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EE2301: Block A Unit 3 1

Block A Unit 3 outline

One port network> Thevenin transformation

> Norton transformation

> Source transformation

Two port network> Hybrid (H) parameters

> Reciprocity theorem

One port networks: G. Rizzoni, “Fundamental of EE”, Chapter 3.6 – 3.7


One-Port Network

Represented as:

One-port Network

+

v–

A one port network is simply a two terminal device (that we can think of as a block or black box), which can be described by its I-V (current-voltage) characteristic.


Thevenin’s Theorem

Any one-port network composed of ideal voltage and current sources, and linear resistors, can be represented by an equivalent circuit consisting of an ideal voltage source VT in series with an equivalent resistance RT

One-port network

+-VT

RT

This is an extremely power method is you want to find the load condition for maximum power transfer.

i.e. what is the RL you need to achieve maximum power transfer for a given one port network?


Norton’s Theorem

Any one-port network composed of ideal voltage and current sources, and linear resistors, can be represented by an equivalent circuit consisting of an ideal current source IN in parallel with an equivalent resistance RN

One-port network IN RN

This is an extremely power method is you want to find the load condition for maximum power transfer.

i.e. what is the RL you need to achieve maximum power transfer for a given one port network?


Finding RN or RT

Method Summary Remove the load (assumption: you need to know what the

load is). Never forget this step! Zero all independent voltage & current sources

> Short-circuit voltage sources (0V across source)

> Open-circuit current sources (0A through source) Calculate the total resistance seen across the load terminals

to obtain equivalent resistance RN or RT

> Combine resistors in series and parallel (REF previous unit)

Applies to both Thevenin & Norton


Zeroing the Sources

That is as if the source never existed

Vs

+

-

IN

Voltage Source

Current Source

No voltage

=> Short

No current

=> Open


Illustration of finding RN or RT

R1

R2

R3

RL+

-vs

Step 1: Remove load

R2+

-vs

R2

R1 R3

Step 2: Zero all independent sources

R1 R3

Step 3

Compute total resistance between the terminals

Thevenin Resistance: RT = R3 + R1||R2


Equivalent R: Example 1

+-VS

R1

R2

+-VS R1

R2

RthR1 R2RthR2

Rth = R1 || R2 Rth = R2


Equivalent R: Example 2

IS

R1

R2 IS R1

R2

RthR2 R1Rth

Rth = R2 Rth = R1 + R2


Finding Thevenin voltage VT

Method Summary Remove load (never forget this step) Define open circuit voltage Voc across the

open load terminals Solve for Voc using any preferred method to

obtain the Thevenin voltage VT = Voc

Definition of VT - The Thevenin equivalent voltage is equal to the open-circuit voltage present at the load terminals (with the load removed)


Finding Thevenin VT

R1

R2

R3

RL+

-vs

Step 1: Remove load!

R2+

-vs

R1 R3+

Voc

-

Step 2: Define open circuit voltage Voc across the open load terminalsStep 3: Solve for Voc using any preferred method

Note that current through R3 = 0 due to open circuit (ie no voltage drop across R3)

Therefore voc = voltage across R2

Voltage divider rule: voc = [R2 / (R1 + R2)] vs

*Thevenin voltage vT = voc


Thevenin example 1

Problem 3.55

Find the Thevenin equivalent circuit as seen by the resistor RL


Thevenin example 1 solution

This slide is meant to blank

1kΩ 1kΩ

1Ω 3Ω

Rth = 504Ω

Note that with RL removed, no current runs through 3Ω resistor

Mesh 1:

10 = 2000i1 - (1000)(0.01)

i1 = 10mA

Mesh 2:

-V = (1001)(0.01) - (1000)(0.01)

V = -10mV

Vth = V = -10V

+ -

504Ω

-10mV


Thevenin example 2

Problem 3.52

Find the voltage across the 3Ω resistor by replacing the remainder with the its Thevenin equivalent

-

V1

+


Thevenin example 2 solution


Equivalent resistance seen by 3Ω resistor is simply 4Ω resistor

Mesh 1:

