# User:Constant314

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### The ideal transformer

Ideal transformer circuit diagram

some statement,.[1] [a]

• Hayt, William; Kemmerly, Jack E. (1993), Engineering Circuit Analysis (5th ed.), McGraw-Hill, ISBN 007027410X

The ideal condition assumptions are:

• The windings of the transformer have negligible resistance, so RP= RS= 0, where RP represents the resistance of the primary winding and RS represents the resistance of the secondary winding. Thus, there is no copper loss in the winding, and hence no voltage drop.
• Flux is confined within the core. Therefore, it is the same flux that links both the windings.
• Permeability of the core is infinitely high which implies that zero mmf (current) is required to set up the flux and that the flux in the core due to the primary winding must be equal and opposite to the flux due to the secondary winding. There is therefore zero net flux in the core.
• The core does not incur any hysteresis or eddy current loss. Hence, no core losses.

If the secondary is an open circuit, an ideal transformer will not allow the flow of primary current.

Consider the ideal, lossless, perfectly-coupled transformer shown in the circuit diagram at right having primary and secondary windings with NP and NS turns, respectively.

The ideal transformer induces secondary voltage ES =VS as a proportion of the primary voltage VP = EP and respective winding turns as given by the equation

Eq. 1 $\frac{V_\text{P}}{V_{\text{S}}} = \frac{E_\text{P}}{E_\text{S}} = \frac{N_\text{P}}{N_\text{S}} =a$,

where,

- VP/VS = EP/ES = a is the voltage ratio and NP/NS = a is the winding turns ratio, the value of these ratios being respectively higher and lower than unity for step-down and step-up transformers,.[3][4][b][c]
- VP designates source impressed voltage,
- VS designates output voltage, and,
- EP & ES designate respective emf induced voltages.[d]

According to the equation shown above, when the number of turns in the primary coil is greater than the number of turns in the secondary coil, the secondary voltage must be less than the primary voltage. On the other hand, when the number of turns in the primary coil is less than the number of turns in the secondary, the secondary voltage must be greater than the primary voltage.

Any load impedance $Z_L$ connected to the ideal transformer's secondary winding causes current to flow without losses from primary to secondary circuits, the resulting input and output apparent power therefore being equal as given by the equation

Eq. 2 $I_P \times V_P = I_S \times V_S$.

Combining the two equations yields the following ideal transformer identity

$\frac{V_P}{V_S} = \frac{I_S}{I_P} = a$.

This formula is a reasonable approximation for the typical commercial transformer, with voltage ratio and winding turns ratio both being inversely proportional to the corresponding current ratio.

The load impedance $Z_L$ is defined in terms of secondary circuit voltage and current as follows

$Z_L = \frac{V_L}{I_L}=\frac{V_S}{I_S}$.

The apparent impedance $Z_L^\prime$ of this secondary circuit load referred to the primary winding circuit is governed by a squared turns ratio multiplication factor relationship derived as follows[6][7]

$Z_L^\prime=\frac{V_P}{I_P}=\frac{aV_S}{I_S/a}=a^2\times\frac{V_S}{I_S}=a^2\times{Z_L}$.

For an ideal transformer, the power supplied in the primary and the power dissipated by the load are both equal to the following equation:[8][9]

$P = \frac{V^2_{\text{S}}}{R_{\text{L}}} = a^2\frac{ {V^2_{\text{p}}}}{{R_{\text{L}}}}$

The primary current is equal to the following equation:[8][9]

$I_{\text{p}} = \frac{P_{\text{R}}}{V_{p}} = a^2\frac{ {V_{\text{p}}}}{{R_{\text{L}}}}$

The equivalent resistance on the source from the load sections and transformer is equal to the following equation:[8][9]

$R_{\text{eq}} = \frac{V_{\text{p}}}{I_{\text{p}}} = a^2 {R_{\text{L}}}$

#### Induction law

The transformer is based on two principles: first, that an electric current can produce a magnetic field and second that a changing magnetic field within a coil of wire induces a voltage across the ends of the coil (electromagnetic induction). Changing the current in the primary coil changes the magnetic flux that is developed. The changing magnetic flux induces a voltage in the secondary coil.

