Journal of Signal Processing, Vol.19, No.5, pp.209-217, September 2015
LECTURE
Circuit Theory Based on New Concepts and Its Application to Quantum Theory 3. Resonance and Eigen-Oscillation Obtained from Commensurate Transmission Line Circuit Nobuo Nagai１(Hokkaido University) and Takashi Yahagi2 (Signal Processing Technology Laboratory) E-mail:
[email protected],
[email protected] Abstract A lossless transmission line of length l is called a unit element. By showing numerical examples, we demonstrate that resonance and eigen-oscillation can be obtained using a unit element as a circuit element. Because resonance and eigen-oscillation are responses in the steady state, the transient response that occurs until the circuit reaches the steady state should be considered. Although the transient response has been conventionally considered a phenomenon of exponential attenuation that depends on a time constant, we numerically demonstrate that the transient response of a unit element is a phenomenon of discrete attenuation that can be expressed using a z-transform. In addition, resonance and eigen-oscillation involve reactive power, which is not related to energy, because the transient response of the circuit using a unit element is closely related to the phases expressed by the z-transform. We also numerically show that the reactive power is confined in the circuit. Keywords: unit element, Laplace transform, z-transform, transcient response, steady-state, lossless transmission line, resonance, active power, reactive power
1. Introduction In general, there are two types of circuit response: transient and steady-state. Conventionally, the transient response has been interpreted as a phenomenon of exponential attenuation that depends on a time constant [1]. However, it is necessary to consider that the reflection and transmission of waves are used in the circuit. In addition, resonance, in which the maximum available power from the power source (input) is consumed by the load (output) at a certain angular frequency, occurs in the steady state. Simply interpreting the transient phenomenon as attenuation that depends on a time constant cannot fully explain the resonance of the circuit in the steady state. Commensurate transmission line circuits include unit elements as circuit elements [2]. Because the length of all unit elements is assumed to be l, the waveform for commensurate transmission line circuits is considered to change at regular intervals. Waveforms that change at regular intervals can be analyzed using a z-transform. Hence, the transient response of commensurate transmission line circuits can be analyzed using a z-transform.
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In this session, we focus on a circuit using a unit element and give numerical examples of the transient phenomenon that occurs until the circuit reaches resonance from a state without an input, as well as those of the voltage and current along the unit element at resonance in the steady state. Thus, some examples of the transient phenomenon independent of a time constant are given. In circuit theory, two circuits can be connected to form another circuit. That is, two cascade matrices can be connected (cascade connection) to form another cascade matrix. A possible state of an integrated circuit is eigenoscillation caused by reactive power, in which the circuit oscillates as an integrated body while its energy is zero [3]. We will demonstrate that eigen-oscillation occurs in commensurate transmission line circuits.
2. Maxwell Equations and Telegrapher’s Equations First, we briefly explain the reason why we use telegrapher’s equations, instead of Maxwell equations, to discuss waves in this lecture series. There are various methods for analyzing Maxwell equations. One of them is vector analysis [4], which does not use the Laplace transform. In contrast, telegrapher’s equations, proposed by
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Heaviside [5], can be used in transmission circuit theory [6]. Therefore, applying the Laplace transform to telegrapher’s equations will determine a total of four waves, i.e., the forward and backward voltage and current waves. A cascade matrix is obtained from these four waves. The maximum available power [6] can be defined in transmission circuit theory and is used to obtain the resonance, under which the energy conservation law holds, in established transmission theory. However, it is reported that the energy conservation law for electromagnetic fields determined by the vector analysis of Maxwell equations is a source of paradoxes [4]. Namely, vector analysis is not necessarily suitable for analyzing the energy conservation law [4]. Then, why is the Laplace transform required in energy transmission theory? Reference [5] includes the following description: “In a 1918 letter referring to Heaviside’s pioneer work (inductive loading) in telegraphy he recalled “… my experiments with knotted clothes lines …done in the backyard at the age of 12 or 13.” What Heaviside was probably getting at are observations on how regularly spaced knots (mass concentrations) on a line affect the propagation of oscillations as one end of the line is wiggled. This is a mechanical analog of electrically exciting a telegraph cable.” On the basis of the regularly spaced knots in the above description, we can think of regularly spaced lumped loading coils, the roles of which in transmission theory should be clarified. In this session, we demonstrate that one of the roles is the Laplace transform in the transient phenomenon. Thus, we use lossless telegrapher’s equations, instead of Maxwell equations, to consider the energy conservation law in circuit theory, similarly to in the first and second sessions of this lecture series. The resonance of a transmission circuit constructed using a unit element obtained from lossless telegrapher’s equations and the role of the unit element during the transient phenomenon are discussed below. 3. Commensurate Transmission Line Circuit Using a Unit Element The unit elements used in commensurate transmission line circuits are circuit elements obtained from lossless telegrapher’s equations. The procedure for deriving unit elements was explained in Sessions 1 and 2. Therefore, only an outline of the procedure is given below. The solutions of the lossless telegrapher’s equations represent circuit elements and are of two types: transient and steady-state. Here, let us use the Laplace transform to consider the transient response. The lossless telegrapher’s equations are expressed using two functions, voltage and current, and can be expressed using a 2×2 matrix, similarly to the Riccati equation [7], as follows [8].
