First-Order Circuits: Time-Domain Solution

Time-Domain Solution of First-Order Circuits

• Response to a DC Forcing Function considers an RC circuit with an initially uncharged capacitor and a switch closed at t = 0.
• KCL application yields the equation: $\frac{v(t) - V_s}{R} + C \frac{dv(t)}{dt} = 0$ which can be rearranged to: $\frac{dv(t)}{dt} + \frac{v(t)}{RC} = \frac{V_s}{RC}$
• The general form of a first-order linear differential equation is: $\frac{dx(t)}{dt} + a x(t) = f(t)$
• $x(t)$ represents the variable being solved for, $a$ is a constant coefficient, and $f(t)$ is the forcing function.
• The homogeneous solution $x_h(t)$ is the solution when $f(t) = 0$,represented as $\frac{dx_h(t)}{dt} + a x_h(t) = 0$
• Assuming $x_h(t) = A e^{st}$ and substituting, we find $s = -a$, resulting in $x_h(t) = A e^{-at}$
• The particular solution $x_p(t)$ satisfies the original equation, and its form depends on $f(t)$.
• Assuming $x_p(t) = B$ due to a constant $f(t)$, substitution yields $B = \frac{f(t)}{a}$, and $x_p(t) = \frac{f(t)}{a}$
• The complete solution is the sum: $x(t) = x_h(t) + x_p(t) = A e^{-at} + \frac{f(t)}{a}$
• $A$ is found using the initial condition $x(0)$, leading to $A = x(0) - \frac{f(0)}{a}$
• The complete solution is $x(t) = \left[ x(0) - \frac{f(0)}{a} \right] e^{-at} + \frac{f(t)}{a}$

Application to the $RC$ Circuit

• For the $RC$ circuit, $a = \frac{1}{RC}$, $f(t) = \frac{V_s}{RC}$, and $v(0) = 0$.
• The voltage solution derives as: $v(t) = V_s (1 - e^{-t/RC})$
• $\tau = RC$ is the time constant, with a larger $\tau$ indicating a slower response.
• After $5\tau$, the response reaches within 1% of its final value.
• General form of a first-order circuit solution with a DC forcing function: $x(t) = x(\infty) + [x(0) - x(\infty)] e^{-t/\tau}$

Example

• A circuit example describes a switch moving from position A to B at $t = 0$, determining $v(t)$ for $t > 0$ involves analysing $x(0)$, $x(\infty)$ and $\tau$
• For $t > 0$, the capacitor discharges through a $6k\Omega$ resistor, $v(\infty) = 0V$ and $\tau = 0.3s$ the voltage $v(t) = 12e^{-t/0.3} V$.

Source-Free $RC$ Circuit

• Consider a circuit with a capacitor initially charged to $V_0$ and the switch opened at $t=0$
• KCL yields: $\frac{v(t)}{R} + C \frac{dv(t)}{dt} = 0$, which rearranges to $\frac{dv(t)}{dt} + \frac{v(t)}{RC} = 0$
• The solution has the form : $v(t) = A e^{-t/RC}$
• $A$ is found using the stating voltage, $v(0) = V_0$, thus $v(t) = V_0 e^{-t/RC}$

Sequential Switching

• Circuits with switches changing positions at different times requires stage analysis.

Example

• A circuit with a switch changing from position A at $t=0$ to position B and at $t = 1s$ the switch is moved back to position A. Requires a separate time domain analysis for each step.

Tension, Compression, and Shear

• Stress $(\sigma)$ is the force per unit area acting on a solid, with the formula $\sigma = \frac{F}{A}$
• Tensile stress is caused by pulling; compressive by pushing; shear from parallel force
• Strain $(\epsilon)$ is the deformation of a solid due to stress expressed as $\epsilon = \frac{\Delta L}{L_0}$
• Tensile strain occurs when stretching, compressive with compression , and shear with parallel surface force.
• Elastic behavior is a material's ability to return to its original shape after stress removal.
• Plastic deformation is permanent deformation after stress removal.
• Yield strength is the stress level before plastic deformation starts.
• Ultimate strength is the maximum stress before fracture.
• Failure occurs when a material fractures.

Elastic Modulus

• Relates stress and strain in the elastic region
• Young's Modulus (E) applies to tensile/compressive stress, and is expressed $E = \frac{\sigma}{\epsilon}$
• Shear Modulus (G) applies to shear stress, expressed $G = \frac{\sigma}{\epsilon}$

Example

• A $1 cm$ steel rod under $10,000 N$ tensile force has a tensile stress of $127 MPa$
• The same rod will have a tensile strain of $0.000635$ (Young's modulus for steel = $200 GPa$).

Practice Problems

• A $2 mm$ copper wire under 500 N results in: $\sigma = 159 MPa$, $\epsilon = 0.00133
• A concrete column with: $\sigma = 10 MPa$, $\epsilon = 0.000333
• A steel bolt with a diameter of $10 mm$, is subjected to a shear force of $5,000 N$ will result in $\sigma = 63.7 MPa$, $\epsilon = 0.000796$

The Electromagnetic Spectrum

• The electromagnetic spectrum is the range of frequencies of electromagnetic radiation.
• The electromagnetic spectrum is comprised of gamma ray, X-Ray, Ultraviolet, Visible, Infrared, Microwave, and Radio

Temperature

• Temperature is a physical property that expresses hot and cold and a manifestation of thermal energy, measured by thermometers in Celsius, Fahrenheit or Kelvin.
• Microscopically, temperature is the average translational kinetic energy of microscopic particles.