Part1: Introduction to 1st order system
Transfer function plays a crucial role when frequency analysis comes into picture. Poles and Zeroes are the most important elements to analyze any loop stability. We know that most of the systems are closed loop only, and if there is a loop, stability analysis is a must. For lower order systems, you can find the poles/zeroes intuitively to some extend, but as the order increases (even in 2nd order system) it becomes very difficult and complicated, so there is no alternatives than to write the transfer function. Now if you go with basic KCL, KVL approach, it will be too lengthy and time consuming. Instead of that, Prof. Ali Hajimiri had come with a "Generalized Time- and Transfer-ConstantCircuit Analysis" which I will be discussing here. If you understand this technique very clearly, you will be able to write higher order transfer function very easily by inspection within fraction of minutes.
Any network with energy-storing (reactive) elements can be represented as a system with external ports (in addition to the input and output) with no frequency-dependent elements inside (e.g., containing only resistors and dependent voltage and current sources) and each reactive element (namely inductors and capacitors) attached to one of the ports, as shown in Figure 1. (If more than one reactive element is connected to the same pair of terminals, each one of them is assumed to have a port of its own with a separate index.)
The transfer function of a linear system with lumped elements can be written as :
\[H\left ( s \right )=\frac{a_0+a_1s+a_2s^2+...+a_ms^m}{1+b_1s+b_2s^2+...+b_ns^n}\]
\[a_{0}=H^{0}\]
\[a_{1}=\sum_{i=1}^{N}\tau _{i}^{0}H^{i}\]
\[a_{2}=\sum_{i}^{1\leqslant i< }\sum_{j}^{j\leqslant N}\tau _{i}^{0}\tau _{j}^{i}H^{ij}\]
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\[a_{n}=\sum_{i}^{1\leqslant i< }\sum_{j}^{j<k}\sum_{k...}^{...\leqslant N }\tau _{i}^{0}\tau _{j}^{i}\tau _{k}^{ij}...H^{ijk...}\]
\[b_{1}=\sum_{i=1}^{N}\tau _{i}^{0}\]
\[b_{2}=\sum_{i}^{1\leqslant i< }\sum_{j}^{j\leqslant N}\tau _{i}^{0}\tau _{j}^{i}\]
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\[b_{n}=\sum_{i}^{1\leqslant i< }\sum_{j}^{j<k}\sum_{k...}^{...\leqslant N }\tau _{i}^{0}\tau _{j}^{i}\tau _{k}^{ij}...\]
Yes! I know these equations are looking really scary but hold on with me. Here, the most important part is the notations. If you clearly able to understand the meanings of the notations, believe me you will be able to write any transfer function within a minute.
- SubScript : denotes the index of the reactive element, from where we have to calculate the $R_{eq}$ in order to find the time constant $\tau$
- SuperScript : denotes the index(es) of the infinite valued element(s). An index 0 in the superscript simply indicates that no reactive element is infinite values i.e. all elements are at their zero values.
Note :
- From the above conventions, I hope it is clear that there can be single or multiple index(es) in the superscript, but there has to be only one index in the subscript.
- For a capacitor $C_1$, $\tau =R_{eq}C_1$
- For an inductor $L_1$, $\tau =\frac{L}{R_{eq}}$
Let's take some examples and try to understand the notation convention first.
- $H^0$ : H when all reactive elements are zero valued.
- $H^i$ : H when ith reactive element is infinite valued and all other reactive elements are zero valued.
- $H^{ij}$ : H when ith & jth reactive elements are infinite valued and all other reactive elements are zero valued.
- $\tau _{i}^{0}$ : Time constant associated with ith reactive element when all other reactive elements are zero valued.
- $\tau _{j}^{i}$ : Time constant associated with jth reactive element when ith reactive element is infinite valued and all other reactive elements are zero valued.
- $\tau _{k}^{ij}$ : Time constant associated with kth reactive element when ith and jth reactive elements are infinite valued and all other reactive elements are zero valued.
Let's jump to some examples.
Example1: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 2a
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$a_{0}=H^{0}=1$
$a_{1}=\tau _{1}^{0}H^{1}=C_1R_1 \times 0=0$
$b_{1}=\tau _{1}^{0}=C_1R_1$
$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}=\frac{1}{1+C_1R_1s}$
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$a_{0}=H^{0}=\frac {R_2}{R_1+R_2}$
$a_{1}=\tau _{1}^{0}H^{1}=C_1\left ( R_1||R_2 \right )$
$b_{1}=\tau _{1}^{0}=C_1\left ( R_1||R_2 \right )$
$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}=\frac{\frac {R_2}{R_1+R_2}+C_1\left ( R_1||R_2 \right )s}{1+C_1\left ( R_1||R_2 \right )s}=\frac {R_2}{R_1+R_2}\frac{1+C_1R_1s}{1+C_1\left ( R_1||R_2 \right )s}$
Now I shall leave a problem for you. Give it a try.
Are you thinking, you can write these first order transfer functions more easily without doing all of these? May be you are right. But, the intention to start with the 1st order systems was solely to make you understand about the notations. In Part 2, we shall move to 2nd order transfer functions, where you will actually understand that how handy and quick this method is.
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