  I L RL    Rth  RL 2

 RL Voltage across load, V L    Rth  RL (b)

2

  Vth  RL    RL  RL 

  RL Vth    RL  RL 

2

 Vth2 R   L 4R L 

 Vth Vth  2 

AC circuit with variable resistive load Rth

Xth IL

Vth

RL

Load current, I L 

Vth  Z th  RL

Vth

R

th

 R L   X th2 2

 Vth2 Power transferred to load, P  I L R L   2  R  R  X th2   L  th 2

For power to be maximum,

dP 0 dRL

 R  L 

  RL dP 2   V dRL th   R  R 2  X 2   th L th  





  2 2   Rth  R L   X th 1   R L  2Rth  R L    0 Vth2  2 2   Rth  RL   X th2  









0   Rth  R L   X th2 1   R L  2 Rth  R L  2

0  Rth2  2Rth R L  R L2  X th2  2Rth R L  2R L2 0  Rth2  X th2  R L2

RL2  Rth2  X th2 RL  Rth2  X th2 RL  Z th

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

16

1. Network Theorems (c)

AC circuit with variable resistive and inductive load Rth

Xth

Load current, I L 

IL

Vth Z th  Z L

Vth



R

 RL    X th  X L  2

th

RL

2

  Vth2  R Power transferred to load, P  I L R L  2 2  L     Rth  RL    X th  X L   2

Vth XL

For power to be maximum, X L  X th For power to be maximum,

dP 0 dRL







RL dP  Vth2   R R 2  X  X 2  dRL  th L     th L





  2 2   Rth  RL    X th  X L  1   RL  2Rth  R L    0 Vth2  2 2 2   Rth  RL    X th  X L    2 2 0    Rth  RL   X th   X th   1  RL  2Rth  R L   









0  Rth2  2Rth RL  RL2  2Rth RL  2RL2 0  Rth2  RL2

RL  Rth So, maximum power transferred to the load when RL  jX L  Rth  jX th Z L  Z th*

(d)

AC circuit with fixed resistive and variable inductive load Rth

Xth

IL

Load current, I L 

Vth Z th  Z L



Vth

R

th

 RL    X th  X L  2

2

RL

  Vth2 R Power transferred to load, P  I L2RL   2 2  L     Rth  RL    X th  X L  

XL

For power to be maximum, X L  X th

Vth

So, maximum power transferred to the load when RL  jX L  Rth  jX th Z L  Z th*

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

17

1. Network Theorems (e)

AC circuit with variable resistive and fixed inductive load Rth

Xth

IL RL

Vth

Load current, I L 

Vth  Z th  Z L

Vth

R

 RL    X th  X L  2

th

2

  Vth2 R Power transferred to load, P  I L2R L   2 2  L     Rth  RL    X th  X L  

XL

For power to be maximum,



dP 0 dRL

  RL dP  Vth2   R R 2  X  X 2  dRL  th L     th L





  2 2   Rth  RL    X th  X L  1   R L  2 Rth  R L    0 Vth2  2 2 2   Rth  RL    X th  X L   





0   Rth  RL    X th  X L  2

2



 1    R  2 R L

th

 RL 

0  Rth2  2Rth RL  RL2   X th  X L   2Rth RL  2RL2 2

0  Rth2   X th  X L   RL2 2

RL  Rth2   X th  X L 

2

RL   Rth  jX th   jX L RL  Z th  jX L

1.13. Millman's theorem 

Statement: Number of voltage sources with their internal resistance are connected in parallel can be replaced by single equivalent voltage source with equivalent internal resistance connected in series.



It is applicable to the circuit that contains only parallel branches with only one resistance and source in a branch.



It is easier to apply theorem to a circuit if all the branches contains same type of source either voltage or current.



It is not applicable to the complex mesh of parallel/series network or to the circuit where resistance elements are connected between the sources.

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

18

1. Network Theorems

I1



Z1

I2

I3

I

In

Z1

Z2

Z3

Zn

Z

E1

E2

E3

En

E

Using source transformation technique each branch voltage source and its internal resistance is replaced with equivalent current source in parallel with internal resistance.

