Source Transformation - KOCWcontents.kocw.net/KOCW/document/2015/korea_sejong/... · 2016-09-09 ·...

CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]

Chapter 9 Sinusoidal Steady-State Analysis 66 9.7 Source Transformations and Thevenin-Norton Equivalent Circuits

Source Transformation

Source transformation in the frequency domain involves transforming a voltage source in series with an impedance to a current source in parallel with an impedance, or vice versa.

As we go from one source type to another, we must keep the following relationship in mind:


Chapter 9 Sinusoidal Steady-State Analysis 67 9.7 Source Transformations and Thevenin-Norton Equivalent Circuits

Thevenin and Norton Equivalent Circuits

Thevenin’s and Norton’s theorems are applied to ac circuits in the same way as they are to dc circuits.

The only additional effort arises from the need to manipulate complex numbers. The frequency-domain version of a Thevenin equivalent circuit is depicted in Fig. 9.25, where a linear circuit is replaced by a voltage source in series with an impedance.

The Norton equivalent circuit is illustrated in Fig.9.26, where a linear circuit is replaced by a current source in parallel with an impedance.

Keep in mind that the two equivalent circuits are related as


Chapter 9 Sinusoidal Steady-State Analysis 68 9.8 The Node-Voltage Method

Nodal Analysis

The basis of nodal analysis is Kirchhoff’s current law. Since KCL is valid for phasors, we can analyze ac circuits by nodal analysis. The following examples illustrate this.


Chapter 9 Sinusoidal Steady-State Analysis 69 9.9 The Mesh-Current Method

Mesh Analysis

Kirchhoff’s voltage law (KVL) forms the basis of mesh analysis. The validity of KVL for ac circuits was shown in Section 9.5 and is illustrated in the following example.


Chapter 9 Sinusoidal Steady-State Analysis 70 9.10 The Transformer

Transformer

A transformer is generally a four-terminal device comprising two (or more) magnetically coupled coils.

Primary winding

Secondary winding

• The primary circuit is the circuit that connected to the input energy source.

• The function of the core is to transfer the changing magnetic flux from the primary coil to the secondary coil.

• The secondary circuit is the circuit that connected to the output of the transformer.



Transformer

Functions of Transformers

• In communication circuits, the transformer is used to match impedances and eliminate dc signals from portion of the system.

• In power circuits, transformers are used to establish ac voltage levels that facilitate the transmission, distribution, and consumption of electric power.

• The linear transformer is found primarily in communication circuits: They are used in radio and TV sets.

• The ideal transformer is used to model the ferromagnetic transformer found in power systems.

Types of Transformers



Linear Transformer

A simple linear transformer is formed when two coils are wound on a single core to ensure magnetic coupling.

The transformer is said to be linear if the coils are wound on a magnetically linear material — a material for which the magnetic permeability is constant. Such materials include air, plastic, Bakelite, and wood. In fact, most materials are magnetically linear.

Linear transformers are sometimes called air-core transformers, although not all of them are necessarily air-core. They are used in radio and TV sets.



• The transformer circuit parameters are

Primary

winding Secondary

winding

𝑅1 : the resistance of the primary winding

𝑅2 : the resistance of the secondary winding

𝐿1 : the self-inductance of the primary winding

𝐿2 : the self-inductance of the secondary winding

𝑀 : the mutual inductance

Circuit Model of a Linear Transformer

Core: no magnetic material



• Applying KVL to the two meshes in Fig.9.38 gives

The Analysis of a Linear Transformer Circuit

𝐕𝑠 = 𝑅1𝐈1 + 𝑗𝜔𝐿1𝐈1 − 𝑗𝜔𝑀𝐈2

= (𝑅1 + 𝑗𝜔𝐿1)𝐈1 − 𝑗𝜔𝑀𝐈2 (9.57)

0 = 𝑅2𝐈2 + 𝑗𝜔𝐿2𝐈2 + Z𝐿𝐈2 − 𝑗𝜔𝑀𝐈1

= (𝑅2 + 𝑗𝜔𝐿2 + Z𝐿)𝐈2 − 𝑗𝜔𝑀𝐈1 (9.58)

• If we let

𝑍11 = 𝑅1 + 𝑗𝜔𝐿1

𝑍22 = 𝑅2 + 𝑗𝜔𝐿2 + Z𝐿

𝐕𝑠 = Z11𝐈1 − 𝑗𝜔𝑀𝐈2

0 = 𝑍22𝐈2 − 𝑗𝜔𝑀𝐈1

Eqs.(9.57) and (9.58) become

(9.59)

(9.60)



