Order of an RC circuit and independent initial conditions

Question

I am trying to determine the order of the transfer function of the RC circuit shown below. I learned that the order of the transfer function is equal to the number of independent initial conditions that I can assign to the capacitors in the circuit, which are also known as independent state variables.

In the circuit below I found that I can assign two independent initial voltage values to both the capacitors. When I obtained the transfer function using voltage division, it is only a first order system with time constant R1*( C1 || C2). Where am I going wrong?

Answer 1

The key is whether or not these voltages are independent of each other, or not. If they are fully independent, then yes. Each represents an order. If they are dependent by nothing more than a constant, let's say, then that rule doesn't apply.

In this case, \$\frac{V_{_{\text{C}_1}}}{V_{_{\text{C}_2}}}=\frac{C_2}{C_1}=k\$. So \$V_{_{\text{C}_1}}=k\cdot V_{_{\text{C}_2}}\$.

And they are not independent. One depends entirely upon the other.

So 1st order.

Obviously, if you had two capacitors in parallel, this would NOT be two more orders rather than one more order. The capacitors could be combined. You know that, I'm sure. So this is a prime example.

But if the voltages across two capacitors are the same by any constant factor (not only a constant factor of 1), then they are still dependent and cannot be counted as two orders. Only one order between them.

Independence is the key to counting orders.

Answer 2

From voltage division we can get:

$$\frac{Vo(s)}{Vi(s)} = \frac{\frac{1}{sC_2}}{\frac{1}{sC_1}+R_1+\frac{1}{sC_2}}$$

then

$$\boxed { \frac{Vo(s)}{Vi(s)} = \frac{sC_1}{R_1C_1C_2s^2+(C_1+C_2)s} \qquad [1] } $$

which represents a second order system.

The problem is the pole-zero cancellation at origin, in such case, that transfer system would be:

$$\frac{Vo(s)}{Vi(s)} = \frac{C_1}{R_1C_1C_2s+C_1+C_2} \qquad [2]$$

That occurs when equations such as (1) are composed by derivatives of input and output. The corresponding differential equation is

$$\frac{dv_o^2(t)}{dt^2}+ \frac{C_1+C_2}{R_1C_1C_2}\frac{dv_o(t)}{dt}= \frac{1}{R_1C_2}\frac{dv_i(t)}{dt}$$

a second order one.

The problem with this cancellation is that the ZIR (Zero Input Response) in \$(1)\$ and \$(2)\$ are different. For a more detailed discussion, also involving initial conditions and controllability, you can refer to my other answer here.