I1

Z2

Z3

I2

Zn

I3

Total current, I  I 1  I 2  I 3  

Z

In

I

In

E1 E2 E3    Z1 Z2 Z3



En Zn

 E 1Y1  E 2Y2  E 3Y3 

 E nYn

n

  E iYi i 1

Total impedance,

1

Z



1

Z1



1

Z2



1

Z3

Y  Y1 Y2 Y3 





1

Zn

 Yn

n

 Yi i 1

Equivalent source, E=IZ 1 =I

Y n



E iYi  i 1

n

Yi  i 1

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

19

1. Network Theorems 1.14. Duality of a network 

Sometimes statement of Kirchhoff’s current law for one network is almost similar to the statement of Kirchhoff’s voltage law of another network i.e. voltage replaced with current or mesh analysis replaced with nodal analysis. R1

v2(t)

L1

v1(t)

v 1 (t )  L1

G2

i2(t)

C1

i1(t)

di 1 1  R1i 1   i 1dt dt C1

i 2(t )  C 2

L2

C2

dv 2 1  G 2v 2  v 2dt dt L2



These two equation are identical mathematical operations, only the part of voltage and current is interchanged. Solution of first equation is the solution of second equation. The similarity between two networks is termed as duality.



Two networks are said to be dual if node equation of one have the same mathematical form as mesh equations of other. The voltage and current variables are not same. Duality of network elements

R and G

 vdt and idt

L and C

L i and v

di dv and C dt dt

1 1 vdt and  idt  L C

q and ψ Steps to draw dual network 

Place a node in each individual mesh and one reference node outside the network i.e. 1, 2, 3 node number in each mesh and 0 node number outside. C2

R1

v(t)

C2

L

C1

3

R1

R2

v(t)

1

C1

L

2

R2

0

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

20

1. Network Theorems 

Join two nodes through each elements at a time. Stay with the same procedure until all possible number of path through each element is considered. Replace each element by its dual element between two connected nodes. L2

C2

G1 3

R1

1

G1

L L1

v(t)

L2

C

1 2

C1

R2

i(t) 0

i(t)

Shital Patel, EE Department

3

L1

C 2

G2

G2

0

Electrical Circuits Analysis (3130906)

21

5. Two Port Network and Network Functions 5.1. Different two port network configuration (a)

T-network When two series arms are connected with one shunt arm it looks like a T-network. Usually T-networks are represented by two ways. Z1

Z1

Z1

Input

Z2

Z3

Input

Output

Symmetrical T-network

Z2

Output

Unsymmetrical T-network

Symmetrical T-network: Impedance of series arm on both side is equal i.e. net series impedance is 2Z1. Unsymmetrical T-network: Impedance of series arm on both side is not equal i.e. net series impedance is (Z1+Z3) (b)

π-network When two shunt arms are connected with one series arm it looks like a π-network. Usually π-networks are represented by two ways. Z1

Input

Z2

Z1

Z2

Input

Output

Z2

Z3

Output

Unsymmetrical π-network

Symmetrical π-network

Symmetrical π-network: Impedance of shunt arm on both side is equal i.e. net shunt impedance is Z2/2. Unsymmetrical π-network: Impedance of series arm on both side is not equal i.e. net shunt impedance is (Z2‖Z3) (c)

L-network When one series arm of unsymmetrical T-network has zero value it looks like an Lnetwork. Z1

Input

Z2

Output

L-network

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

1

5. Two Port Network and Network Functions (d)

Lattice network When two diagonal arm/cross arms are connected with one series arm it looks like a Lattice network. Usually lattice networks are represented by two ways. Z1

Z

Input

Input

Output Z

Z

Output Z3

Z4

Z

Z2

Symmetrical lattice network

Unsymmetrical lattice network

Symmetrical lattice network: Impedance of diagonal arm/cross arms and series arm are equal. Unsymmetrical lattice network: Impedance of diagonal arm/cross arms and series arm are not equal. (e)

Bridge T-network When series arm of T-network is bridged by shunt arm, it looks like a Bridge T-network. Usually Bridge T-networks are represented by two ways. Z3/2 Z1/4

Z3 Z1/2

Z1/2

Input

Z2

Z1/4

Input

Output

Z2

Z1/4

Output

Z1/4

Z3/2 Unbalance bridge T-network

(f)

Balance bridge T-network

Ladder network A cascade of repeated section of network is called ladder network. Z1

Input

Z1

Z1

Z2 Z1

Z1

Z2 Z1

Z1

Z2 Z1

Z1

Output

Z1

Ladder network

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

2

5. Two Port Network and Network Functions 5.2. Open circuit impedance parameters (Z-parameters) I1

I1

Z22

I2

I2 V1

V2

V1



Z11

Z12I2

Z21I1

V2

When current I1 and I2 are selected as independent variable and V1 and V2 are dependent variable, network can be characterized by following set of equation.

