Fundamentals of Digital Measurement - Part 18: "Transfer Functions and Impedance"

A transfer function is a function that represents the relationship between the input and output of a system, and is defined as the ratio of the Laplace transforms of the input and output signals. That is, in Figure 1 on the right, if the Laplace transforms of the input and output signals are X(s) and Y(s), then the transfer function H(s) is:

······························ (1)

; This is how it works. "s" is called the Laplace operator (complex variable, s = σ + jω), and H(s) is generally the rational function (complex function) of "s". The transfer function is an extremely important function used in numerical models of electrical systems (circuit analysis, filter design, automatic control systems) and mechanical systems. In equation (1), if we set s = jω (or σ = 0), then H(jω) becomes a function of angular frequency ω, and is called the frequency transfer function (or frequency response function). Physically, it represents the frequency response of the system and is a quantity that can actually be measured with frequency analyzers such as FFT analyzers. In circuit systems, it shows the frequency characteristics of the circuit with respect to a steady sinusoidal signal. Hereafter, it will be written as H(ω).

<Note>
In previous discussions, we used "i" to represent the imaginary unit, but this time we will use "j," which is more common in engineering.

Next, let's consider impedance.

In Figure 2 on the right, when an AC voltage (complex sine wave) V is applied to a one-port pair circuit (linear time-invariant circuit), and an AC current (complex sine wave) I flows, the impedance Z is the same as in a DC circuit;

······························ (2)

The unit is ohms (Ω).

Furthermore, the reciprocal of Z is called admittance Y;

·····································(3)

The unit is Siemens (S). In Figure 2, Z in equation (2) is specifically defined as the drive point impedance and
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<Note>
All uppercase variables represent complex quantities (which have both magnitude and phase).

Since impedance Z is a complex number;

                    Z= R+jX                  ·····································(4)

The real part R is called resistance, and the imaginary part X is called reactance (positive for inductive and negative for capacitive).
Furthermore;

·································· (5)

When expressed as a vector, it can be represented as a vector (length and argument) on the complex plane. Also, admittance Y and impedance Z are reciprocals of each other;

·····································(6)

Once one is determined, the other can be easily found.

In the electrical circuit system shown in Figure 3

V1/I1: Drive point impedance
V2/I1: Transfer impedance
V2/V1: Voltage transfer function
I2/I1: Current transfer function

These terms are used, and while impedance and transfer function actually have different physical meanings, they are unified using the same complex number ratio in the Laplace transform domain (Laplace operator s) or frequency domain (jω).
It can be treated as such.

Table 1 summarizes the impedance and admittance of basic circuit elements for a steady sinusoidal signal with angular frequency ω.

Table 1 Impedance and Admittance of LCR
(The imaginary part of this value is obtained by adding 'j' to it; this is the actual value.)

○ Circuit Example 1: RC Series Circuit (Figure 4)

The impedance Z is in series;

·····················(7)

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If we consider Z in equation (7) as a vector and plot its locus on the complex plane with ω as the variable, then ω = 0
Since Z = -∞ and ω = ∞, Z = R, so we get Figure 5.

Such a trajectory is called a vector trajectory. In frequency analyzers, it is also called a Nyquist plot.
It is. A characteristic feature of the impedance trajectory is that the real part is usually a pure resistance (R > 0), so it is negative.
It never becomes that way, and its trajectory is always either in the first quadrant (inductive reactance) or the fourth quadrant (capacitive reactance).
This means it is limited to tance. Such a complex plane is specifically called the impedance plane.
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Next, considering Figure 4 as a transfer system with input V1 and output V2, its voltage transfer function H(ω) is:

·····················(8)

This is the result. Also, the gain and phase are;

·····················(9)

The vector locus (a) and Bode plot (gain characteristics: (b), phase characteristics: (c)) of H(ω) are shown in Figure 6 below.
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The transfer function in equation (8) is called a first-order lag element in control systems.
As such, it is commonly called an RC low-pass filter (first order), and its time constant is T = CR. Angular frequency
The point where ω = 1/T corresponds to the cutoff frequency (-3dB).
Furthermore, as shown in Figure 6, the vector locus traces a semicircle. The geometric explanation for this is as follows:
In equation (8), if we let x be the real part of H(ω) and y be the imaginary part, then;

···················(10)

from;

···················(11)

Thus, we can see that the locus of P(x, y) is a circle with center (1/2, 0) and radius 1/2.
Alternatively, if we consider it as a complex number;

···················(12)

It can be expressed as follows, and the complex number equation (12) refers to a circle with center (1/2, 0) and radius 1/2, as above.
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Another explanation is that if we let H'(ω) be the reciprocal of H(ω), then from equation (8):

H '(ω ) = 1 +jω CR                                              ···················(13)

Therefore, the vector locus of H'(ω) starts from a point A(1,0) on the real axis and is parallel to the imaginary axis.
It will be a straight line (only in the first quadrant). Draw a semicircle with diameter OA in the fourth quadrant, and on H'(ω)
Let Q be the point of interest. Draw a line such that the angle θ between OQ and the real axis is equal to this line, and let P be the point of intersection with the semicircle.
Thus, OP = 1/OQ. That is, this point P is any point on H(ω), and at ω = 0, point A,
We draw the locus of the origin O (a semicircle in the fourth quadrant) where ω = ∞.

As in this example, generally, the reciprocal of a complex number whose vector locus is a straight line on the complex plane is a complex number
The trajectory of a cleric is always a circle.
Next, let's consider the Laplace transform region (s-plane) by setting jω = s in H(ω) of equation (1).
Then H(s) is;

···················(14)

Thus, the solution to the denominator of equation (14) is -1/T (called a pole of H(s)), and if we call this point Q, then Q lies on the real axis in the s-plane. (Figure 8) Setting s = jω and varying ω from 0 to ∞ is equivalent to moving point P on the imaginary axis in Figure 8, and the vector of the denominator (complex number) of H(ω) corresponds to QP;

···················(15)

···················(16)

*T = CR
The hyphen (*) is used because a complex number is in the denominator.

;And, just like in equation (9), we can find the gain and phase of H(ω).
Therefore, the poles of the transfer function H(s) are important parameters for analyzing the characteristics of H(s).

○ Circuit Example 2: RC Series-Parallel Circuit (Figure 9)

In Figure 9, the drive point impedance Z is;

(17)

(18) 

At ω=0, x = R1+R2, y = 0
ω=∞, x = R1, y = 0

Comparing equation (8) and equation (17), the vector locus of Z in equation (17) clearly has a diameter of R².
It can be seen that this is equivalent to moving the semicircle by R1 in the real axis direction (Figure 10).

The circuit in Figure 9 is a basic equivalent circuit for the electrochemical impedance of a battery or similar device.
The vector locus of a circuit without inductive reactance (L component) will only be in the fourth quadrant.
Therefore, in the field of electrochemical impedance, only the imaginary part is inverted, and the vector is placed in the first quadrant.
The most common method is to draw the trajectory. This display is called a call-call plot (see example in Figure 11).

This concludes our series, "Fundamentals of Digital Measurement." Thank you for reading my humble writing.
Thank you very much.

Starting next time, a different person will be writing the article on the topic of audio-related applications.
We hope you will continue to enjoy reading our work.

○ References

1. "Basic Electrical Circuits I" (by Masamitsu Kawakami), Corona Publishing Co., Ltd.
2. "Electrical Circuit Theory" (by Masanao Ariga), Morikita Publishing.

(Excerpt from the email newsletter issued on March 18, 2009)