• The solutions for I1 and I2 from the Eqs. above

The Analysis of a Linear Transformer Circuit

𝐕𝑠 = Z11𝐈1 − 𝑗𝜔𝑀𝐈2

0 = 𝑍22𝐈2 − 𝑗𝜔𝑀𝐈1

(9.61)

(9.62)

𝐈1 =𝑍22

𝑍11𝑍22 + 𝜔2𝑀2 𝐕𝑠

𝐈2 =𝑗𝜔𝑀

𝑍11𝑍22 + 𝜔2𝑀2 𝐕𝑠 =𝑗𝜔𝑀

𝑍22𝐈1

• The input impedance at the terminal a-b, is

𝑍𝑖𝑛 = 𝑍𝑎𝑏 =𝐕𝑠𝐈1

=𝑍11𝑍22 +𝜔2𝑀2

𝑍22= 𝑍11 +

𝜔2𝑀2

𝑍22

𝑍11 = 𝑅1 + 𝑗𝜔𝐿1

𝑍22 = 𝑅2 + 𝑗𝜔𝐿2 + Z𝐿

= 𝑅1 + 𝑗𝜔𝐿1 +𝜔2𝑀2

𝑅2 + 𝑗𝜔𝐿2 + Z𝐿

Notice that the input impedance is independent of the magnetic polarity of the transformer and comprises two terms.

(9.63)

(9.64)



Reflected Impedance

(9.64)

𝑍11 = 𝑅1 + 𝑗𝜔𝐿1

𝑍22 = 𝑅2 + 𝑗𝜔𝐿2 + Z𝐿

𝑍𝑖𝑛 = 𝑍𝑎𝑏 = 𝑅1 + 𝑗𝜔𝐿1 +𝜔2𝑀2


• The primary impedance

𝑅1 + 𝑗𝜔𝐿1

𝑍𝑅 =𝜔2𝑀2


• The reflected impedance

The second term is due to the coupling between the primary and secondary windings.

It is as though this impedance is reflected to the primary.

If we express the load impedance in rectangular form,

𝑍𝐿 = 𝑅𝐿 + 𝑗𝑋𝐿


𝑅2 + 𝑗𝜔𝐿2 + (𝑅𝐿 + 𝑗𝑋𝐿) =

𝜔2𝑀2

(𝑅2 + 𝑅𝐿) + 𝑗(𝜔𝐿2 + 𝑋𝐿)

(9.65)



Reflected Impedance

𝑍11 = 𝑅1 + 𝑗𝜔𝐿1 𝑍22 = 𝑅2 + 𝑗𝜔𝐿2 + Z𝐿


𝑅2 + 𝑗𝜔𝐿2 + (𝑅𝐿 + 𝑗𝑋𝐿) =

𝜔2𝑀2

(𝑅2 + 𝑅𝐿) + 𝑗(𝜔𝐿2 + 𝑋𝐿)

=𝜔2𝑀2 𝑅2 + 𝑅𝐿 − 𝑗(𝜔𝐿2 + 𝑋𝐿)

(𝑅2 + 𝑅𝐿)2+(𝜔𝐿2 + 𝑋𝐿)

2

=𝜔2𝑀2 𝑅2 + 𝑅𝐿 − 𝑗(𝜔𝐿2 + 𝑋𝐿)

(𝑅2 + 𝑅𝐿)2+(𝜔𝐿2 + 𝑋𝐿)

2

=𝜔2𝑀2 𝑅2 + 𝑅𝐿 − 𝑗(𝜔𝐿2 + 𝑋𝐿)

𝑍222

𝑍𝐿 = 𝑅𝐿 + 𝑗𝑋𝐿

=𝜔2𝑀2

𝑍222 𝑅2 + 𝑅𝐿 − 𝑗(𝜔𝐿2 + 𝑋𝐿)

The self-impedance of the secondary circuit is reflected into the primary circuit by a scaling factor of 𝜔𝑀 𝑍22 2, and that the sign of the reactive component is reversed.

Thus the linear transformer reflects the conjugate of the self-impedance of the secondary circuit into the primary winding by a scalar multiplier.

(9.66)

𝑍22 = 𝑅2 + 𝑗𝜔𝐿2 + 𝑅𝐿 + 𝑗𝑋𝐿

𝑍22∗ = 𝑅2 + 𝑅𝐿 − 𝑗(𝜔𝐿2 + 𝑋𝐿)


Chapter 9 Sinusoidal Steady-State Analysis 78 9.11 The Ideal Transformer

Ideal Transformer

An ideal transformer consists of two (or more) coils with a large number of turns wound on a common core of high permeability.

