The understanding of transient phenomena in direct current (DC) circuits is essential for professionals across various Kenyan industries. Transient responses occur when a circuit experiences sudden changes, such as switching on or off, and they significantly influence the performance and safety of electrical systems in places like county hospitals, banks, and educational institutions. This chapter explores how voltages and currents evolve over time in resistive-inductive (R-L) and resistive-capacitive (R-C) circuits, providing the necessary theoretical foundation and practical calculation methods. Mastery of these concepts ensures effective management of electrical equipment and prevents damage caused by unexpected transient behaviors.

3.1 Derivation of Growth and Decay Equations in R-L and R-C Circuits

The transient response of electrical circuits containing inductors and capacitors differs fundamentally from that of purely resistive circuits due to energy storage elements. When a DC voltage is applied or removed, the current or voltage does not change instantaneously but follows exponential growth or decay patterns governed by differential equations. Understanding these patterns is crucial for professionals managing complex electrical systems, such as those in county government offices or retail businesses, where inductive motors or capacitive filters are common.

3.1.1 Derivation of Growth Equation in R-L Circuits

The growth equation describes how current increases when an inductive load is connected to a DC supply through a resistor. Upon closing the switch, the inductor initially opposes changes in current due to its stored magnetic energy, causing a gradual rise.

The governing differential equation for the current i(t) in an R-L circuit connected to a DC voltage V is:

V = L (di/dt) + R i

Rearranging:

L (di/dt) = V - R i

Dividing both sides by L:

di/dt = (V/L) - (R/L) i

This is a first-order linear differential equation. Using integrating factor methods, the solution for current growth starting from zero at time t=0 is:

i(t) = (V/R)(1 - e^(-Rt/L))

where:

• V/R is the final steady-state current,

• e is the base of natural logarithms,

• Rt/L is the dimensionless time variable.

The term τ = L/R is called the time constant of the circuit, representing the time for current to reach approximately 63.2% of its final value.

In practical terms, this means that in a hospital's emergency power system, the current through an inductive load such as a motor will not instantly reach its full value but will increase exponentially, affecting the design of protective devices.

3.1.2 Derivation of Decay Equation in R-L Circuits

When the supply voltage is suddenly removed from an R-L circuit, the current does not fall to zero instantly because the inductor releases stored energy. The current decays exponentially according to:

L (di/dt) + R i = 0

Rearranged:

di/dt = -(R/L) i

This homogenous differential equation has the solution:

i(t) = I₀ e^(-Rt/L)

where I₀ is the initial current at the moment the supply is disconnected.

The current decreases to 36.8% of its initial value after one time constant τ = L/R.

3.1.3 Derivation of Growth Equation in R-C Circuits

In an R-C circuit, when a DC voltage is applied, the capacitor charges through the resistor causing the voltage across the capacitor to rise exponentially.

The voltage across the capacitor v_c(t) satisfies:

V = v_c + R i

Since i = C (dv_c/dt), substituting:

V = v_c + R C (dv_c/dt)

Rearranged:

dv_c/dt + (1/RC) v_c = V / (RC)

The solution with zero initial voltage at t=0 is:

v_c(t) = V (1 - e^(-t/RC))

Here, τ = RC is the time constant representing the time for the capacitor voltage to reach 63.2% of the supply voltage.

In a retail business using power factor correction capacitors, understanding this charging behavior helps avoid sudden voltage changes that could damage equipment.

3.1.4 Derivation of Decay Equation in R-C Circuits

When the supply is disconnected, the capacitor discharges through the resistor, and the voltage across it decays exponentially according to:

dv_c/dt + (1/RC) v_c = 0

The solution is:

v_c(t) = V₀ e^(-t/RC)

where V₀ is the initial voltage across the capacitor at the moment of disconnection.

The voltage falls to 36.8% of its initial value after one time constant τ = RC.

3.1.5 Sketching of Growth and Decay Curves in R-L and R-C Circuits

Graphical representation of transient responses aids in visualizing how current or voltage changes over time, reinforcing understanding.

In R-L circuits:

• The growth curve of current starts at zero and asymptotically approaches the steady-state value V/R.

• The decay curve starts from initial current I₀ and exponentially decreases toward zero.

In R-C circuits:

• The voltage growth curve begins at zero and approaches the supply voltage V asymptotically.

• The voltage decay curve starts at initial voltage V₀ and exponentially decreases toward zero.

The time constant τ defines the characteristic time scale on these curves, with the value at t = τ marking approximately 63.2% of growth or 36.8% of decay.

3.1.6 Calculations Involving Growth and Decay Based on Time Constants

Calculating transient responses requires applying the growth or decay formulas with given circuit parameters.

Example calculation:

Given an R-L circuit with R = 20 Ω, L = 0.1 H, and supply voltage V = 240 V, find the current after 0.005 seconds.

Step 1: Calculate time constant

τ = L / R = 0.1 / 20 = 0.005 seconds

Step 2: Apply growth formula

i(t) = (V / R) (1 - e^(-t / τ))

Substitute values:

i(0.005) = (240 / 20) (1 - e^(-0.005 / 0.005))

Calculate exponent:

-0.005 / 0.005 = -1

Calculate exponential:

e^(-1) ≈ 0.3679

Calculate current:

i(0.005) = 12 (1 - 0.3679) = 12 × 0.6321 = 7.585 A

This calculation is crucial for sizing circuit breakers in county government offices to handle transient currents safely.

Chapter Summary

This chapter focused on the behavior of transients in direct current circuits, specifically examining the derivation of growth and decay equations for resistor-inductor (R-L) and resistor-capacitor (R-C) circuits. It covered how to sketch the growth and decay curves that represent voltage or current changes over time in these circuits. Calculations based on time constants were explored to quantify the rate at which these transients occur, providing a mathematical understanding of circuit responses. The chapter then discussed the practical application of time constants when switching inductive and capacitive loads, highlighting how these factors influence circuit performance and stability. Finally, the analysis of passive and active filters was introduced, demonstrating how these filters manipulate signal frequencies in electrical systems. Together, these topics build a comprehensive understanding of transient phenomena and their significance in designing and controlling DC circuits.