1. Polar to Rectangular

From the diagram above, these formulas convert polar coordinates to rectangular coordinates:

x = r cosq,   y = r sinqSo the polar point: (r,q) can be converted to rectangular coordinates like this: r cosq, r sinq )  ð  ( x, y ) 2. Rectangular to PolarAgain, from the diagram above, these formulas convert rectangular coordinates to polar coordinates:By the rule of Pythagoras:

Tan  = ^y/_x  ,  so therefore: = tan^-1( ^y/_x )So the rectangular point: (x,y) can be converted to polar coordinates like this:( , tan^-1( ^y/_x ) )      ( r ,  ).

16. Ideal current source.

In circuit theory, an ideal current source is a circuit element where the current through it is independent of the voltage across it. It is amathematical model, which real devices can only approach in performance. If the current through an ideal current source can be specified independently of any other variable in a circuit, it is called an independent current source. Conversely, if the current through an ideal current source is determined by some other voltage or current in a circuit, it is called a dependent or controlled current source.

 No real current source is ideal (no unlimited energy sources exist) and all have a finite internal resistance (none can supply unlimited voltage). However, the internal resistance of a physical current source is effectively modeled in circuit analysis by combining a non-zero resistance in parallel with an ideal current source (the Norton equivalent circuit). The connection of an ideal open circuit to an ideal non-zero current source does not represent any physically realizable system.

An ideal current source is a simple model for many current sources.  It is reminiscent of the ideal voltage source - but with voltage and current interchanged.  Here is the story.

There is a special circuit symbol for an ideal current source.  See below.

I_L = constant, no matter what the terminal voltage is.

The plot of load current against terminal voltage is similar to the plot for an ideal voltage source, but voltage and current are interchanged.  Here is the plot.

 

  17. Inductive reactance and inductive susceptance

Inductive reactance   is proportional to the signal frequency   and the inductance  .

An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the counter-emf   (voltage opposing current) due to a rate-of-change of magnetic flux density   through a current loop.

For an inductor consisting of a coil with N loops this gives.

The counter-emf is the source of the opposition to current flow. A constant direct current has a zero rate-of-change, and sees an inductor as a short-circuit (it is typically made from a material with a lowresistivity). An alternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

Susceptance is the reciprocal of reactance, X and is given the symbol B.

In AC circuits susceptance is defined as the ease at which a reactance (or a set of reactances) allows current to flow when a voltage is applied. Susceptance has the opposite sign to reactance so capacitive susceptance BC is positive, +ve in value and inductive susceptance BL is negative, -ve in value.

18. Power balance in DC Circuits

Resistors dissipate energy in the form of heat, and the rate at which they dissipate energy is called power. The power dissipated by resistors is delivered by the voltage and/or current sources in the circuit.

The unit of power is the watt (one joule/second).

There are several ways of calculating the power of resistors.

19.Phasor Diagrams. Impedance and Reactance.

A phasor is a vector that contains both magnitude and direction or amplitude and phase information. It can be used to represent AC quantities. Since phasors have magnitudes and directions, they can be represented as complex numbers. A phasor notation or phasor domain is a method that uses complex numbers to represent the sinusoidal quantities for analysing AC circuits. It can represent sine waves in terms of their peak value (magnitude) and phase angle (direction). The peak value can be easily converted to the RMS value. The phasor notation can simplify the calculations for AC sinusoidal circuits; therefore, it is widely used in circuit analysis and calculations.

The complex number has two main forms, the rectangular form and the

polar form.

● Rectangular form: A =x + jy ( j=√-1)

where x is the real part and y is the imagery part of the complex number A; j is called the imagery unit.

Note: The symbol i is used to represent imagery unit in mathematics. Since I has been used to represent AC current in the circuit analysis, j is used to denote the imagery unit rather than i to avoid confusion.

● Polar form:

This is the abbreviated form of the exponential form , in which a is called modulus of the complex number, and the angle φ is called argument of the complex number.

Impedance (symbol Z) is a measure of the overall opposition of a circuit to

Impedance can be split into two parts:

• Resistance R (the part which is constant regardless of frequency)

• Reactance X (the part which varies with frequency due to capacitance and inductance)

Impedance, Z =	V
	I

Reactance (symbol X) is a measure of the opposition of capacitance and inductance to current. Reactance varies with the frequency of the electrical signal. Reactance is measured in ohms, symbol  .

There are two types of reactance: capacitive reactance (Xc) and inductive reactance (X_L).

The total reactance (X) is the difference between the two:    X = X_L - Xc

Capacitive reactance, Xc 

Xc =1/2пfC

		where:	Xc = reactance in ohms ( )  f    = frequency in hertz (Hz) C   = capacitance in farads (F)