2022-09-19 15:56 Status: #reference # Linear Circuit Transfer Functions an Introduction To Fast Analytical Techniques ## 1 Electrical Analysis – Terminology and Theorems [Extra Element Theorem (EET)](Extra%20Element%20Theorem%20(EET).md) and [nEET](Extra%20Element%20Theorem%20(EET).md) make extensive use of time constant methodology. There are six TF's:  ### 1.2 Theorems not to forget! 1. Voltage divider 2. Current divider 3. [thevenin's theorem](thevenin's%20theorem.md) #### DC Conditions Cap goes to open, inductor to short. [SPICE](SPICE.md) uses this to calculate bias points prior to .TRAN or .AC analysis. #### Output Nulls [Null Double Injection (NDI)](Null%20Double%20Injection.md) uses abstraction which translates mathematical zero to the Laplace-transformed world. An output null can form aty 0-Hz when you have a zero at the origin (shared b/w real and im axes) - cap in series with signal or inductor in parallel. Or highly underdamped notch filter. Takeaways: >1. A transfer function is a mathematical relationship linking an excitation signal (the input) to a response signal (the output). Excitation and response can appear at different terminals or ports but can also be observed across a common port. Thisisthe case for impedance and admittance transfer functions. >2. A transfer function is usually made of a numerator N and a denominator D but not always. When written in the form of a fraction, the zeros of the transfer function are the numerator roots while poles are the denominator roots. >3. A network featuring storage elements such as capacitors and inductors involve time constants. These time constants imply a resistive term R that ‘drives’ the concerned capacitor or inductor. This resistance can be observed, in certain conditions, by ‘looking’ into the considered element terminals while the said element is removed from the circuit. A time constant involving a capacitive term is τ = RC while a time constant characterizing an inductive term is τ = L/R. >4. When the port output resistance is evaluated, we have seen that the input source does not play a role in the resistance expression. When evaluating a port output resistance, the excitation voltage source is turned off (set to 0 V) and is replaced by a short circuit (a strap). For the dual case, if the excitation source is a current generator, it must be set to 0 A or become an open circuit. >5. Fast Analytical Circuits Techniques (FACTs) consist of expressing a transfer function with the above time constants and gainsin a clear and ordered form. Thisform issaid to be of lowentropy if you can tell where poles, zeros, and gains are located without having to rework the equation. >6. There are several important analysis techniques that you must know and be at ease with to start manipulating complex networks: the voltage divider, the current divider and Thévenin’s/Norton’s theorems. Superposition sets the foundations for the Extra Element Theorem we will discover in the next chapter. >7. By applying some of the simple techniques explored in this chapter, we were able to derive a transfer function without writing a single equation. In other words, we derived the transfer function by inspection. When circuits are not too complex, writing the transfer function by inspection is a real pleasure! ## 2 Transfer Functions >When you see a capacitor in series with the excitation signal or in series somewhere in the path to the output, then the dc component is blocked and it is likely that you find one or several zeros at the origin, depending how many dc blocks you can spot. The same applies for an inductor when in dc it shorts the signal path or the load to ground. >An underdamped 2nd-order low-pass filter can be physically realized to produce a really high output voltage at resonance ### 2.3 #### 2.3.1 Low Entropy Expressions  #### 2.3.3 Second Order polynomial      For: - Q<0.5, overdamped, 2 distinct, real roots - Q=0.5, equal real roots (critically damped) - Q>0.5, underdamped - Q->$\infty$, undamped  >As a conclusion, when the quality factor is low, well below 1, you can consider a 2nd-order denominator(or numerator, it also works for zeros) as two cascaded 1st-order filters.     Degenerative cases:  This capacitive loop circuit is only 3rd order, even though 4 caps  Same for the inductive node - only 4th order. #### 2.3.7 Zeros in the Network Conditions for null response - open along the stimulus path or short to redirect the stimulus.   ### 2.4 First Step Towards a Generalized 1st-order Transfer Function       where $\tau_1$ is the time constant associated with element 1 ($\beta_1 C_1$). #### 2.4.2 Obtaining the Zero with the Null Double Injection Nulling the output is different from shorting it (since a short still has an AC current!).  Similar to the 'virtual ground' of an op amp. Have a virtual 'test' generator while the source is still connected - this is the *double injection*. >A few remarks now: despite its presence in the schematic, the input source Vin plays no role in the expression of the zero position. This makessense: why would changing the modulation amplitude or the bias level would affect the zero position after all? Unless, of course, if there is one or several controlled-sourceslinked to Vin (which is not the case in the example). Apart from this particular case, Vin has no role in determining the capacitor/inductor resistance during a NDI Zeros change with probing position, poles do not. #### 2.4.3 Checking Zeros Obtained in Null Double Injection with SPICE For finding an input/output impedance or admittance using NDI,  >In Chapter 1, we have seen how SPICE could help us verify that our analytical results were correct.By biasing the energy-storing component terminals with a dc current source (1 A for practical reasons), measuring the voltage at the injection node while the excitation source was set to 0 gave us the resistance seen by the capacitor or the inductor in this particular condition. This is extremely useful, in particular for high-order transfer functions involving controlled sources and complex architectures. I have been able several times to track an error I made in a particular time constant derivation by comparing the value Mathcad® gave me and what the SPICE dc analysis delivered.  ### 2.4.4 Network Excitation The denominator stays constant for a voltage source applied in series or a current source in parallel, as these do not affect the original network:  >This property is interesting when you need to calculate several transfer functions pertinent to a network like gain or output impedance. Once you have the denominator, you can keep it for other calculations aslong asthe excitation source installed for the next transfer function determination does not change the structure when turned off ## 3.2 The Extra Element Theorem   >This theorem tells us that the gain of a linear system considering an extra element Z is equal to the gain of the system with the element Z physically disconnected further multiplied by a correction factor involving the extra element Z and two impedances seen from the extra element port when the output is nulled (Zn) and the excitation is zeroed (Zd)  >1. Identify the extra element Z. It can be an energy-storage element L or C but also a resistor R. The EET also works for dependent sources but this option will not be explored here. The extra element is usually selected as the element that ‘annoys’ you, with which, the transfer function becomes more complex to solve. >2. Decide whether you can short or remove the extra element. In some cases, if you remove the element, the transfer function may go to zero and (3.52) cannot be applied. It is the case for circuits having a zero at the origin for instance. Short it instead and use (3.55). Once the energy-storing element is put in its reference state (open or shorted), calculate the leading term Ajz0 or Ajz1. This term is named the reference gain. >3. Apply the techniques we have seen in Chapters 1 and 2. Set the excitation source to 0 and evaluate the resistance seen from the port created when the extra element is removed. You have Zd. >4. Use the Null Double Injection (NDI) to determine the resistance offered by the port created when the extra element is removed and the response is nulled. You have Zn. >5. If the reference circuit is purely resistive, Zd=Rd and Zn=Rn are resistances. The correction factor directly gives the corner frequencies of the circuit under study --- # Source [Linear_Circuit_Transfer_Functions_-_An_Introduction_to_Fast_Analytical_Techniques](../../Spaces/Career/Companies/Millibeam/docs/references/Linear%20Analysis/Linear_Circuit_Transfer_Functions_-_An_Introduction_to_Fast_Analytical_Techniques.pdf)