How to write Transfer Function by inspection

 Part2: Introduction to 2nd order system

In Part1, I have given you a glimpse of how to write the transfer function by inspection. As I have told you, the sole intention of the Part1 was only to talk about the notations (Subscript & Superscript) because they might be little bit confusing at starting. Once you gain a grip on that, rest will be a cakewalk for you. 

In Part1, I also have given some 1st order circuit example. There you could have argued that, it's way easy to mathematically derive the TF instead of this method. I agree, but those examples were just to make you habituated with the process. Here, in Part2, I have shown you some 2nd order circuits which will actually highlight the importance and handiness of this method. I will suggest you to first give them a try, then only you look at the solutions.

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First let's quickly recap the notations once again.

• 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 $L_1$, $\tau =\frac{L}{R_{eq}}$

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Let's take Example 3, which I have given as an assignment and then we shall move to 2nd order systems.

Example 3: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 3a

Figure 3: a) An RC filter, b) Model for $H^0$ calculation, c) Model for $H^1$ calculation, d) Model for $H^2$ calculation, e) Circuit for $\tau _{1}^{0}$ calculation, f) Circuit for $\tau _{2}^{0}$ calculation.

$$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}$$

1. $H^0$ :  H when $C_1=C_2=0$. 

 From Figure 3b, $H^0=\frac {R_2}{R_1+R_2}$

2. $H^1$ :  H when $C_1\rightarrow \infty$ and $C_2=0$ 

 

From Figure 3c, $H^1=1$

3. $H^2$ :  H when $C_2\rightarrow \infty$ and $C_1=0$ 

 

From Figure 3d, $H^2=0$

4. $\tau _{1}^{0}$ : Time constant associated with $C_1$ when $C_2=0$

From Figure 3e, $\tau _{1}^{0}=C_1 \times R _{1}^{0}=C_1\left ( R_1||R_2 \right )$

5. $\tau _{2}^{0}$ : Time constant associated with $C_2$ when $C_1=0$

From Figure 3f, $\tau _{2}^{0}=C_2 \times R _{1}^{0}=C_2\left ( R_1||R_2 \right )$

$a_{0}=H^{0}=\frac {R_2}{R_1+R_2}$

$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1\left ( R_1||R_2 \right )$

$b_1=\tau_1^0+\tau_2^0=\left(C_1+C_2\right)\left(R_1||R_2\right)$

$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}=\frac{\frac {R_2}{R_1+R_2}+C_1\left ( R_1||R_2 \right )s}{1+\left(C_1+C_2\right)\left ( R_1||R_2 \right )s}=\frac {R_2}{R_1+R_2}\frac{1+C_1R_1s}{1+\left(C_1+C_2\right)\left ( R_1||R_2 \right )s}$

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Example 4: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 3a

Figure 4: a) A 2nd order RC low-pass filter. b) Model for $H^0$ calculation, c) Model for $H^1$ calculation, d) Model for $H^2$ calculation, e)Model for $H^{12}$ calculation, f) Circuit for $\tau _{1}^{0}$ calculation, g) Circuit for $\tau _{2}^{0}$ calculation, h) Circuit for $\tau _{2}^{1}$ calculation.

$$H\left ( s \right )=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}$$

1. $H^0$ :  H when $C_1=C_2=0$. 

 From Figure 4b, $H^0=1$

2. $H^1$ :  H when $C_1\rightarrow \infty$ and $C_2=0$ 

 

From Figure 4c, $H^1=0$

3. $H^2$ :  H when $C_2\rightarrow \infty$ and $C_1=0$ 

 

From Figure 4d, $H^2=0$

4. $H^{12}$ :  H when $C_1,C_2\rightarrow \infty$ 

 

From Figure 4e, $H^{12}=0$

5. $\tau _{1}^{0}$ : Time constant associated with $C_1$ when $C_2=0$

From Figure 4f, $\tau _{1}^{0}=C_1 \times R _{1}^{0}=C_1R_1$

6. $\tau _{2}^{0}$ : Time constant associated with $C_2$ when $C_1=0$

From Figure 4g, $\tau _{2}^{0}=C_2 \times R _{2}^{0}=C_2\left ( R_1+R_2 \right )$

7. $\tau _{2}^{1}$ : Time constant associated with $C_2$ when $C_1 \rightarrow \infty$

From Figure 4h, $\tau _{2}^{1}=C_2 \times R _{2}^{1}=C_2R_2$

$a_{0}=H^{0}=1$

$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1R_1 \times 0+C_2\left(R_1+R_2\right) \times 0=0$

