Question

An RC circuit used to produce time delay is shown in Figure 10.40. For a step input voltage ${\mathit{V}}_{\mathbf{I}}\left(t\right)\mathbf{=}\mathbf{2}\mathit{u}\left(t\right)$and $\mathit{C}\mathbf{=}\mathbf{10}\mathit{\mu }\mathit{F}$, determine ${\mathit{T}}_{\mathbf{d}\mathbf{e}\mathbf{l}\mathbf{a}\mathbf{y}}$for the following cases: (a) $\mathit{R}\mathbf{=}\mathbf{100}\mathbf{}\mathit{k}\mathbf{}\mathit{\Omega }$and (b) $\mathit{R}\mathbf{=}\mathbf{1}\mathbf{}\mathit{M}\mathbf{}\mathit{\Omega }$. Sketch the output ${\mathit{V}}_{\mathbf{0}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$versus time for cases (a) and (b).

Short answer

Answer

(a) The value of the time delay $T_{delay}$for $100k\Omega$is 0.693 s.

(b) The value of the time delay $T_{delay}$for $1M\Omega$is 6.93 s.

The graph between the output voltage $v_0\left(t\right)$and time.

Step-by-step solution

Write the given data from the question.

The input voltage,$V_i\left(t\right)=2u\left(t\right)$

The capacitor, $C=10\mu F$

determine the equation to calculate the time delay.

The equation to calculate the time constant is given as follows.

$\mathit{\zeta }\mathbf{=}\mathit{R}\mathit{C}$ …… (1)

The equation to calculate the output voltage is given as.

role="math" localid="1656396263857" ${\mathit{v}}_{\mathbf{0}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{v}}_{\mathbf{i}}\left(\mathbf{1}\mathbf{-}{\mathbf{e}}^{\mathbf{-}\frac{\mathbf{t}}{\mathbf{\zeta }}}\right)$ …… (2)

Determine the value of the time delay when resistance is .

(a)

The resistance, $R=100k\Omega$

The output voltage after the time delay is equal to 1.

Calculate the time constant.

Substitute$100k\Omega$for R and $10\mu F$for C into equation (1).

$$\begin{gathered} \zeta =100\times {10}^3\times 10\times {10}^{-6} \\ \zeta =1s \end{gathered}$$

Calculate the time delay.

Substitute $1V$ for $v_0\left(t\right),2V$for $v_i\left(t\right),T_{delay}$ for t and 1s for $\zeta$into equation (2).

$$\begin{gathered} 1=2\left(1-e^{-\frac{T_{delay}}{1}}\right) \\ 0.5=1-e^{-T_{delay}} \\ {e}^{-T_{delay}}=1-0.5 \\ {e}^{-T_{delay}}=0.05 \end{gathered}$$

Solve further as,

$$\begin{gathered} -T_{delay}=In\left(0.5\right) \\ {T}_{delay}=-In\left(0.5\right) \\ {T}_{delay}=0.693s \end{gathered}$$

Hence, the value of the time delay $T_{delay}$for $100k\Omega$is 0.693 s.

Determine the value of the time delay when resistance is 1 MΩ.

(b)

The resistance, $R=1M\Omega$

The output voltage after the time delay is equal to 1.

Calculate the time constant.

Substitute $1M\Omega$for R and $10\mu F$for C into equation (1).

$$\begin{gathered} \zeta =1\times {10}^6\times {10}^{-6} \\ \zeta =10s \end{gathered}$$

Calculate the time delay.

Substitute 1 V for $v_0\left(t\right),2V$for $v_i\left(t\right),T_{delay}$ for t and 10 s for $\zeta$into equation (2).

$$\begin{gathered} 1=2\left(1-e^{-\frac{T_{delay}}{10}}\right) \\ 0.5=1-e^{-\frac{T_{delay}}{10}} \\ {e}^{-\frac{T_{delay}}{10}=1-0.5} \\ {e}^{-\frac{T_{daley}}{10}}=0.5 \end{gathered}$$

Solve further as,

$$\begin{gathered} -\frac{T_{delay}}{10}=In\left(0.5\right) \\ \frac{T_{delay}}{10}=-In\left(0.5\right) \\ {T}_{delay}=10\times 0.693 \\ {T}_{delay}=6.93s \end{gathered}$$

Hence the value of the time delay $T_{delay}$for $1M\Omega$is 6.93 s.

The graph between the output voltage $v_0\left(t\right)$and time is shown below.