V1  Z11I1  Z12I2 V2  Z21I1  Z22I2 V 1   Z11  V   Z  2   21

Z12   I1  Z22   I2 

Where,

Z11  Input driving point impedance Z22  Output driving point impedance Z12  Reverse transfer impedance Z21  Forward transfer impedance 

When output of two port network is open circuited i.e. I2 = 0

V1  Z11 I1  Z12(0)

V2  Z21 I1  Z22(0) 

 Z21 

V1 I1

I2 0

V2 I1

I2 0

When input of two port network is open circuited i.e. I1 = 0

V1  Z11 (0)  Z12I2

V2  Z21 (0)  Z22I2 

 Z11 

 Z12 

 Z22 

V1 I2

I1 0

V2 I2

I1 0

Condition for symmetry Network is said to be symmetrical if two port can be interchanged without affecting port voltage and current. It is possible when

V1 I1 

 I2 0

V2 I2

 Z11  Z22 I1 0

Condition for reciprocity Network is said to be reciprocal when I1’ = I2’ i.e. Z12=Z21

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

3

5. Two Port Network and Network Functions I1

I2

I1

V2

V1=VS

I2

I1

I2 V2=VS

V1

From circuit  V1  Vs , V2  0, I2  -I2'

From circuit  V1  0, V2  Vs , I1  -I1'

V1  Z11 I1  Z12I2

 VS  Z11 I1  Z12 (-I2' )

V1  Z11 I1  Z12I2

 0  Z11 (-I1' )  Z12I2

V2  Z21 I1  Z22I2

 0  Z 21 I1  Z 22(-I2' )

V2  Z21 I1  Z22I2

 VS  Z21 (-I1' )  Z 22I2

Z 21    I2'   VS Z Z Z Z  11 22 21 12 

Shital Patel, EE Department

Z12    I1'   VS Z Z Z Z  11 22 21 12 

Electrical Circuits Analysis (3130906)

4

5. Two Port Network and Network Functions 5.3. Short circuit admittance parameters (Y-parameters) I1 I1

I2

V1

V2

V1



I2 Y11

Y12V2

Y21V1

Y22

V2

When voltage V1 and V2 are selected as independent variable and I1 and I2 are dependent variable, network can be characterized by following set of equation.

I1  Y11V1  Y12V2 I2  Y21V1  Y22V2  I 1  Y11 Y12  V1   I   Y Y  V   2   21 22   2  Where,

Y11  Input driving point admittance Y22  Output driving point admittance Y12  Reverse transfer admittance Y21  Forward transfer admittance 

When output of two port network is short circuited i.e. V2 = 0

I1  Y11V1  Y12(0)

I2  Y21V1  Y22(0) 

 Y21 

I1 V1

V2 0

I2 V1

V2 0

When input of two port network is short circuited i.e. V1 = 0

I1  Y11 (0)  Y12V2

I2  Y21 (0)  Y22V2 

 Y11 

 Y12 

 Y22 

I1 V2

V1 0

I2 V2

V1 0

Condition for symmetry Network is said to be symmetrical if two port can be interchanged without affecting port voltage and current. It is possible when

I1 V1 

 V2 0

I2  Y11  Y22 V2 V 0 1

Condition for reciprocity Network is said to be reciprocal when I1’ = I2’ i.e. Y12=Y21

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

5

5. Two Port Network and Network Functions I1

I2

I1

V2

V1=VS

I2

I1

V1

I2 V2=VS

From circuit  V1  Vs , V2  0, I2  -I2' I1  Y11V1  Y12V2

 I1  Y11 VS   Y12(0)

From circuit  V1  0, V2  Vs , I1  -I1' I1  Y11V1  Y12V2

 -I1'  Y11 (0)  Y12 Vs 

I2  Y21V1  Y22V2

 -I2'  Y21 VS   Y22(0)

I2  Y21V1  Y22V2

 I2  Y21 (0)  Y22 Vs 

 I2'  Y21VS

Shital Patel, EE Department

 I1'  Y12VS

Electrical Circuits Analysis (3130906)

6

5. Two Port Network and Network Functions 5.4. Relation between Z- parameters and Y-parameters 

For many network problem solution, it becomes necessary to convert one set of parameters to another set for the ease of mathematics.



