 Source Transformation 

 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:

𝐕

= 𝐙

𝐈

𝐈

= 𝐕

𝐙

 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

𝐕

= 𝐙

𝐈

𝐙

= 𝐙

 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.

 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.

 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.

Figure 9.38 The frequency domain circuit model for a transformer used to connect a load to a source.

• 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

𝐕

= 𝑅

𝐈

+ 𝑗𝜔𝐿

𝐈

− 𝑗𝜔𝑀𝐈

= (𝑅

+ 𝑗𝜔𝐿

)𝐈

− 𝑗𝜔𝑀𝐈

0 = 𝑅

𝐈

+ 𝑗𝜔𝐿

𝐈

+ Z

𝐈

− 𝑗𝜔𝑀𝐈

= (𝑅

+ 𝑗𝜔𝐿

+ Z

)𝐈

− 𝑗𝜔𝑀𝐈

• If we let

𝑍

= 𝑅

+ 𝑗𝜔𝐿

𝑍

= 𝑅

+ 𝑗𝜔𝐿

+ Z

𝐕

= Z

𝐈

− 𝑗𝜔𝑀𝐈

0 = 𝑍

𝐈

− 𝑗𝜔𝑀𝐈

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 𝐕

= Z

𝐈

− 𝑗𝜔𝑀𝐈

0 = 𝑍

𝐈

− 𝑗𝜔𝑀𝐈

(9.61)

(9.62)

𝐈

= 𝑍

𝑍

𝑍

+ 𝜔

𝑀

𝐕

𝐈

= 𝑗𝜔𝑀

𝑍

𝑍

+ 𝜔

𝑀

𝐕

= 𝑗𝜔𝑀 𝑍

𝐈

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

𝑍

= 𝑍

= 𝐕

𝐈

= 𝑍

𝑍

+ 𝜔

𝑀

𝑍

= 𝑍

+ 𝜔

𝑀

𝑍

𝑍

= 𝑅

+ 𝑗𝜔𝐿

𝑍

= 𝑅

+ 𝑗𝜔𝐿

+ Z

= 𝑅

+ 𝑗𝜔𝐿

+ 𝜔

𝑀

𝑅

+ 𝑗𝜔𝐿

+ Z

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)

𝑍

= 𝑅

+ 𝑗𝜔𝐿

𝑍

= 𝑅

+ 𝑗𝜔𝐿

+ Z

𝑍

= 𝑍

= 𝑅

+ 𝑗𝜔𝐿

+ 𝜔

𝑀

𝑅

+ 𝑗𝜔𝐿

+ Z

• The primary impedance

𝑅

+ 𝑗𝜔𝐿

𝑍

= 𝜔

𝑀

𝑅

+ 𝑗𝜔𝐿

+ Z

• 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,

𝑍

= 𝑅

+ 𝑗𝑋

𝑍

= 𝜔

𝑀

𝑅

+ 𝑗𝜔𝐿

+ (𝑅

+ 𝑗𝑋

) = 𝜔

𝑀

(𝑅

+ 𝑅

) + 𝑗(𝜔𝐿

+ 𝑋

)

(9.65)

 Reflected Impedance

𝑍₁₁ = 𝑅₁ + 𝑗𝜔𝐿₁ 𝑍₂₂ = 𝑅₂+ 𝑗𝜔𝐿₂+ Z_𝐿

𝑍

= 𝜔

𝑀

𝑅

+ 𝑗𝜔𝐿

+ (𝑅

+ 𝑗𝑋

) = 𝜔

𝑀

(𝑅

+ 𝑅

) + 𝑗(𝜔𝐿

+ 𝑋

)

= 𝜔

𝑀

𝑅

+ 𝑅

− 𝑗(𝜔𝐿

+ 𝑋

) (𝑅

+ 𝑅

)

+(𝜔𝐿

+ 𝑋

)

= 𝜔

𝑀

𝑅

+ 𝑅

− 𝑗(𝜔𝐿

+ 𝑋

) (𝑅

+ 𝑅

)