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d V x 0 sL V x 0 (3.1) dx I x sC 0 I x 0 Denoting the eigenvalue of the matrix in Eq. (3.1) as γ, we obtain (3.2) s LC The term γ is expressed in a different manner depending on the form of the equations. For example, s (3.3) j j u u Here, γ is the propagation constant, s is the variable of the Laplace transform, u is the propagation velocity, β is the phase constant, and ω is the angular frequency. Using two constants of integration, Ka(s) and Kb(s), the voltage-based solutions of the lossless telegrapher’s equations [Eq. (3.1)] are expressed as (3.4a) V x K a s exp su 1 x K b s exp su 1 x K s K s (3.4b) I x a exp su 1 x b exp su 1 x R0 R0 As mentioned above, the circuit element of length l obtained from the lossless telegrapher’s equations is called a unit element. Using Eqs. (3.4a) and (3.4b), the cascade matrix of the unit element is given by jR0 sin l cosl V 0 V l (3.5) j I l l l sin cos I 0 R 0 Only commensurate transmission line circuits constructed using unit elements can be expressed using a z-transform. Hence, transient responses can be obtained only for such circuits. Then, we focus on a transmission circuit constructed using a unit element, as shown in Fig. 3.1, and discuss the resonance of the circuit and the transient response that occurs until the circuit reaches resonance.
RG
R0
e(t)
0
x
RR
l
Fig.3.1 A commensurate transmission-line circuit using a unit element
Because the transient response is obtained using the Laplace transform, we assume that all the elements of the circuit shown in Fig. 3.1 have been subjected to the Laplace transform. Namely, a voltage source is used as the power source, where the voltage is given by
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(3.6) et EG exp jt To discuss the transient response, the voltage is assumed to be given by e(t) in general so that it can be applied to arbitrary waveforms such as rectangular waves. The Laplace transform of e(t) is written as E(s). L [e(t )] E ( s) (3.7) For the circuit shown in Fig. 3.1, the two constants of integration, Ka(s) and Kb(s), in Eqs. (3.4a) and (3.4b) are obtained as follows by considering the circuit’s termination conditions. K a ( s ) tG
E (s) 1 2 1 rR rG exp( su 1 2l )
(3.8a)
K b ( s ) tG
E ( s ) rR exp( su 1 2l ) 2 1 rR rG exp( su 1 2l )
(3.8b)
Here, E(s)/2 is half the voltage from the voltage source and indicates the incident voltage. The term tG is the instantaneous transmission coefficient for the voltage wave traveling from the voltage source to the unit element and is expressed as 2 R0 (3.9) tG R0 RG Moreover, the terms rR and rG are the instantaneous reflection coefficients for the voltage waves traveling from the unit element to the load and to the voltage source, respectively, and are expressed by R R0 (3.10a) rR R RR R0 R R0 (3.10b) rG G RG R0 Therefore, we obtain (3.10c) t G 1 rG
Let us express the values Ka(s) [Eq. (3.8a)] and Kb(s) [Eq. (3.8b)] using a z-transform, as used in signal processing. In Eqs. (3.8a) and (3.8b), u represents the propagation velocity of the voltage wave and u−12l is the time required for the wave to travel a distance of 2l. Therefore, we assume 2l (3.11) T u Using Eq. (3.11), we can define (3.12) exp( su 1 2l ) exp( sT ) z 1 Equation (3.12) indicates that the voltage wave lags by a time T when it makes a round trip along the unit element. Substituting Eq. (3.12) into Eqs. (3.8a) and (3.8b) will give the z-transforms of Ka(s) and Kb(s), respectively.