Referring to the two figures here, current passing through the primary coil creates a magnetic field. The primary and secondary coils are wrapped around a core of very high magnetic permeability, usually iron,[e] so that most of the magnetic flux passes through both the primary and secondary coils. Any secondary winding connected load causes current and voltage induction from primary to secondary circuits in indicated directions.

Ideal transformer and induction law

The voltage induced across the secondary coil may be calculated from Faraday's law of induction, which states that:

$V_\text{S} = E_\text{S} = N_\text{S} \frac{\mathrm{d}\Phi}{\mathrm{d}t}.$

where Vs = Es is the instantaneous voltage, Ns is the number of turns in the secondary coil, and dΦ/dt is the derivative[f] of the magnetic flux Φ through one turn of the coil. If the turns of the coil are oriented perpendicularly to the magnetic field lines, the flux is the product of the magnetic flux density B and the area A through which it cuts. The area is constant, being equal to the cross-sectional area of the transformer core, whereas the magnetic field varies with time according to the excitation of the primary. Since the same magnetic flux passes through both the primary and secondary coils in an ideal transformer,[6] the instantaneous voltage across the primary winding equals

$V_\text{P} = E_\text{P} = N_\text{P} \frac{\mathrm{d}\Phi}{\mathrm{d}t}.$

Taking the ratio of the above two equations gives the same voltage ratio and turns ratio relationship shown above, that is,

$\frac{V_\text{P}}{V_\text{S}} = \frac{E_\text{P}}{E_\text{S}} = \frac{N_\text{P}}{N_\text{S}}=a$.

The changing magnetic field induces an emf across each winding.[10] The primary emf, acting as it does in opposition to the primary voltage, is sometimes termed the counter emf.[11] This is in accordance with Lenz's law, which states that induction of emf always opposes development of any such change in magnetic field.

As still lossless and perfectly-coupled, the transformer still behaves as described above in the ideal transformer.

#### Polarity

Instrument transformer, with polarity dot and X1 markings on LV side terminal

A dot convention is often used in transformer circuit diagrams, nameplates or terminal markings to define the relative polarity of transformer windings. Positively-increasing instantaneous current entering the primary winding's dot end induces positive polarity voltage at the secondary winding's dot end.[12][13][14][g][h]

## Photos of Doppler 019

Vestibule and Quonset hut housing a Transit satellite tracking station 019. 1. Triad satellite magnetometer down load antenna. 2. flag pole, 3. Utility pole in background, 4 Revolving light temperature alarm, 5 VLF antenna, 6-9 Doppler satellite tracking antennas, 10. stove pipe for heater, 11 Flood light for low visibility conditions, 12 fuel tank.

Some of the equipment inside Transit satellite tracking station 019. 1. Automatic Control Unit, 2. timer-counter, 3. Time burst detector, 4. time conversion chart, 5. satellite ephemeris, 6. tracking receiver, 7. time display, 8 Header-Tailer programmer, 9. Digitizer and main clock, 10. master oscillator, 11. strip chart recorder, 12. paper tape punch, 13. short wave receiver. Out of site: VLF receiver, refraction correction unit, backup battery system, power supplies, AC voltage regulators.

## Wien bridge oscillator

### Analyzed from Loop Gain

In this version of the oscillator, Rb is a small incandescent lamp. Usually R1 = R2 = R and C1 = C2 = C. In normal operation, Rb self heats to the point where its resistance is Rf/2.

According to Schilling[17] the loop gain of the Wien bridge oscillator is given by

$T = ( \frac { R_1 / (1 + sC_1 R_1) } {R_1 / (1 + sC_1 R_1) + R_2 + 1/(sC_2)} - \frac {R_b} {R_b + R_f } )A_0 \,$

where $A_0 \,$ is the frequency dependent gain of the op-amp. (Note, the component names in Schilling have been replace with the component names in the figure.)

Schilling further says that the conditon of oscillation is $T = 1 \,$. Which, assuming $R_1 = R_2 = R\,$ and $C_1 = C_2 = C \,$ is satisfied by

$\omega = \frac {1} {R C} \rightarrow F = \frac {1} {2 \pi R C}\,$

and

$\frac {R_f} {R_b} = \frac {2 A_0 + 3} {A_0 - 3} \,$ with $\lim_{A_0\rightarrow \infin} \frac {R_f} {R_b} = 2 \,$

## Miller effect diagrams

An ideal voltage inverting amplifier with an impedance connecting output to input.