V1 = 2(4+2) - i (2)

2i + V1 = 12

Mesh 2:

3 = i(2+2) - 2 (2)

i = 1.75A

V1 = 8.5V

Vth = (1.5)(2) - 8.5 = -5V

+ -

4Ω

-5V

V3Ω = 3/(3+4) * -5 = -5.14V


Finding Norton current IN

Method Summary Replace load with short circuit (SC) Define SC current isc to be Norton equivalent

current Solve for isc using any preferred method to

obtain IN = isc

Definition of IN - The Norton equivalent current is equal to the short-circuit current that would flow if the load were replaced by a short circuit


Finding Norton IN (1)

R1

R2

R3

RL+

-vs

Step 1: Short circuit load

R2+

-vs

R1 R3

Step 2: Define short circuit current isc

Step 3: Solve for isc using any preferred method

isc

sa

aaas

vRRRRRR

RRv

R

v

R

v

R

vv

133221

32

321

0

Apply KCL at node A:

3R

vi asc

A


Norton example 1

Problem 3.51 (modified)

Find the Norton equivalent circuit to the left of the 3Ω resistor


Norton example 1 solution


5Ω 4Ω

1Ω

RN = 3.22Ω

Norton current is given by the current through 1Ω resistor

V966.436

54||1

4||11

RV

A966.4111 RVII RRN


Norton example 2

Problem 3.53

Find the Norton equivalent circuit to the left of the 2Ω resistor


Norton example 2 solution


KCL at node 1:

637

1312

21

2111

VV

VVVV

KCL at node 2:

643

321

21

221

VV

VVV

Solve for V2: V2 = -1.263V

IN = V2 / 3 = -0.42A

1Ω 3Ω

1Ω 3Ω

Rth = 4.75Ω


Source Transformation

+

-

RT

RTVT IN

We can transform between the two equivalent circuits, observing each time that:

VT = IN RT

For example using one of the previous circuits as shown below:

R2+

-

R1 R3

VS

R2R1

R3

I1

VS and R1 form a Thevenin circuit which we can transform to a Norton circuit defined by current source I1 = VS/R1 and parallel resistance R1. Note that R1 and R2 and be combined.


Transformation illustration

R1||R2

R3

I1

+

-

R3

V1

The parallel combination of R1 & R2 (R1||R2) and current source I1 form a new Norton circuit (with a different value of parallel resistance than in the first instance), which can in turn be transformed back into a Thevenin circuit as follows. This Thevenin circuit comprises of a series resistor R1||R2, and a voltage source of V1 = I1(R1||R2).

R1||R2

Finally, it will now become obvious that this Thevenin resistor is in series with R3, and therefore can be easily combined. Therefore, in summary we see thatVT = V1 = [R2/(R1+R2)]VS and RT = R1||R2 + R3 (same result as before)

Note that V1 is NOT the same as VS. Hence the transformed circuit values are not to be confused with the original values – never make this mistake!

Source Transformation: Proof


Comes directly from the theorem:Thevenin to Norton – Any network can be composed of a current source in parallel with a resistor; this would therefore also include a Thevenin circuitNorton to Thevenin – Any network can be composed of a voltage source in series with a resistor; this would therefore also include a Norton circuit

Two-port network


Looking back

A port refers to a pair of terminals through which a current may enter or leave a network

We have focused only on one-port networks so far, where we consider the voltage across or current through a single pair of terminal

The rest of this unit deals with two-port networks

A two-port network is an electrical network with two separate ports for input and output

We will see examples of two-port networks (op amps and transistor circuits) later on in this course

Like for a one-port network, knowing the parameters of a two-port network enables use to treat it as a “black-box” placed within a larger network.

In a two-port network, we need to relate V1, V2, I1, I2

The terms relating these currents and voltages are known as parameters

Impedance parameters


As shown in the figure, a two-port network may be voltage-driven or current-driven.

The terminal voltages and currents represent 4 variables, of which two are independent.