$a_{2}=\tau _{1}^{0}\tau_{2}^{1}H^{12}=C_1R_1 \times C_2R_2 \times 0=0$

$b_1=\tau_1^0+\tau_2^0=C_1R_1+C_2\left(R_1+R_2\right)$

$b_{2}=\tau _{1}^{0}\tau_{2}^{1}=C_1R_1 C_2R_2$

$H\left ( s \right )=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{1}{1+\left[C_1R_1+C_2\left(R_1+R_2\right)\right]s+C_1R_1C_2R_2s^2}$

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Example5: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 5a

Figure 5: a) A 2nd order RC band-pass filter. b) Model for $H^0$ calculation, c) Model for $H^1$ calculation, d) Model for $H^2$ calculation, e)Model for $H^{12}$ calculation, f) Circuit for $\tau _{1}^{0}$ calculation, g) Circuit for $\tau _{2}^{0}$ calculation, h) Circuit for $\tau _{2}^{1}$ calculation.

$$H\left ( s \right )=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}$$

1. $H^0$ :  H when $C_1=C_2=0$. 

 From Figure 5b, $H^0=0$

2. $H^1$ :  H when $C_1\rightarrow \infty$ and $C_2=0$ 

 

From Figure 5c, $H^1=0$

3. $H^2$ :  H when $C_2\rightarrow \infty$ and $C_1=0$ 

 

From Figure 5d, $H^2=\frac{R_2}{R_1+R_2}$

4. $H^{12}$ :  H when $C_1,C_2\rightarrow \infty$ 

 

From Figure 5e, $H^{12}=0$

5. $\tau _{1}^{0}$ : Time constant associated with $C_1$ when $C_2=0$

From Figure 5f, $\tau _{1}^{0}=C_1 \times R _{1}^{0}=C_1R_1$

6. $\tau _{2}^{0}$ : Time constant associated with $C_2$ when $C_1=0$

From Figure 5g, $\tau _{2}^{0}=C_2 \times R _{2}^{0}=C_2\left ( R_1+R_2 \right )$

7. $\tau _{2}^{1}$ : Time constant associated with $C_2$ when $C_1 \rightarrow \infty$

From Figure 5h, $\tau _{2}^{1}=C_2 \times R _{2}^{1}=C_2R_2$

$a_{0}=H^{0}=0$

$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1R_1 \times 0+C_2\left(R_1+R_2\right) \times \frac{R_2}{R_1+R_2}=C_2R_2$

$a_{2}=\tau _{1}^{0}\tau_{2}^{1}H^{12}=C_1R_1 \times C_2R_2 \times 0=0$

$b_1=\tau_1^0+\tau_2^0=C_1R_1+C_2\left(R_1+R_2\right)$

$b_{2}=\tau _{1}^{0}\tau_{2}^{1}=C_1R_1 C_2R_2$

$H\left ( s \right )=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{C_2R_2s}{1+\left[C_1R_1+C_2\left(R_1+R_2\right)\right]s+C_1R_1C_2R_2s^2}$

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Example6: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 6a

Figure 6: a) A 2nd order LC trap with CS amp b) Model for $H^0$ calculation, c) Model for $H^L$ calculation, d) Model for $H^C$ calculation, e)Model for $H^{LC}$ calculation, f) Circuit for $\tau _{L}^{0}$ calculation, g) Circuit for $\tau _{C}^{0}$ calculation, h) Circuit for $\tau _{C}^{L}$ calculation.

$$H\left ( s \right )=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}$$

1. $H^0$ :  H when $C=L=0$. 

 From Figure 6b, $H^0=-g_m\left(R_1||R_2\right)$

2. $H^L$ :  H when $L\rightarrow \infty$ and $C=0$ 

 

From Figure 6c, $H^L=0$

3. $H^C$ :  H when $C\rightarrow \infty$ and $L=0$ 

 

From Figure 6d, $H^C=-g_m\left(R_1||R_2\right)=H^0$

4. $H^{LC}$ :  H when $L,C\rightarrow \infty$ 

 

From Figure 6e, $H^{LC}=-g_m\left(R_1||R_2\right)=H^0$

5. $\tau _{L}^{0}$ : Time constant associated with $L$ when $C=0$

From Figure 6f, $\tau _{L}^{0}=\frac{L} {R _{L}^{0}}=\frac{L}{R_1+R_2}$

6. $\tau _{C}^{0}$ : Time constant associated with $C$ when $L=0$

From Figure 6g, $\tau _{C}^{0}=C \times R _{C}^{0}=C\times0=0$

7. $\tau _{C}^{L}$ : Time constant associated with $C$ when $L \rightarrow \infty$

