Circuit Theory/First Order Circuits

First Order Circuits

First order circuits are circuits that contain only one energy storage element (capacitor or inductor), and that can therefore be described using only a first order differential equation. The two possible types of first-order circuits are:

1. RC (resistor and capacitor)
2. RL (resistor and inductor)

RL and RC circuits is a term we will be using to describe a circuit that has either a) resistors and inductors (RL), or b) resistors and capacitors (RC). These circuits are known as "First Order" circuits, because the solution to the circuit can be written as a first-order differential equation.

RL Circuits

An RL parallel circuit

An RL Circuit has at least one resistor (R) and one inductor (L). These can be arranged in parallel, or in series. Inductors are best solved by considering the current flowing through the inductor. Therefore, we will combine the resistive element and the source into a Norton Source Circuit. The Inductor then, will be the external load to the circuit. We remember the equation for the inductor:

$v(t)=L\frac{di}{dt}$

If we apply KCL on the node that forms the positive terminal of the voltage source, we can solve to get the following differential equation:

$i_{source}(t)=\frac{L}{R_n}\frac{d{i}_{inductor}(t)}{dt}+i_{inductor}(t)$

We will show how to solve differential equations in a later chapter.

RC Circuits

A parallel RC Circuit

No, RC does not stand for "Remote Control". An RC circuit is a circuit that has both a resistor (R) and a capacitor (C). Like the RL Circuit, we will combine the resistor and the source on one side of the circuit, and combine them into a thevenin source. Then if we apply KVL around the resulting loop, we get the following equation:

$v_{source}=RC\frac{d{v}_{capacitor}(t)}{dt}+v_{capacitor}(t)$

We will talk about the general solutions to these equations below.

First Order Solution

RL and RC circuits will both produce a first-order differential equation. The reader does not, however, require a prior knowledge of differential equations to read this topic, because we work through to the general solution of the equation. To understand the material fully, you would need a knowledge of derivatives and integrals. We will replace the capacitor voltages and the inductor currents in the previous equations with an x to signify that this will be a general solution to either type of problem. Here, we will consider a general equation of the form:

$\frac{dx(t)}{dt}+\frac{x(t)}{T_c}=k$

Where k is a constant value that corresponds to the source value (current for RL and voltage for RC circuits), possibly scaled by a certain factor based on the resistance, inductance, and/or capacitance of the circuit, when we divide through. T_c is a value known as the "Time Constant".

If we separate out the variables, we can get all the x terms on one side of the equation, and all the t terms on the other:

$\frac{dx}{x-k{T}_c}=-\frac{dt}{T_c}$

We can integrate both sides of this equation. The left side can be integrated with respect to x, and the left side can be integrated with respect to t. Performing the integrations gives us the following equation:

$\ln (x-k{T}_c)=-\frac{t}{T_c}+D$

Where D is an arbitrary constant of integration. If we raise both sides to e (to get rid of the natural log function) we will get the following final result:

$x(t)=k{T}_c+A{e}^{-\frac{t}{T_c}}$

$A=e^D$

It turns out that A is also the value of the initial condition of the circuit, x(0). Also kT_c is equal to the value of the steady-state value of the function. Combining this knowledge, we get the following equation:

$x(t)=x(\mathrm{\infty })+[x(0)-x(\mathrm{\infty })]e^{-\frac{t}{T_c}}$

Where:

• $x(\mathrm{\infty })$

Is the steady state value of x. This is our general result. Remember that x gets replaced by the function for either the capacitor voltage or the inductor current, to get the solution to an RC or an RL circuit, respectively.

Time Constant

The Time Constant, T_c, is an indicator of the amount of time it takes for a system to react to an input. The Time Constant is based on the amount of total resistance, capacitance, and inductance of a circuit. In general, the Time constant for an RL circuit is:

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

and the time constant for an RC circuit is:

$T_c=RC$

In general, from an engineering standpoint, we say that the system is at steady state after a time period of five time constants.