Electrostatics - Transient Currents

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Transient Currents

When an electric circuit containing an ohmic resistance only is switched on the electric current acquires its maximum value almost in zero time. Similarly when such a circuit is switched off the current reduces to zero almost in zero time. Similarly when such a circuit is switched off the current reduces to zero almost in zero time however when the electric circuit contains an inductor or a capacitor or both, the growth as well as decay of current are opposed by the induced. Therefore electric current takes some finite time to reach its maximum value when the circuit is switched on similarly when the circuit is switched off the electric current takes again some finite time to decay to zero value. 

Such electric current which vary for a small finite time while growing from zero to maximum value or while decaying from maximum value to zero value are called transient currents.

Growth and decay of current in an inductor
    
Growth of current.

Consider an ohmic resistance R and a coil of inductance L connected to a battery E through a mores key K 

On pressing K the battery is connected. Current grows in the R-L circuit. Due to self induction, an induced is set up across L by lens’s law this induced opposes the growth of current. 

If is strength of current at any instant t and dI/dt is rate of growly of current at this instant then 

Potential difference across R= IR

Potential difference across L = L dI/dt

As E is the of the battery,

∴ e = IR + L (dI/dt)

When current reaches its maximum value

I = I₀ (dI/dt) = 0

From E I₀ R using this value in we 
 
I₀ R = IR + L (dI/dt)

R ( I₀ – I) = L (dI/dt)

Or (dI/I₀) – I = (R/L) dt

Integrating both sides we get

∫(dI/I₀) – I = ∫(R/L) dt

-log (I₀ – I) = R/L (t + k)

Where l is constant of integration.

During growth I = (A₀ at t) = 0 

∴ form – log Io = (o + k)

Or k = - log I₀

Put in – log (I₀ – I) = (R/L) t = log I₀

Or log (I₀ – I) – log I₀ = - (R / L) t

Or log I₀ – (I/I₀) = - (R/L) t

Or I₀ – (I/I₀) = e 

I – (I/_I0) = w –( R/ L)

Or I / I₀ = (I – e) – (R/L) t 

I = I₀ ( I – e – R/ L ) = I₀ (I – e – t/t)

Where 

τ = L/R

This is Helmholtz equation governing growth of current in L.R circuit

It shows that growth of current in an inductor is exponential.

Time constant. The quantity = L/R is called time constant or inductive time constant of LR circuit. This is because dimensions of r = L/R are those of time and for a given LR circuit its value is constant.

If t = L/R, then from

I = I₀ (1 – e-1) =I₀ (1 – 1 /e) =I₀ (1 – ½.718)

I =I₀ (1 – 0.368) = 0.632I0

I = 63.2% I.

We may define time constant of Kr circuit as the time in which current in the circuit grows to 63.2% of the maximum value of current.

Again from Eqn. we find that for 

I=I₀. e t/r = 0 or t = ∞

Current in LR circuit would attain maximum value only after infinite time. However, practically current reaches its maximum value after a time which is roughly five times the time constant.
    
Decay of current

In when Morse key K is released the battery is cut off. The current in the LR circuit decays. If I is the current at any time t during break then as battery is disconnected putting E = 0 in we get.

IR + L (dI/dt) = 0

L (dI/dt)= - Ir 

Or dI/I = - (R/L) dt 

Integrating both sides we get 

∫dI /I = - ∫ (R/L) dt

Log I = R/L (t + K)

Where K is a constant of integrating.

At t= 0, I = I₀

∴ From log I₀ = O + K

Or K = log I₀

Put in log I = - R/Lt + log I₀

Log I – log I₀ = - R/Lt

Log I/I₀ =- R/Lt or I/I₀ = e –R/L

Or , I = I₀e ^– R/L t = I₀e^{- t/ r}

Where, r = L/R

This is the Helmholtz equation for decay of current in LR circuit.

Time constant. The quantity r = L/R is the time constant of LR circuit as said already.

If t = L/R then from the above equation,

I = I₀e ^{- 1} = I₀/e = I₀ /2.718 = 0.368 I0 = 36.8% I₀

Hence we may define time constant of LR circuit as the item in which current decays to 36.8% of the maximum value.

Again from (8) for I = 0

e-t/r = 0 or t = ∞

Current reduces to zero only after infinite time. The decay of current with time has been shown in physical significance of time constant 

If r = L/R is small then form (6) I attains its final value Io more rapidly ; and from (8) current decays to zero also more rapidly. If L/R is large both the growth and decay of current in LR circuit are slow. Thus time constant of LR circuit determines the rate at which current grows or decays in the LR circuit. Smaller the values of time constant faster are the growth as well as decay of current in the circuit. The reverse is also true. 

 

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