Notes › Fundamentals of Electric Circuits (Sadiku) Lecture 7

First-Order Circuits

calendar_today July 4, 2025 article 820 words access_time 8 min replay Modified July 4, 2025

Table of Contents

First-Order Circuit

Definition 07.1 (First-Order Circuit).

A circuit containing one energy storage device (resistor/capacitor) with a resistor. Characterized by a first-order differential equation.

Two ways to excite 1st order circuit:
1. Source-free circuit: assume the energy storage element holds all the energy via initial condition
  - Although there are no independent sources, there may be dependent sources
2. Independent sources

Source-Free RC

Voltage response is an exponential decay of $V_o$
- This is called the natural response, since it is due to the energy storage unit itself

$$v(t) = V_{0}e^{-\frac{t}{RC}}$$

Definition 07.2 (Natural Response).

The behavior of voltage and current with no external excitation.

Definition 07.3 (Time Constant).

The time required for the response to decay to a factor of $\frac{1}{e}$ or 36.8% of its initial value.

$$\tau = RC$$

Circuits with a small $\tau$ reach steady state (complete energy dissipation) quickly
Current response by Ohm’s Law:

$$i(t) = \frac{V_{0}}{R}e^{-\frac{t}{\tau}}$$

Power dissipated by $R$:

$$p(t) = \frac{V_{0}^2}{R}e^{-\frac{2t}{\tau}}$$

Energy absorbed by $R$:

$$w(t) = \frac{1}{2}CV_{0}^2 (1-e^{-\frac{2t}{\tau}})$$

$R$ is often $R_{\text{th}}$ at the terminals of the capacitor

Example 07.4.

(Practice Problem 7.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 282])

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Example 07.5.

(Practice Problem 7.2 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 283])

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Source-Free RL

Natural response is an exponential decay of $I_0$

$$i(t) = I_{0}e^{-t \frac{R}{L}}$$

Time constant

$$\tau = \frac{L}{R}$$

Voltage response by Ohm’s Law:

$$v(t) = I_{0}R e^{-\frac{t}{\tau}}$$

Power dissipated by $R$:

$$p(t) = I_{0}^2 R e^{-2\frac{t}{\tau}}$$

Energy absorbed by $R$:

$$w(t) = \frac{1}{2} LI_{0}^2 (1 - e^{-2\frac{t}{\tau}})$$

Example 07.6.

(Practice Problem 7.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 287])

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Example 07.7.

(Practice Problem 7.4 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 288])

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Example 07.8.

(Practice Problem 7.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 289])

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Singularity Functions

Definition 07.9 (Singularity Functions).

Functions that are discontinuous or have discontinuous derivatives.

Unit Step

$$u(t) = \begin{cases} 0,& t < 0 \\ 1,& t>0 \end{cases}$$

Represents an abrupt change in voltage/current

Unit Impulse

$$\delta(t) = \frac{\mathrm{d}}{\mathrm{d}t}u(t) = \begin{cases} 0,& t<0 \\ \text{Undefined},& t=0 \\ 0,& t>0 \end{cases}$$

Zero everywhere except at $t=0$, where it spikes to a value, which is equal to its area:

$$\int_{0^-}^{0^+} \delta (t) \, \mathrm{d}t = 1$$

Any function integrated with the impulse function is “sampled” at the point of impulse

$$\int_{a}^{b}f(t)\delta (t - t_{0}) \, \mathrm{d}t = f(t_{0})$$

Unit Ramp

$$r(t) = \int_{-\infty}^{t} u(\tau) \, \mathrm{d}\tau = \begin{cases} 0,& t \le 0 \\ t,& t \ge 0 \end{cases}$$

Complete Response

The complete response is the sum of the response from the stored energy (natural response) and the response from the independent source (forced response)

$$v = v_{n} + v_{f}$$

where $v_n = V_0 e^{-t\frac{t}{\tau}}$ and $v_f = V_s(1 - e^{-\frac{t}{\tau}})$

Alternatively: the sum of the transient response and the steady-state response
- $v_t$ decays to zero as $t \rightarrow \infty$
- $v_{ss}$ remains after $v_t$ has died

$$v = v_{t} + v_{ss}$$

where $v_t = (V_0 - V_s)e^{-\frac{t}{\tau}}$ and $v_{ss} = V_s$

Final equation (valid from both perspectives):

$$v(t) = v(\infty) + [v(0) - v(\infty)]e^{-\frac{t}{\tau}}$$

where $v(0)$ is the voltage at $t = 0^+$ and $v(\infty)$ is the steady-state value

Example 07.10.

(Example 7.10 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 301])

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Example 07.11.

(Practice Problem 7.10 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 302])

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Example 07.12.

Example 7.11 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 302]

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Example 07.13.

Practice Problem 7.11 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 303]

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RC Step Response

Complete response:

$$v(t) = \begin{cases} V_{0},& t < 0 \\ V_{s} + (V_{0}-V_{s})e^{-\frac{t}{\tau}},& t > 0 \end{cases}$$

If capacitor uncharged:

$$v(t) = V_{s}(1-e^{-\frac{t}{\tau}})u(t)$$

Current through capacitor:

$$\frac{V_{s}}{R} e^{-\frac{t}{\tau}}u(t)$$

RL Step Response

Complete response:

$$i(t) = \frac{V_{s}}{R} + \left( I_{0} - \frac{V_{s}}{R} \right) e^{-\frac{t}{\tau}}$$

If there is no $I_0$:

$$i(t) = \frac{V_{s}}{R} (1-e^{-\frac{t}{\tau}})u(t)$$

Voltage across inductor:

$$v(t) = V_{s}e^{-\frac{t}{\tau}}u(t)$$

Complete response (easy form):

$$i(t) = i(\infty) + [i(0) - i(\infty)]e^{-\frac{t}{\tau}}$$

Example 07.14.

(Example 7.12 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 306])

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Example 07.15.

(Practice Problem 7.12 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 306])

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Example 07.16.

(Example 7.13 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 307])

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Example 07.17.

(Practice Problem 7.13 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 308])

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First-Order Op Amps

Analyze with nodal analysis or reduce to Thevenin equivalent circuit

Example 07.18.

(Example 7.14 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 309])

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Example 07.19.

(Practice Problem 7.14 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 310])

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Example 07.20.

Example 7.15 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 310]

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Example 07.21.

(Practice Problem 7.15 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 311])

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Example 07.22.

(Example 7.16 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 312])

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Example 07.23.

(Practice Problem 7.16 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 313])

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Sources

Alexander & Sadiku, Fundamentals of Electric Circuits

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