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.
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 $L_1$, $\tau =\frac{L}{R_{eq}}$
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
.png)
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, $H^1=1$
.png)
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.
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
)$
$b_1=\tau_1^0+\tau_2^0=\left(C_1+C_2\right)\left(R_1||R_2\right)$
$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}=\frac{\frac
{R_2}{R_1+R_2}+C_1\left ( R_1||R_2 \right
)s}{1+\left(C_1+C_2\right)\left ( R_1||R_2 \right )s}=\frac
{R_2}{R_1+R_2}\frac{1+C_1R_1s}{1+\left(C_1+C_2\right)\left ( R_1||R_2
\right )s}$
$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1\left ( R_1||R_2 \right )$
$b_1=\tau_1^0+\tau_2^0=\left(C_1+C_2\right)\left(R_1||R_2\right)$
$H\left ( s \right )=\frac{a_0+a_1s}{1+b_1s}=\frac{\frac
{R_2}{R_1+R_2}+C_1\left ( R_1||R_2 \right
)s}{1+\left(C_1+C_2\right)\left ( R_1||R_2 \right )s}=\frac
{R_2}{R_1+R_2}\frac{1+C_1R_1s}{1+\left(C_1+C_2\right)\left ( R_1||R_2
\right )s}$
Example 4: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 3a
.png)
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, $H^1=0$
.png)
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.
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
$
$a_{0}=H^{0}=1$
$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1R_1 \times
0+C_2\left(R_1+R_2\right) \times 0=0$
$a_{2}=\tau _{1}^{0}\tau_{2}^{1}H^{12}=C_1R_1 \times C_2R_2 \times
0=0$
$b_1=\tau_1^0+\tau_2^0=C_1R_1+C_2\left(R_1+R_2\right)$
$b_{2}=\tau _{1}^{0}\tau_{2}^{1}=C_1R_1 C_2R_2$
$H\left ( s \right
)=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{1}{1+\left[C_1R_1+C_2\left(R_1+R_2\right)\right]s+C_1R_1C_2R_2s^2}$
$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1R_1 \times
0+C_2\left(R_1+R_2\right) \times 0=0$
$a_{2}=\tau _{1}^{0}\tau_{2}^{1}H^{12}=C_1R_1 \times C_2R_2 \times
0=0$
$b_1=\tau_1^0+\tau_2^0=C_1R_1+C_2\left(R_1+R_2\right)$
$b_{2}=\tau _{1}^{0}\tau_{2}^{1}=C_1R_1 C_2R_2$
$H\left ( s \right
)=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{1}{1+\left[C_1R_1+C_2\left(R_1+R_2\right)\right]s+C_1R_1C_2R_2s^2}$
Example5: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 5a
.png)
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, $H^1=0$
.png)
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.
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
$
$a_{0}=H^{0}=0$
$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1R_1 \times
0+C_2\left(R_1+R_2\right) \times \frac{R_2}{R_1+R_2}=C_2R_2$
$a_{2}=\tau _{1}^{0}\tau_{2}^{1}H^{12}=C_1R_1 \times C_2R_2 \times
0=0$
$b_1=\tau_1^0+\tau_2^0=C_1R_1+C_2\left(R_1+R_2\right)$
$b_{2}=\tau _{1}^{0}\tau_{2}^{1}=C_1R_1 C_2R_2$
$H\left ( s \right
)=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{C_2R_2s}{1+\left[C_1R_1+C_2\left(R_1+R_2\right)\right]s+C_1R_1C_2R_2s^2}$
$a_{1}=\tau _{1}^{0}H^{1}+\tau _{2}^{0}H^{2}=C_1R_1 \times
0+C_2\left(R_1+R_2\right) \times \frac{R_2}{R_1+R_2}=C_2R_2$
$a_{2}=\tau _{1}^{0}\tau_{2}^{1}H^{12}=C_1R_1 \times C_2R_2 \times
0=0$
$b_1=\tau_1^0+\tau_2^0=C_1R_1+C_2\left(R_1+R_2\right)$
$b_{2}=\tau _{1}^{0}\tau_{2}^{1}=C_1R_1 C_2R_2$
$H\left ( s \right
)=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{C_2R_2s}{1+\left[C_1R_1+C_2\left(R_1+R_2\right)\right]s+C_1R_1C_2R_2s^2}$
Example6: Determine $H\left(s\right)=\frac{v_{out}}{v_{in}}$ from Figure 6a
.png)
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, $H^L=0$
.png)
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.
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) $
$a_{0}=H^{0}$
$a_{1}=\tau _{L}^{0}H^{L}+\tau _{C}^{0}H^{C}=\frac{L}{R_1+R_2} \times
0+0 \times H^0=0$
$a_{2}=\tau _{L}^{0}\tau_{C}^{L}H^{CL}=\frac{L}{R_1+R_2} \times
C\left(R_1+R_2\right) \times H^0=LCH^0$
$b_1=\tau_L^0+\tau_C^0=\frac{L}{R_1+R_2}$
$b_{2}=\tau _{L}^{0}\tau_{C}^{L}=LC$
$H\left ( s \right
)=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{H^0+LCH^0s^2}{1+\frac{L}{R_1+R_2}s+LCs^2}=-g_m\left(R_1||R_2\right)\frac{1+LCs^2}{1+\frac{L}{R_1+R_2}s+LCs^2}$
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.
$a_{1}=\tau _{L}^{0}H^{L}+\tau _{C}^{0}H^{C}=\frac{L}{R_1+R_2} \times
0+0 \times H^0=0$
$a_{2}=\tau _{L}^{0}\tau_{C}^{L}H^{CL}=\frac{L}{R_1+R_2} \times
C\left(R_1+R_2\right) \times H^0=LCH^0$
$b_1=\tau_L^0+\tau_C^0=\frac{L}{R_1+R_2}$
$b_{2}=\tau _{L}^{0}\tau_{C}^{L}=LC$
$H\left ( s \right
)=\frac{a_0+a_1s+a_2s^2}{1+b_1s+b_2s^2}=\frac{H^0+LCH^0s^2}{1+\frac{L}{R_1+R_2}s+LCs^2}=-g_m\left(R_1||R_2\right)\frac{1+LCs^2}{1+\frac{L}{R_1+R_2}s+LCs^2}$
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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