Admittance and Susceptance

Admittance :

In electrical engineering, the admittance (Y) is the inverse of the impedance (Z). The SI unit of admittance is the siemens. Oliver Heaviside coined the term in December 1887.Admittance is a measure of how much current is admitted in a circuit. The admittance has its most obvious utility in dealing with parallel AC circuits.

where
Y is the admittance, measured in siemens
Z is the impedance, measured in ohms.

Therefore the expression for admittance in terms of voltage and current can also be wriiten as
Y = I/V = G + j B
G, the real part of the admittance, is the conductance of the circuit, and B, the imaginary part of the admittance, is the susceptance of the circuit. The units of admittance are called siemens or mhos (reciprocal ohms).

Substituting the expression of Z = R + j X in Y = 1/Z after simplifying and equating to G + j B the expressions are



The magnitude of admittance is given by:


Susceptance :

In electrical engineering, the susceptance (B) is the imaginary part of the admittance. In SI units, the susceptance is measured in siemens. Oliver Heaviside first defined this property, which he called permittance, in June 1887.Susceptance is the measure of how much a circuit is susceptible to conducting a changing current.

It can also be defined as the opposite of reactance.Just as there's capacitive reactance and inductive reactance, so too there is capacitive susceptance (BC) and inductive susceptance (BL). Just like with conductance, both of these are the reciprocal of their corresponding reactances. That is to say, capacitive susceptance is 1 divided by the capacitive reactance, and inductive susceptance is 1 divided by the inductive reactance.

Thus, capacitive susceptance can be simplified into the following equation: BC = 2*pi*f*C.

Inductive susceptance, meanwhile, becomes exactly like the formula for capacitive reactance, except that it of course uses inductance rather than capacitance: BL = 1/2*pi*f*L

Faraday's Law and Lenz's Law

Faraday's Law :

 

Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.

Lenz's Law :

When an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant.

Electrical Resonance and Resonant frequency

Electrical Resonance :

Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedance between the input and output of the circuit is at a minimum (or when the transfer function is at a maximum). Often this happens when the impedance between the input and output of the circuit is zero and when the transfer function equals one.

Resonance with capacitors and inductors :
Resonance of a circuit involving capacitors and inductors occurs because the collapsing magnetic field of the inductor generates an electric current in its windings that charges the capacitor, and then the discharging capacitor provides an electric current that builds the magnetic field in the inductor, and the process is repeated continually. An analogy is a mechanical pendulum. In some cases, resonance occurs when the inductive reactance and the capacitive reactance of the circuit are of equal magnitude, causing electrical energy to oscillate between the magnetic field of the inductor and the electric field of the capacitor.

At resonance, the series impedance of the two elements is at a minimum and the parallel impedance is a maximum. Resonance is used for tuning and filtering, because it occurs at a particular frequency for given values of inductance and capacitance. It can be detrimental to the operation of communications circuits by causing unwanted sustained and transient oscillations that may cause noise, signal distortion, and damage to circuit elements.

Parallel resonant or near-to-resonance circuits can be used to prevent the wastage of electrical energy, which would otherwise occur while the inductor built its field or the capacitor charged and discharged. As an example, asynchronous motors waste inductive current while synchronous ones waste capacitive current. The use of the two types in parallel makes the inductor feed the capacitor, and vice versa, maintaining the same resonant current in the circuit, and converting all the current into useful work.

Since the inductive reactance and the capacitive reactance are of equal magnitude, ωL = 1/ωC, so:

where ω = 2πf, in which f is the resonant frequency in hertz, L is the inductance in henries, and C is the capacitance in farads when standard SI units are used.