Let, set of Z-parameters and Y-parameters are, Z-parameters to Y-parameters V1  Z11I1  Z12I2

Y-parameters to Z-parameters I1  Y11V1  Y12V2

V2  Z21I1  Z22I2

I2  Y21V1  Y22V2

V 1   Z11 Z12   I1   V   Z    2   21 Z22   I2  Solving Z- parameters for I1 and I2  I1  1   V1 Z12  V Z22   2  Z11 Z12  Z   21 Z22  

V1 Z22  V2 Z12 Z11 Z22  Z12 Z21



Z22 Z12      V1     V2  Z11 Z22  Z12 Z21   Z11 Z22  Z12 Z21   V1  2   Z11 V1  Z V2   21   Z11 Z12  Z   21 Z22  

 I 1  Y11 Y12  V1   I   Y Y  V   2   21 22   2  Solving Y- parameters for V1 and V2  V1  1   I1 Y12  I Y  22   2 Y Y  11 12  Y   21 Y22 

V2 Z11  V1 Z21 Z11 Z22  Z12 Z21

I1Y22  I2Y12 Y11Y22  Y12Y21

Y22 Y12      I1      I2  Y11Y22  Y12Y21   Y11Y22  Y12Y21   V1  2  Y11 I1  Y I2   21  Y11 Y12  Y   21 Y22  

I2Y11  I1Y21 Y11Y22  Y12Y21

Y21 Y11 Z21 Z11          I1    V1   V2   I2    Y11Y22  Y12Y21   Y11Y22  Y12Y21   Z11 Z22  Z12 Z21   Z11 Z22  Z12 Z21  Comparing these equation with the equation Comparing these equation with the equation of Y-parameters of Z-parameters Z 22 Y22 Y11  Z11  Z11 Z 22  Z12 Z 21 Y11Y22  Y12Y21

Y12  

Z12 Z11 Z 22  Z12 Z 21

Z12  

Y12 Y11Y22  Y12Y21

Y21  

Z 21 Z11 Z 22  Z12 Z 21

Z 21  

Y21 Y11Y22  Y12Y21

Y22 

Z11 Z11 Z 22  Z12 Z 21

Shital Patel, EE Department

Z 22 

Y11 Y11Y22  Y12Y21

Electrical Circuits Analysis (3130906)

7

5. Two Port Network and Network Functions 5.5. Hybrid parameters (h-parameters) I1 I1

I2

I2

V1

V2

V1



h11

h12V2

h21I1

h22

V2

When current I1 and voltage V2 are selected as independent variable and voltage V1 and current I2 are dependent variable, network can be characterized by following set of equation.

V1  h11 I1  h12V2 I2  h21 I1  h22V2 V 1  h11 h12   I1   I   h    2   21 h22  V2  Where,

h11  Short circuit input impedance h22  Open circuit output admittance h12  Open circuit reverse voltage gain h21  Short circuit forward current gain 

When output of two port network is short circuited i.e. V2 = 0

V1  h11 I1  h12  0

I2  h21I1  h22  0 

V1 I1

V2 0

I2 I1

V2 0

 h21 

When input of two port network is open circuited i.e. I1 = 0

V1  h11  0  h12V2

I2  h21  0  h22V2 

 h11 

 h12 

 h22 

V1 V2

I1 0

I2 V2

I1 0

Condition for symmetry Network is said to be symmetrical if two port can be interchanged without affecting port voltage and current. It is possible when

V1 I1

 I2 0

V2 I2

 h11h22  h12h21  1 I1 0

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

8

5. Two Port Network and Network Functions Let, I2  h21 I1  h22V2

Let, I2  h21 I1  h22V2 I2  h21  0  h22V2

0  h21 I1  h22V2 V2  

h21 I1 h22



V1  h11 I1  h12V2

V2 I2

 I1 0

1 h22

Condition for symmetry,

 h  V1  h11 I1  h12   21 I1   h22 

V1 I1

 h h h h  V1   11 22 12 21  I1 h22   V h h h h  1  11 22 12 21 I1 I 0 h22



 I2 0

V2 I2

I1 0

h11h22  h12h21 1  h22 h22

 h11h22  h12h21  1

2



Condition for reciprocity Network is said to be reciprocal when I1’ = I2’ i.e. h12= -h21

