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I am going through the following tutorial: Time-Response Identification of a Resistor–Capacitor (RC) Circuit

Why do we call the RC circuit a first order system?

I understand the order of a system is based on the highest derivative involved in its governing equation. The input governing equation of an RC circuit is described by equation 1 in the tutorial:

$$e_i-iR-\frac{1}{C}\int i \;dt =0$$

I understand the equation. But, I'm confused because there is no derivative term in it and yet it is called a first order system.

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The order of system gives the number of memory storing elements in the circuit.

What you have written above is a KVL equation of the circuit. But when you determine the order of the system you need to find the transfer function of the system. Laplace domain is normally used. The highest power of $s$ in the denominator in a strictly proper transfer function gives the order of the system.

So in the $s$-domain,say you take the output across the capacitor,

$\dfrac{V_o(s)}{V_i(s)} = \dfrac{\frac{1}{sC}}{R+\frac{1}{sC}}$

simplifying we get

$\dfrac{V_o(s)}{V_i(s)}= H(s)=\dfrac{\frac{1}{RC}}{s+\frac{1}{RC}}$

So the highest power of $s$ is given by 1. So the order of the system is 1.

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$$R\frac{di}{dt}-\frac1Ci=\frac{de_i}{dt}.$$

Isn't that a nice first order differential equation ?

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Another way to see it is rather than look for a 'derivative term' look for relative levels of differentiation, which is what you've got with $i$ and $\int i dt$.

Differentiate those two wrt $t$:

$\frac{di}{dt}$ and $i$.

Exactly what you're looking for...

But keep in mind the differential relationship defined in the ODE is the same.

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