The terminal voltages can be related to the terminal currents

V1 = z11I1 + z12I2

V2 = z21I1 + z22I2

These z terms are known as impedance parameters (since there are defined by V over I), or simply z parameters.

2

1

2221

1211

2

1

I

I

zz

Zz

V

V

It can also be expressed in matrix form:

The voltages V1 and V2 are the dependent variables in this case

Handling parameter subscripts


11: Input from port 1 and output back to port 1

21: Input from port 1 and output to port 2

12: Input from port 2 and output to port 1

22: Input from port 2 and output back to port 2

z parameters


The values of the parameters can be obtained by setting:

I1 = 0 (input port open-circuited) or

I2 = 0 (output port open-circuited)

Therefore:

0I2

222

0I1

221

0I2

112

0I1

111

12

12

I

Vz,

I

Vz

I

Vz,

I

Vz

z11: Open-circuit input impedance

z12: Open-circuit transfer impedance from port 1 to port 2

z21: Open-circuit transfer impedance from port 2 to port 1

z22: Open-circuit output impedance

Deriving z parameters


We can see from the definition of the z parameters, we obtain z11 and z21 by connecting V1 to port 1 and leaving port 2 open-circuited (I2 = 0)

[Referring to Figure (a)]:

z11 = V1/I1, z21 = V2/I1

Likewise, we obtain z12 and z22 by connecting V2 to port 2 and leaving port 1 open-circuited (I1 = 0)

[Referring to Figure (b)]:

z12 = V1/I2, z22 = V2/I2

z11 and z22 are known as driving-point impedances

z21 and z12 are known as transfer impedances

Z parameter example 1


Determine the Z-parameters for the following circuit

We first apply V1 at port 1 and open-circuit port 2 (I2 = 0) [Figure (a)]

z11 = V1/I1 = (20 + 40)I1/I1 = 60Ω

z21 = V2/I1 = 40I1/I1 = 40Ω

Next we apply V2 at port 2 and open-circuit port 1 (I1 = 0) [Figure (a)]

z12 = V1/I2 = 40I2/I2 = 40Ω

z22 = V2/I1 = (40 + 30)I2/I2 = 70Ω

Z parameter example 2


Determine the Z-parameters for the following circuit

z11 = 14, z12 = z21 = z22 = 6Ω

Hybrid parameters


The z parameters do not exist for all two-port networks, so there is a need to develop an alternative set of parameters to describe such types of two-port networks. This particular set of parameters is based on making V1 and I2 the dependent variables.

V1 = h11I1 + h12V2

I2 = h21I1 + h22V2

These h terms are known as hybrid parameters, since there are a hybrid (mix) of ratios, or simply h parameters.

This set of parameters is very useful for describing electronic devices such as transistors (which we will cover in Block C); namely because they are easier to measure compared to z or y parameters.

0I2

222

0V1

221

0I2

112

0V1

111

12

12

V

Ih,

I

Ih

V

Vh,

I

Vh

The h parameters are defined as:

h parameters


We can see that:

h11 represents an impedance; h12 represents a voltage ratio

h21 represents a current ratio; h22 represents an admittance

This is why they are known as hybrid (which means a mixture) parameters

h11: Short-circuit input impedance

h12: Open-circuit reverse voltage gain

h21: Open-circuit forward current gain

h22: Short-circuit output admittance

To find h11 and h21:

Short-circuit port 2 and apply I1 to port 1


Open-circuit port 1 and apply I2 to port 2

h parameter example 1


Find the h parameters for the following circuit.


Short-circuit port 2 and apply I1 to port 1 [Fig (a)]

h11 = V1/I1 = (2 + 3||6)I1/I1 = 4Ω

h21 = I2/I1 = -2/3 (current divider)


Open-circuit port 1 and apply V2 to port 2 [Fig (b)]

h12 = V1/V2 = 6/(6+3) = 2/3 (voltage divider)

h22 = I2/V2 = I2/(3+6)I2 = 1/9S

h parameter example 2


Find the h parameters for the following circuit.

h11 = 1.2Ω, h12 = 0.4, h21, = -0.4, h22 = 0.4S

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