Actual / Virtual Resistance / Conductance

 

Actual / Virtual Resistance / Conductance

When electricity flow in the conductor, it experience obstruction. It is like biker facing difficulty while driving through crowded road. Difficulty level depends on the density of the crowd. But difficulty level may be less with the same crowd if road is wider. This phenomenon of obstruction to flow of current in conductor is a characteristic of the conductor material. This characteristic is material related depending on its atomic structure and is known as Resistivity or Specific Resistance and ρ (rho) as symbol. Actual resistance of the particular conductor depends on resistivity, length and cross section of the conductor.

R = ρ × L / A

Where            L = Length of the conductor

A = Cross section area of conductor

ρ = Resistivity of the conductor material.

Relation with supply voltage and current is given by Ohm’s Law.

Current I in the circuit = Applied Voltage / Resistance of the circuit.

I = V / R  = V/1 × 1/R  = V/1 × G/1  = VG      where G is conductance.

Resistance is a factual obstruction to current in the conductor itself. Hence it is Actual Resistance.

Electric current flowing in the conductor produces magnetic field around the conductor. This magnetic field around the conductor remains steady while steady direct current is flowing in the conductor. But this magnetic field is varying and reversing direction with alternating current in the conductor. The magnetic flux is linked to the conductor itself. So varying magnetic flux induces electro motive force in the conductor itself by electromagnetic induction. This EMF impedes the effect of applied voltage and consequently effective voltage is reduced to some extent and so also the current in the circuit.

Resistance as per ohm’s law, R = V/I  seems to have increased.

Reduction in current perceives as if circuit resistance has increased. This implies that it is not resistance only. Increased entity is identified as impedance Z. This Z is mix of actual resistance R of the conductor and additional entity acting as if resistance. This new entity is identified as reactance X_L.

Reactance is not the property of the conductor. But it depends on the layout of the conductor and associated factors as under

N = numbers of turns of the coil. (Straight conductor can be considered as one turn)

A = Area of the coil.                        (Circular coil = πR²)                        (Rectangle coil = L × B)

l = Length of the coil. (Compact coil has more reactance compared to loose (helical) coil)

µ = Absolute permeability of core material

X_{L =} 2πfL = 2πfµAN²/l    and current I = V/ X_L  = Vl / 2πfµAN²

This shows no relation with material of conductor of the coil. It means inductive reactance of identical coils is the same irrespective of copper or aluminum conductor.

On one hand it depends on other factors related to coil configuration like numbers of turns, coil area, coil length, and type of the core. On the other hand it depends on frequency of the supply.

Resistance R obstructs the flow of current internally due to atomic structure of conductor material.

Inductive Reactance X_L imitates to obstruct the flow of current but is due to magnetic effect.

Therefore it is Virtual Resistance

 

Device consisting two metal plates separated by insulation (dielectric) is known as capacitor. It is a non conducting device because of dielectric in the path. Therefore current cannot flow in the circuit with capacitor.

However current seems to flows for short time when DC is applied. This is a charging current on account of accumulation of positive and negative charge on the plates. Voltage built up due to accumulated charge in capacitor and is opposite to applied voltage. So effective voltage is applied voltage minus back voltage. During charging process back voltage goes on increasing and current is decreasing. Ultimately current stops after some time as capacitor voltage rise equals to supply voltage when capacitor is fully charged. So practically capacitor acts as non conducting path after initial charging current.   

But with AC supply the process is different. Because the supply voltage reverses before charging current drops substantially. So current continue in reverse direction, first discharging and further charging the capacitor in reverse direction. As voltage of AC is continuously reversing, this phenomenon continues with illusion that current is continuously flowing in the circuit. In fact the current is not flowing through the capacitor. Capacitor is a non conducting device as there is a dielectric between two plates. Conductivity of dielectric is almost zero. So there is no through conducting path. But due to charging, discharging and reverse charging of capacitor emulates that current is flowing through the circuit. Therefore it is Virtual Conductance.

Capacitance is related to following parameters of the device.

A = Plate area     

D = Distance between plates

ɛ = Absolute Permittivity of Dielectric

Capacitance C = ɛA/D

Capacitive reactance Xc = 1 / 2πfC

Current in pure capacitive reactance = V / Xc = V × 2πfC = V2πfɛA / D

As Current I = Voltage V × Conductance G

So  2πfɛA / D  is Virtual Conductance of Capacitor.

 

It is also observed that current in Virtual Resistance i.e. Inductive reactance is

 I = Vl / 2πfµAN²

It means current is low at high frequency and vice a versa.

This can be visualize from inductor charging current curve as under.

                 

 Consider two frequencies f₁ > f₂

t₁ and t₂ represent time period for reversal of current i.e. half cycle time.

As frequency   f₁ > f₂, time t₁ < t₂

During time period t₀ to t₁   current rise from I₀ to I₁  and its average current is I_a1

During time period t₀ to t₂   current rise from I₀ to I₂ and its average current is I_a2

It is observed that I_a1 < I_a2

At higher frequency f_1, average current I_a1  is less than average current I_a2  at lower frequency f₂

This is because at high frequency average current is low because of short time for current to grow before reversal.

Whereas at low frequency average current is high because of long time for current to grow before reversal.

 

Similarly current in capacitor I = V ×  2πfɛA / D          

It means current is high at high frequency and vice a versa.

This can be visualize from capacitor charging current curve as under.

  Consider two frequencies f₁ > f₂

t₁ and t₂ represent time period for reversal of current i.e. half cycle time.

As frequency   f₁ > f₂, time t₁ < t₂

During time period t₀ to t₁   current fall from I₀ to I₁ and its average current is I_a1

During time period t₀ to t₂   current fall from I₀ to I₂ and its average current is I_a2

It is seen that I_a1 > I_a2

At higher frequency f_1, average current I_a1  is high than average current I_a2  at lower frequency f₂

This is because at high frequency average current is high because of short time for current to drop before reversal.

Whereas at low frequency average current is low because of long time for current to drop before reversal.

 

Inductance and capacitor are virtual entity and opposite in nature as Resistance and Conductance. So they cancel each other’s effect when in the circuit. So the circuit having both inductor and capacitor has net effective reactance as difference of two. Also at particular frequency net reactance may be zero.