Because of this high permeability of the core, the flux links all the turns of both coils, thereby resulting in a perfect coupling.

An deal transformer has three properties:

(a) Ideal transformer, (b) circuit symbol for ideal

transformers.

Iron-core transformers are close approximations to ideal transformers. These are used in power systems and electronics.

• The coefficient of coupling is unity(𝑘 = 1)

• The self-inductance of each coil is infinite (𝐿1 = 𝐿2 = ∞)

• The coil losses due to parasitic resistance are negligible (𝑅1 = 0 = 𝑅2).



Determining of the Voltage Ratio


𝐕1 = 𝑗𝜔𝐿1𝐈1 (1)

𝐕2 = 𝑗𝜔𝑀𝐈1 (2)

• From Eq.(1),

𝐈1 =𝐕1𝑗𝜔𝐿1

(3)

• Substituting this in Eq.(2) gives

𝐕2 = 𝑗𝜔𝑀𝐈1 = 𝑗𝜔𝑀𝐕1𝑗𝜔𝐿1

=𝑀𝐕1𝐿1

• But 𝑀 = 𝐿1𝐿2 for perfect coupling (k=1). Hence,

(4) 𝐕2 =

𝐿2𝐿1

𝐕1 = 𝑎𝐕1 or 𝐕2𝐕1

= 𝑎 =𝑁2

𝑁1

𝑎 =𝑁2

𝑁1=

𝐿2𝐿1

(5)

: Turns ratio or transformation ratio.




𝐕2𝐕1

= 𝑎 =𝑁2

𝑁1

By choosing the turns ratio, we now have the ability to change any ac voltage to any other ac voltage.

• 𝑎 = 1 : we generally call the transformer an isolation transformer

• 𝑎 > 1 : a step-up transformer, as the voltage is increased from primary

to secondary (V2 > V1).

• 𝑎 < 1 : a step-down transformer, since the voltage is decreased from primary to

secondary (V2 < V1 ).

The ratings of transformers are usually specified as V1/V2.

A transformer with rating 2400/120 V should have 2400 V on the primary and 120 in the secondary (i.e., a step-down transformer). Keep in mind that the voltage ratings are in rms.

(5)




step-up transformer step-down transformer

http://www.google.co.kr/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRw&url=http://www.askamathematician.com/2011/07/q-can-light-be-used-to-transfer-energy-instead-of-power-lines/&ei=P0cbVaKSC4bc8AXLj4CIBQ&psig=AFQjCNGpdeP_9bdj_ufKdE_UDojlLXeCgQ&ust=1427937405268337



Determining of the Current Ratio


𝐕1 = 𝑗𝜔𝐿1𝐈1 − 𝑗𝜔𝐿1𝐈1

0 = −𝑗𝜔𝑀𝐈1 + 𝑗𝜔𝐿2𝐈2

• From Eq.(2),

(1)

(2)

𝐈2𝐈1=𝑗𝜔𝑀

𝑗𝜔𝐿2=

𝐿1𝐿2

=1

𝑎

𝐈2𝐈1=𝑁1𝑁2

or

𝑁1𝐈1 = 𝑁2𝐈2 𝑀 = 𝐿1𝐿2 for perfect coupling (k=1).

(3)

(4)

(5)

(9.85)

(9.86)

(9.85)

(9.84)



Determining the Polarity of the Voltage and Current Ratios

It is important that we know how to get the proper polarity of the voltages and the direction of the currents for the transformer.

If the polarity of V1 or V2 or the direction of I1 or I2 is changed, 𝑎 in Eqs. (1), (2) may need to be replaced by − 𝑎.

The two simple rules to follow are:

Relating primary and secondary quantities in an ideal transformer.

𝐕2𝐕1

= 𝑎

𝐈2𝐈1

=1

𝑎

(1)

(2)

• If V1 and V2 are both positive or both negative at the dotted terminals, use +𝑎 in Eq.(1). Otherwise, use −𝑎 .

• If I1 and I2 both enter into or both leave the dotted terminals, use −𝑎 in Eq.(2). Otherwise, use +𝑎 .

𝑎 =𝑁2

𝑁1



Applications: Transformer as an Isolation Device

Electrical isolation between two devices : transfer power without any physical connection between them.

In a transformer, energy is transferred by magnetic coupling, without electrical connection between the primary circuit and secondary circuit.

Three simple practical examples A transformer used to isolate an ac supply from a rectifier. (1) A transformer is often used to couple the ac supply to

the rectifier.

(2) A transformer is often used to couple two stages of an

amplifier, to prevent any dc voltage in one stage from

affecting the dc bias of the next stage.