Resonant freqency :
A resonant frequency is a natural frequency of vibration determined by the physical parameters of the vibrating object. This same basic idea of physically determined natural frequencies applies throughout physics in mechanics, electricity and magnetism, and even throughout the realm of modern physics.The lowest resonant frequency of a vibrating object is called its fundamental frequency. Some of the implications of resonant frequencies are:

1. It is easy to get an object to vibrate at its resonant frequencies, hard to get it to vibrate at other frequencies.
2. A vibrating object will pick out its resonant frequencies from a complex excitation and vibrate at those frequencies, essentially "filtering out" other frequencies present in the excitation.
3. Most vibrating objects have multiple resonant frequencies.

Resonant frequency is given by the expression :

Vector Diagrams - RLC Series and Parallel circuit

Vector Diagrams :

RLC Series circuits
When we add a resistance to a series LC circuit, as shown in the schematic diagram to the right, the behavior of the circuit is similar to the behavior of the LC circuit with no resistance, but there are some variations.We wil see the affects of added resistance with the parameters given below

• f = 1 MHz
• e = 10 vrms
• L = 150 µh
• C = 220 pf
• R = 100 Ω

With these measured voltages across R L and C we get are

• vL = 39.1v
• vC = 30.0v
• vR = 4.15v

We must take into account the different phase angles between voltage and current for each of the three components in the circuit. The vector diagram to the right, while not to scale, illustrates this concept.
Since this is a series circuit, the current is the same through all components and is therefore our reference at a phase angle of 0°. This is shown in red in the diagram. The resistor's voltage, vR, is in phase with the current and is shown in green. The blue vector shows vL at +90°, while the gold vector represents vC, at -90°. Since they oppose each other diametrically, the total reactive voltage is vL - vC. It is this difference vector that is combined with vR to find vT (shown in cyan in the diagram).
We already know that vT = 10 vrms. Now we can see that vT is also the vector sum of (vL - vC) and vR. In addition, because of the presence of R, the phase angle between vT and i will be arctan((vL-vC)/vR), and can vary from -90° to +90°.

RLC Parallel circuits
The schematic diagram to the right shows three components connected in parallel, and to an ac voltage source: an ideal inductance, and ideal capacitance, and an ideal resistance. We will use the following values for our components

• VAC = 10 vrms.
• f = 1 MHz. ( = 6283185.3 rad/sec)
• L = 150 µh. (XL = 942.4778 )
• C = 220 pf. (XC = 723.43156 )
• R = 1000

According to Ohm's Law:
iL =vL/XL = 10/942.4778 = 0.01061033 = 10.61033 mA.
iC =vC/XC = 10/723.43156 = 0.013823008 = 13.823008 mA.
iR = vR/R = 10/1000 = 0.01 = 10 mA.

If we measure the current from the voltage source, we find that it supplies a total of 10.503395 mA to the combined load — only about half a milliamp more than iR alone.
So we now have 10 mA of resistive current and just over 3.2 mA of reactive current, and yet the measured total current is just over 10.5 mA.

As usual, the vectors, shown to the right, tell the story. Since this is a parallel circuit, the voltage, v, is the same across all components. It is the current that has different phases and amplitudes within the different components.

Since voltage is the same throughout the circuit, we use it as the reference, at 0°. Current through the resistor is in phase with the voltage dropped across that resistor, so iR also appears at 0°.

Current through an inductor lags the applied voltage, so iL appears at -90°. Current through a capacitor leads the applied voltage, so iC appears at +90°. Since iC is greater than iL, the net reactive current is capacitive, so its phase angle is +90°.
Now the total current, iT, is the vector sum of reactive current and resistive current. Since iR is significantly greater than the difference, iC - iL, the total impedance of this circuit is mostly resistive, and the combined vector for iT is at only a small phase angle, as shown in the diagram.


The above are the cases of ideal inductor and capacitor.If those are not ideal the vector diagram wil be as shown below


(a)Series RLC circuit


(b)In (a) XL > XC, and the driving voltage (V) leads the current by a phase angle of f.


(c)In (b) XC > XL, and the driving voltage (V) lags the current by a phase angle of f.

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