From Figure 6h, $\tau _{C}^{L}=C \times R _{C}^{L}=C\left(R_1+R_2\right)$

$a_{0}=H^{0}$

$a_{1}=\tau _{L}^{0}H^{L}+\tau _{C}^{0}H^{C}=\frac{L}{R_1+R_2} \times 0+0 \times H^0=0$

$a_{2}=\tau _{L}^{0}\tau_{C}^{L}H^{CL}=\frac{L}{R_1+R_2} \times C\left(R_1+R_2\right) \times H^0=LCH^0$

$b_1=\tau_L^0+\tau_C^0=\frac{L}{R_1+R_2}$

$b_{2}=\tau _{L}^{0}\tau_{C}^{L}=LC$

$H\left ( s \right )=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{H^0+LCH^0s^2}{1+\frac{L}{R_1+R_2}s+LCs^2}=-g_m\left(R_1||R_2\right)\frac{1+LCs^2}{1+\frac{L}{R_1+R_2}s+LCs^2}$

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In Part3, I will take some 2nd order examples with active elements (MOSFETs). Till then, try to have a good grip on this technique and also revise the Look in impedance of MOSFET circuits.

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How to write Transfer Function by inspection

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.

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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.)

Figure 1:  Network with N ports in addition to the input and output with all the inductors and capacitors presented at the additional ports and no energy storing elements inside. 

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.

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• 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 $L_1$, $\tau =\frac{L}{R_{eq}}$

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Let's take some examples and try to understand the notation convention first.

1. $H^0$ :  H when all reactive elements are zero valued.
2. $H^i$ :  H when ith reactive element is infinite valued and all other reactive elements are zero valued.
3. $H^{ij}$ :  H when ith & jth reactive elements are infinite valued and all other reactive elements are zero valued.
4. $\tau _{i}^{0}$ : Time constant associated with ith reactive element when all other reactive elements are zero valued.
5. $\tau _{j}^{i}$ : Time constant associated with jth reactive element when ith reactive element is infinite valued and all other reactive elements are zero valued.
6. $\tau _{k}^{ij}$ : 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.

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Example1: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 2a

Figure 2: a) An RC low-pass filter. b) Model for $H^0$ calculation, c) Model for $H^1$ calculation, d) Circuit for $\tau _{1}^{0}$ calculation.

$$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}$$

1. $H^0$ :  H when $C_1=0$. 

 From Figure 2b, $H^0=1$

2. $H^1$ :  H when $C_1\rightarrow \infty$ 

 

From Figure 2c, $H^1=0$

3. $\tau _{1}^{0}$ : Time constant associated with $C_1$

From Figure 2d, $\tau _{1}^{0}=C_1 \times R _{1}^{0}=C_1R_1$

$a_{0}=H^{0}=1$

$a_{1}=\tau _{1}^{0}H^{1}=C_1R_1 \times 0=0$

$b_{1}=\tau _{1}^{0}=C_1R_1$

$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}=\frac{1}{1+C_1R_1s}$

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Example2: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 3a

Figure 3: a) An RC high-pass filter. b) Model for $H^0$ calculation, c) Model for $H^1$ calculation, d) Circuit for $\tau _{1}^{0}$ calculation.

$$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}$$

1. $H^0$ :  H when $C_1=0$. 

 From Figure 3b, $H^0=\frac {R_2}{R_1+R_2}$

2. $H^1$ :  H when $C_1\rightarrow \infty$ 

 

From Figure 3c, $H^1=1$

3. $\tau _{1}^{0}$ : Time constant associated with $C_1$

From Figure 3d, $\tau _{1}^{0}=C_1 \times R _{1}^{0}=C_1\left ( R_1||R_2 \right )$

$a_{0}=H^{0}=\frac {R_2}{R_1+R_2}$

$a_{1}=\tau _{1}^{0}H^{1}=C_1\left ( R_1||R_2 \right )$

$b_{1}=\tau _{1}^{0}=C_1\left ( R_1||R_2 \right )$

$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}=\frac{\frac {R_2}{R_1+R_2}+C_1\left ( R_1||R_2 \right )s}{1+C_1\left ( R_1||R_2 \right )s}=\frac {R_2}{R_1+R_2}\frac{1+C_1R_1s}{1+C_1\left ( R_1||R_2 \right )s}$

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Now I shall leave a problem for you. Give it a try.

Example3: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 4a

Figure 4a

Hint: Try to find the following terms: $H^0$,$H^1$,$H^2$,$\tau _{1}^{0}$,$\tau_2^0$

In the next blog, I will give the solution.

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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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