I1

I2

I1

V2

V1=VS

I2

From circuit  V1  Vs , V2  0, I2  -I2'

I1

V1

I2 V2=VS

From circuit  V1  0, V2  Vs , I1  -I1'

V1  h11 I1  h12V2

 Vs  h11I1  h12(0)

V1  h11 I1  h12V2

  0  h11  -I1'   h12 Vs 

I2  h21 I1  h22V2

 -I2'  h21 I1  h22(0)

I2  h21 I1  h22V2

 I2  h21  -I1'   h22 Vs 

 I2'  

h21 Vs h11

Shital Patel, EE Department

 I1' 

h12 VS h11

Electrical Circuits Analysis (3130906)

9

5. Two Port Network and Network Functions 5.6. Relation between h-parameters and Z- parameters 

For many network problem solution, it becomes necessary to convert one set of parameters to another set for the ease of mathematics.



Let, set of Z-parameters, and h-parameters are, h-parameters to Z-parameters V1  h11 I1  h12V2

Z-parameters to h-parameters V1  Z11I1  Z12I2

I2  h21 I1  h22V2

V2  Z21I1  Z22I2

V 1  h11 h12   I1   I   h    2   21 h22  V2  Solving h- parameters for V1 and V2 I2  h21 I1  h22V2

V2  

h21 1 I1  I2 h22 h22

V1  h11 I1  h12V2 1   h  h11 I1  h12   21 I1  I2 h22   h22 h h h  h11 I1  12 21 I1  12 I2 h22 h22

V 1   Z11  V   Z  2   21

Z12   I1  Z22   I2 

Solving Z-parameters for V1 and I2 V2  Z 21 I1  Z 22I2

 I2  

Z 21 1 I1  V2 Z 22 Z 22

V1  Z11 I1  Z12I2 1  Z   Z11 I1  Z12   21 I1  V2  Z 22   Z 22 Z Z Z  Z11 I1  12 21 I1  12 V2 Z 22 Z22

h Z Z Z Z Z   h h h h    11 22 12 21  I1  12 V2   11 22 12 21  I1  12 I2 h22 Z 22 h22 Z 22     Comparing these equation with the equation Comparing these equation with the equation of Z-parameters of h-parameters h h h h Z Z Z Z Z11  11 22 12 21 h11  11 22 12 21 h22 Z 22

Z12 

h12 h22

Z 21   Z 22 

h21 h22

1 h22

Shital Patel, EE Department

h12 

Z12 Z 22

h21   h22 

Z 21 Z 22

1 Z 22

Electrical Circuits Analysis (3130906)

10

5. Two Port Network and Network Functions 5.7. Relation between h-parameters and Y-parameters 

For many network problem solution, it becomes necessary to convert one set of parameters to another set for the ease of mathematics.



Let, set of Y-parameters and h-parameters are, h-parameters to Y-parameters V1  h11 I1  h12V2

Y-parameters to h-parameters I1  Y11V1  Y12V2

I2  h21 I1  h22V2

I2  Y21V1  Y22V2

V 1  h11 h12   I1   I   h    2   21 h22  V2  Solving h- parameters for I1 and I2 V1  h11 I1  h12V2

 I1 

h 1 V1  12 V2 h11 h11

I2  h21 I1  h22V2 h  1   h21  V1  12 V2   h22V2 h11   h11 h h h  21 V1  12 21 V2  h22V2 h11 h11 

h21  h h h h  V1   11 22 12 21 V2 h11 h11  

 I 1  Y11 Y12  V1   I   Y Y  V   2   21 22   2  Solving Y-parameters for V1 and I2 I1  Y11V1  Y12V2 V1 

Y 1 I1  12 V2 Y11 Y11

I2  Y21V1  Y22V2 Y  1   Y21  I1  12 V2   Y22V2 Y11   Y11 Y Y Y  21 I1  12 21 V2  Y22V2 Y11 Y11 

Y21  Y Y Y Y  I1   11 22 12 21 V2 Y11 Y11  

Comparing these equation with the equation Comparing these equation with the equation of Y-parameters of h-parameters 1 1 Y11  h11  Y11 h11

Y12  

h12 h11

h12  

Y12 Y11

Y21 

h21 h11

h21 

Y21 Y11

Y22 

h11h22  h12h21 h11

h22 

Y11Y22  Y12Y21 Y11

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

11

5. Two Port Network and Network Functions 5.8. Transmission parameters (ABCD parameters) I1

I2 V2

V1



When voltage V2 and current I2 are selected as independent variable and voltage V1 and current I1 are dependent variable, network can be characterized by following set of equation.

