Introduction:

The charge control equations are given for forward active mode of transistor as:

i_C = Q_F/τ_F – dQ_BC/dt

i_B = Q_F/τ_BF + dQ_F/dt + dQ_BC/dt + dQ_BE/dt

i_E = Q_F/τ_F + Q_F/τ_BF + dQ_F/dt + dQ_BE/dt

Where, all currents and charges are intrinsically taken as instantaneous functions of time.

Delay Time, t_d

The above equations do not apply to calculate the delay time, as the transistor is not in forward active mode throughout this time. The delay time is, though, very small in relation to the other switching times and is therefore ignored.

Fall-Time, t_f : Transistor Turn-On

The fall-time, t_f, for the output voltage is obtained by resolving the charge control equations for this condition. To resolve the above charge control equations in this concern, the currents require being associated in terms of circuit parameters. As the emitter terminal is grounded, this can’t be done for the emitter current and therefore its charge control equation is of little advantage. Throughout transistor turn-on in the forward active mode, the base of transistor can be taken as sitting at the potential of V_{BE ON} and hence the base current can be supposed constant throughout this period as shown in figure below.

Therefore, i_B = (V_CC - V_BEON)/R_B

And, dQ_BE/dt = C_BE (dV_BE/dt) = 0

Figure: Constant Voltage on Base throughout Transistor Turn-On

In this case, the charge control equations are simplified to:

i_C = Q_F/τ_F - dQ_BC/dt

i_B = (Q_F/τ_BF) + (dQ_F/dt) + (dQ_BC/dt) = (V_CC – V_{BE ON})/R_B

The equation for i_B is expressed in terms of circuit parameters and is therefore clearly the one to be solved. Though, since a solution for i_c is needed, substitutions for Q_F and dQ_F/dt terms in terms of i_c should be found. Let first consider a figure shown below:

Figure: The Effect of Collector-Base Junction Capacitance

From the circuit:

dQ_BC/dt = C_BC (dV_BC/dt) = C_BC d/dt[V_{BE ON} – Vo(t)]

Therefore, dQ_BC/dt = - C_BC [dV_o(t)]/dt

However,

Vo(t) = VCC - iCRC

Therefore,

dV_o/dt = -R_C (di_C/dt)

Hence,

dQ_BC/dt = C_BC R_C (di_C/dt)……………… (a)

Secondly, consider:

i_C = Q_F/τ_F - dQ_BC/dt

Q_F = τ_F i_C + τ_F (dQ_BC/dt)

And hence,

Q_F = τ_F i_C + τ_F C_BC R_C (di_C/dt)…………. (b)
   
Thirdly,

dQ_F/dt = τ_F (di_C/dt) + τ_F C_BC R_C (d²i_C/dt²)

All through turn-on, the collector current starts to rise towards βF IB however is limited on reaching saturation to I_CMAX << β_FI_B. The increase in collector current can be seen to contain an almost constant rate throughout this time as shown in figure below.

Then, if (di_C/dt) ≈ constant => (d²i_C/dt²) → 0 and avoiding the second order derivative gives:

dQ_F/dt ≈ τ_F (di_C/dt)……………. (c)

Figure: Rise in Collector Current throughout Transistor Turn-On

Substituting the equations (a), (b) and (c) to replace the charge terms in the equation for base current provides:

As τ_F/τ_BF = τ_F/β_Fτ_F = 1/β_F then, this equation can be rearranged as:

(1/βF) iC + [(1/βF) CBC RC + τ_F + CBC RC] (diC/dt) = (VCC - VBEON)/RB

On multiplying by β_F gives:

i_C + (C_BC R_C + β_F τ_F + β_F C_BC R_C) (di_C/dt) = β_F [(V_CC - V_{BE ON})/R_B]

If β_F >> 1 then (β_F +1) C_BC R_C ≈ β_F C_BC R_C and hence:

i_C + β_F (τ_F + C_BC R_C) (di_C/dt) = β_F [(V_CC - V_{BE ON})/R_B]

This is a first order linear differential equation in ic which can be solved by taking the Laplace Transform. Note that the Laplace Transforms of constant, K and a derivative of a function are as follows:

L {K} = K/s

L {dy/dt} = s L {y} – y (t = 0) = sY(s) – y (t = 0)

And hence,

I_C(s) + β_F (τ_F + C_BC R_C) [s I_C (s) - i_C (t = 0)] = β_F [(V_CC - VBE ON)/sR_B]

i_C(t = 0) is the initial value of collector current throughout turn-on that should be taken as zero. Then,

The solution can be easily obtained by taking Inverse Laplace Transform where the Inverse Laplace transforms of the term:

This equation exhibits that in linear forward active region, throughout transistor turn-on, the collector current increases exponentially from zero towards the value of   β_F [(V_CC - V_{BE ON})/R_B)]

Time constant of the expression can be seen to depend on transistor properties β_F, τ_F and C_BC and also the value of the external resistor R_C.

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