+(𝜔𝐿

+ 𝑋

)

= 𝜔

𝑀

𝑅

+ 𝑅

− 𝑗(𝜔𝐿

+ 𝑋

)

𝑍

= 𝜔

𝑀

𝑍

𝑅

+ 𝑅

− 𝑗(𝜔𝐿

+ 𝑋

)

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+ 𝑅_𝐿 + 𝑗𝑋_𝐿 𝑍₂₂^∗ = 𝑅₂+ 𝑅_𝐿 − 𝑗(𝜔𝐿₂ + 𝑋_𝐿)

 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

• Applying KVL to the two meshes in Fig.9.41 gives

𝐕

= 𝑗𝜔𝐿

𝐈

𝐕

= 𝑗𝜔𝑀𝐈

• From Eq.(1),

𝐈

= 𝐕

𝑗𝜔𝐿

• Substituting this in Eq.(2) gives

𝐕

= 𝑗𝜔𝑀𝐈

= 𝑗𝜔𝑀 𝐕

𝑗𝜔𝐿

= 𝑀𝐕

𝐿

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

(4)

𝐕

= 𝐿

𝐿

𝐕

= 𝑎𝐕

or 𝐕

𝐕

= 𝑎 = 𝑁

𝑁

𝑎 = 𝑁

𝑁

= 𝐿

𝐿

(5)

: Turns ratio or

transformation ratio.

 Determining of the Voltage Ratio 𝐕

𝐕

= 𝑎 = 𝑁

𝑁

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

• 𝑎 = 1

• 𝑎 > 1

• 𝑎 < 1

 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)

 Determining of the Voltage Ratio

step-up transformer step-down transformer

 Determining of the Current Ratio

• Applying KVL to the two meshes in Fig.9.41 gives

𝐕

= 𝑗𝜔𝐿

𝐈

− 𝑗𝜔𝐿

𝐈

0 = −𝑗𝜔𝑀𝐈

+ 𝑗𝜔𝐿

𝐈

• From Eq.(2),

(1) (2)

𝐈

𝐈

= 𝑗𝜔𝑀

𝑗𝜔𝐿

= 𝐿

𝐿

= 1 𝑎 𝐈

𝐈

= 𝑁

𝑁

or

𝑁

𝐈

= 𝑁

𝐈

1𝐿₂ 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.

𝐕₂

𝐕₁ = 𝑎 𝐈₂

𝐈₁ = 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 = 𝑁

𝑁

 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

𝐕

𝐕

= 𝑎 = 𝑁

𝑁

𝐈

𝐈

= 1

𝑎 = 𝑁

𝑁

𝐙

= 𝐕

𝐈

= 1 𝑎

𝐕

𝐈

= 𝐙

𝑎

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 R_L must be matched with the source resistance R_s.

 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.

 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.

𝐕

𝐕

= 𝑁

+ 𝑁

𝑁

= 1 + 𝑁

𝑁

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

𝐈

𝐈

= 𝑁

𝑁

+ 𝑁

 Ideal Autotransformers

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

𝐢

𝐢

= 𝑁

+ 𝑁

𝑁

= 1 + 𝑁

𝑁

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

𝐯

𝐕

= 𝑁

𝑁

+ 𝑁

 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.

 Ideal Autotransformers

 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

 Phasor Diagram

Re Im

𝐈 𝐕

𝐈 = 𝐕 𝑗𝜔𝐶 1

= 𝑗𝜔𝐶𝐕

𝐈 = 𝐕 𝑅 𝐈 = 𝐕

𝑗𝜔𝐿 = −𝑗 𝐕

𝜔𝐿

 Impedance Diagram

𝑍 = 𝑅 + 𝑗𝑋

Re Im

𝑅 𝑗𝜔𝐿

−𝑗 1 𝜔𝐶

𝑍 = 𝑅 𝑍 = 𝑗𝜔𝐿

𝑍 = 1 𝑗𝜔𝐿

90°

−90°