V1  AV2  BI2 I1  CV2  DI2 V 1   A B   V2  I     1   C D    I2  Where,

A  Open circuit reverse voltage gain B  Short circuit transfer impedance C  Open circuit transfer admittance D  Short circuit reverse current gain 

When output of two port network is open circuited i.e. I2 = 0

V1  AV2  B  0

I1  CV2  D  0 

C 

V1 V2

I2 0

I1 V2

I2 0

When output of two port network is short circuited i.e. V2 = 0

V1  A  0  BI2

B

I1  C  0  DI2 

 A

D

V1  I2

V2 0

I1  I2

V2 0

Condition for symmetry Network is said to be symmetrical if two port can be interchanged without affecting port voltage and current. It is possible when

V1 I1

 I2 0

V2 I2

 AD I1 0

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

12

5. Two Port Network and Network Functions Let,

Let, V1  AV2  BI2  0  CV2  DI2

V1  AV2  B  0 I1  CV2  D  0 

V1 I1

 I2 0

CV2  DI2

AV2 A  CV2 C



V2 I2

 I1 0

D C

Condition for symmetry, V1 V  2 I1 I 0 I2 I 0 2

1

A D   C C A  D 

Condition for reciprocity Network is said to be reciprocal when I1’ = I2’ i.e. AD-BC=1

I1

I2

I1

V2

V1=VS

I2

From circuit  V1  Vs , V2  0, I2  -I2'

I1

V1

I2 V2=VS

V1  AV2  BI2

 Vs  A  0  B(-I2' )

From circuit  V1  0, V2  Vs , I1  -I1' V1  AV2  BI2   0  A(Vs )  BI2

I1  CV2  DI2

 I1  C  0  D(-I2' )

I1  CV2  DI2

1  I2'  VS B

 -I1'  C(Vs )  DI2

 AV   AD  BC   I1'  C(Vs )  D  s    Vs  B   B 

Condition for reciprocity, I1'  I2'

1  AD  BC   VS   Vs B  B   AD  BC  1

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

13

5. Two Port Network and Network Functions 5.9. Relation between ABCD parameters and Z-parameters 

For many network problem solution, it becomes necessary to convert one set of parameters to another set for the ease of mathematics.



Let, set of ABCD parameters and Z-parameters are, ABCD parameters to Z-parameters V1  AV2  BI2

Z-parameters to ABCD parameters V1  Z11I1  Z12I2

I1  CV2  DI2

V2  Z21I1  Z22I2

V 1   A B   V2  I     1   C D    I2  Solving ABCD parameters for V1 and V2 I1  CV2  DI2 1 D I1  I2 C C V1  AV2  BI2 V2 

D  1  A  I1  I2   BI2 C  C 

A AD I1  I2  BI2 C C

V 1   Z11  V   Z  2   21

Z12   I1  Z22   I2 

Solving Z-parameters for V1 and I1 V2  Z21 I1  Z22I2

 I1 

Z 1 V2  22 I2 Z21 Z21

V1  Z11 I1  Z12I2 Z  1   Z11  V2  22 I2   Z12I2 Z21   Z21 Z Z Z  11 V2  11 22 I2  Z12I2 Z21 Z21

Z Z Z Z Z  A  AD  BC   11 V2   11 22 12 21  I2 I1    I2 Z21 Z21 C C     Comparing these equation with the equation Comparing these equation with the equation of Z-parameters of ABCD parameters A Z Z11  A  11 C Z 21 AD  BC Z Z Z Z Z12  B  11 22 12 21 C Z 21 1 Z 21  1 C C Z 21 D Z 22  Z C D  22 Z 21



Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

14

5. Two Port Network and Network Functions 5.10. Relation between ABCD parameters and Y-parameters 

For many network problem solution, it becomes necessary to convert one set of parameters to another set for the ease of mathematics.