(3) A transformer providing isolation between the power

lines and the voltmeter.

A transformer providing dc isolation between two amplifier stages

A transformer providing isolation between the power lines and the voltmeter



Applications: Transformer as an Impedance Matching Device

Input impedance

𝐕2𝐕1

= 𝑎 =𝑁2

𝑁1

𝐈2𝐈1=1

𝑎=𝑁1𝑁2

𝐙𝒊𝒏 =𝐕1𝐈1

=1

𝑎2𝐕2𝐈2

=𝐙𝐿𝑎2

The input impedance is also called the reflected impedance, since it appears as if the load impedance is reflected to the primary side.

This ability of the transformer to transform a given impedance into another impedance provides us a means of impedance matching to ensure maximum power transfer.



Applications: Transformer as an Impedance Matching Device

For maximum power transfer, the load resistance RL must be matched with the source resistance Rs.

In most cases, the two resistances are not matched; both are fixed and cannot be altered. However, an iron core transformer can be used to match the load resistance to the source resistance. This is called impedance matching.

For example, to connect a loudspeaker to an audio power amplifier requires a transformer, because the speaker’s resistance is only a few ohms while the internal resistance of the amplifier is several thousand ohms.

Transformer used as a matching device.


Chapter 9 Sinusoidal Steady-State Analysis 87 9.12 The Ideal Autotransformer

Ideal Autotransformers

Unlike the conventional two-winding transformer we have considered so far, an autotransformer has a single continuous winding with a connection point called a tap between the primary and secondary sides.

The tap is often adjustable so as to provide the desired turns ratio for stepping up or stepping down the voltage.

This way, a variable voltage is provided to the load connected to the autotransformer.

(a) Step-down autotransformer, (b) step-up autotransformer.

An autotransformer is a transformer in which both the primary and the secondary are in a single winding.

𝐕1𝐕2

=𝑁1 +𝑁2

𝑁2= 1 +

𝑁1𝑁2

For the step-down autotransformer circuit of Fig.(a),

𝐈1𝐈2=

𝑁2

𝑁1 + 𝑁2




(a) Step-down autotransformer, (b) step-up autotransformer.

𝐢1𝐢2=𝑁1 + 𝑁2

𝑁1= 1 +

𝑁2

𝑁1

For the step-up autotransformer circuit of Fig.(a),

𝐯1𝐕2

=𝑁2

𝑁1 +𝑁2

A major difference between conventional transformers and autotransformers is that the primary and secondary sides of the autotransformer are not only coupled magnetically but also coupled conductively.

The autotransformer can be used in place of a conventional transformer when electrical isolation is not required.




Autotransformers are often used to step up or step down voltages in the 110-115-120 V range and voltages in the 220-230-240 V range.

Examples :

- Provide 110 V or 120 V (with taps) from 230 V input, allowing equipment designed for 100 or 120 V to be used with a 230 V supply.

Korea electrical equipments:220 V

US electrical equipment: 110 V

European appliances: 230 V

In all cases the supply and the autotransformer must be correctly rated to supply the required power

http://www.google.co.kr/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRw&url=http://www.mpja.com/500VA-0-130VAC-Variable-Power-Transformer/productinfo/15162 TR/&ei=S3QbVcP_EMPRmAWMvYDIAw&psig=AFQjCNEdJDu_XA0BoPfxmFrK0AOSFKRkPg&ust=1427948948574160


Chapter 9 Sinusoidal Steady-State Analysis 90 9.13 Phasor Diagrams

Phasor Diagram

Re

Im

𝐕 𝐈

𝐈 =𝐕

1𝑗𝜔𝐶

= 𝑗𝜔𝐶𝐕

𝐈 =𝐕

𝑅

𝐈 =𝐕

𝑗𝜔𝐿= −𝑗

𝐕

𝜔𝐿


Chapter 9 Sinusoidal Steady-State Analysis 91 9.13 Phasor Diagrams

Impedance Diagram

𝑍 = 𝑅 + 𝑗𝑋

Re

Im

𝑅

𝑗𝜔𝐿

−𝑗1

𝜔𝐶

𝑍 = 𝑅

𝑍 = 𝑗𝜔𝐿

𝑍 =1

𝑗𝜔𝐿

90°

−90°

Source Transformation - KOCWcontents.kocw.net/KOCW/document/2015/korea_sejong/... · 2016-09-09 ·...

Documents

Transcript of Source Transformation - KOCWcontents.kocw.net/KOCW/document/2015/korea_sejong/... · 2016-09-09 ·...