Let, set of ABCD parameters and Y-parameters are, ABCD parameters to Y-parameters V1  AV2  BI2

Y-parameters to ABCD parameters I1  Y11V1  Y12V2

I1  CV2  DI2

I2  Y21V1  Y22V2

V 1   A B   V2  I     1   C D    I2  Solving ABCD parameters for I1 and I2 V1  AV2  BI2 1 A  I2   V1  V2 B B I1  CV2  DI2 A   1  CV2  D   V1  V2   B B  

D AD V1  CV2  V2 B B

 I 1  Y11 Y12  V1   I   Y Y  V   2   21 22   2  Solving Y-parameters for V1 and I1 I2  Y21V1  Y22V2 V1  

Y22 1 V2  I2 Y21 Y21

I1  Y11V1  Y12V2 1   Y  Y11   22 V2  I2  Y12V2 Y21   Y21 Y Y Y   11 22 V2  11 I2  Y12V2 Y21 Y21

Y  Y Y Y Y  D  AD  BC     11 22 12 21 V2  11 I2 V1   V2 Y21 Y21 B  B    Comparing these equation with the equation Comparing these equation with the equation of Z-parameters of ABCD parameters D Y Y11  A   22 B Y21 AD  BC 1 Y12   B B Y21 1 Y21   Y Y Y Y B C   11 22 12 21 Y21 A Y22  Y B D   11 Y21



Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

15

5. Two Port Network and Network Functions 5.11. Relation between ABCD parameters and h-parameters 

For many network problem solution, it becomes necessary to convert one set of parameters to another set for the ease of mathematics.



Let, set of ABCD parameters and h-parameters are, ABCD parameters to h-parameters V1  AV2  BI2

h-parameters to ABCD parameters V1  h11 I1  h12V2

I1  CV2  DI2

I2  h21 I1  h22V2

V 1   A B   V2  I     1   C D    I2  Solving ABCD parameters for V1 and I2 I1  CV2  DI2 1 C  I2   I1  V2 D D V1  AV2  BI2 C   1  AV2  B   I1  V2   D D   AV2 

B BC I1  V2 D D

V 1  h11 h12   I1   I   h    2   21 h22  V2  Solving h-parameters for V1 and I1 I2  h21 I1  h22V2

 I1  

h22 1 V2  I2 h21 h21

V1  h11 I1  h12V2 1   h  h11   22 V2  I2  h12V2 h21   h21 h h h   11 22 V2  11 I2  h12V2 h21 h21

h  h h h h     11 22 12 21 V2  11 I2 B  AD  BC  I1   V h21 h21  2   D  D  Comparing these equation with the equation Comparing these equation with the equation of ABCD parameters of h-parameters h h h h B A   11 22 12 21 h11  h21 D h AD  BC B   11 h12  h21 D 1 h h21   C   22 D h21 C 1 h22  D D h 

21

Shital Patel, EE Department

Electrical Circuits Analysis (3130906)

16

5. Two Port Network and Network Functions Parameter Conversion [Z]

[Z]

 Z11 Z  21

[Y]

Z22   Z Z Z Z  11 22 12 21  Z21   Z11 Z22  Z12 Z21

[Y]

Y22   Y Y Y Y  11 22 12 21  Y21   Y11Y22  Y12Y21

Z12  Z 22 

Z12  Z11 Z22  Z12 Z21    Z11  Z11 Z22  Z12 Z21 



[h]

 Z11 Z22  Z12 Z21  Z22   Z  21  Z22 

[T]

 Z11 Z  21  1   Z21

Z12  Z22   1   Z22 

Z11 Z22  Z12 Z21   Z21   Z22  Z21 

Shital Patel, EE Department

[h]

Y12  Y11Y22  Y12Y21    Y11  Y11Y22  Y12Y21 



1 Y  11  Y21   Y11

Y12 Y11

   Y11Y22  Y12Y21   Y11  

Y   22  Y21   Y11Y22  Y12Y21  Y21 

 h11h22  h12h21  h22   h  21  h22  1 h  11  h21   h11

Y11 Y12  Y   21 Y22 

1  Y21   Y11    Y21 



[T]

h12  h22   1   h22 

A C  1  C

   h11h22  h12h21   h11 

 D  B   1  B



 h11 h  21

h12 h11

 B  D   1  D

h12  h22 

 h11h22  h12h21  h21   h  22  h21 

h11  h21   1    h21 



AD  BC   C  D   C 

AD  BC   B  A   B

AD  BC   D  C   D A B C D  

Electrical Circuits Analysis (3130906)

17