4. Numerical Examples For the circuit constructed using a unit element, as shown in Fig. 3.1, the voltage and current waves are represented using the instantaneous transmission and reflection coefficients. Therefore, both the transient and steady-state responses can be determined. In this section, the resonance
Journal of Signal Processing, Vol. 19, No. 5, September 2015
and the transient phenomenon that occurs until the circuit reaches resonance are explained using numerical examples. 4.1 Circuit configuration A problem given as an example in Section 5 of Ref. [3] is cited to obtain numerical examples. The problem focuses on a transient phenomenon in which waves are reflected seven or eight times along a unit element before the circuit reaches the steady state. We assume the instantaneous reflection coefficient to be 0.65 and adopt resonance condition (i) described in Session 2 to discuss the resonance of the circuit. The internal resistance of the voltage source (input port, RG) and the load resistance (output port, RR) are assumed to be 1 Ω as follows. (3.13a) RG RR 1 Because RG is not equal to the characteristic resistance of the unit element R0, the instantaneous reflection coefficient can be calculated. R0 is assumed as follows so that the instantaneous reflection coefficient is 0.65 and the voltage along the unit element takes a large value. (3.13b) R0 4.714 Therefore, rR and rG in Eq. (3.10) are equal to each other and we assume (3.13c) rR rG 0.65 The input voltage E(s)/2 is assumed to be 1. In this case, (3.13d) t G 1 rG 1.650 Figure 3.2 shows the circuit obtained by substituting the above numerical values into the equations.
1
・ 1
4.714
2ejωt
0 Fig.3.2
x
l
A circuit for a numerical example
The maximum available power P0 of the circuit shown in Fig. 3.2 is given by
EG
2
22 (3.14) 1 4 RG 4 For the circuit with the numerical values shown in Fig. 3.2, Ka(s) and Kb(s) given by Eqs. (3.8a) and (3.8b), respectively, are rewritten as E ( s) 1 1.65 (3.15a) K a ( s) t G 2 1 rR rG z 1 1 0.4225z 1 P0
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E ( s ) rR z 1 1.0725 z 1 (3.15b) 1 2 1 rR rG z 1 0.4225 z 1 Equations (3.15a) and (3.15b) can be used to determine frequency characteristics. However, the obtained characteristics are not based on effective values because these equations are complex functions expressed by a ztransform. K b ( s) t G
4.2 Voltage and current along unit element at resonance Frequency characteristics can be determined using the values obtained above. In the following, we focus on the voltage and current along the unit element of a circuit that achieves resonance in the steady state as well as on the transient phenomenon that occurs until the circuit reaches resonance. First, we examine the voltage and current at resonance in the steady state. By substituting the numerical values into Eq. (3.4), the voltage and current are respectively expressed as V x
1.65 1.0725 z 1 exp su 1 x exp su 1 x 1 1 0.4225 z 1 0.4225 z 1
0.35 0.2275 z 1 exp su 1 x exp su 1 x 1 1 0.4225 z 1 0.4225 z 1
Fig.3.3 Voltage and current on the unit element
(3.16a)
I x
(3.16b) For this circuit, resonance is assumed to occur when the phase for the length of the unit element is equal to π under resonance condition (i) described in Session 2. Therefore, z at resonance is given by (3.17) z 1 exp j 2 1 When the forward voltage wave given by Eq. (3.16a) is multiplied by the voltage transmission coefficient (= 0.35), the voltage at the load resistance is 1. When the forward current wave given by Eq. (3.16b) is multiplied by the current transmission coefficient (= 1.65), the current at the load resistance is 1. Therefore, the active power is 1 W, which is equal to the maximum available power, meaning that the circuit is resonant. The voltage and current along the unit element at resonance are expressed by Eqs. (3.16a) and (3.16b), respectively. Figure 3.3 shows the absolute values of voltage and current along the unit element when its length is assumed to be π. In this case, the product of the voltage and current is the apparent power, which is divided into active and reactive powers, as shown in Fig. 3.4. The active power is 1 at any point in the unit element, meaning that the unit element forms a lossless circuit. In Fig. 3.3, the voltage along the unit element has a peak, showing that a standing voltage wave exists even at resonance.
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Fig.3.4 Active and reactive powers on the unit element
4.3 Transient phenomenon The voltage and current in the steady state can be determined by substituting the complex numbers related to the line length for z−1 in the expressions for Ka(s) [Eq. (3.15a)] and Kb(s) [Eq. (3.15b)], respectively. Next, Ka(s) and Kb(s) are expanded as polynomials of z−1 as follows. (3.18a) K a s 1.65 0.697z 1 0.295z 2 K b s 1.0725 z 1 0.453 z 2 0.191z 3 (3.18b)
Table 3.1 Voltage values of forward and backward wave in unit element
1st term 2nd term 3rd term 4th term 5th term 6th term 7th term 8th term
Ka(s) 1.650 0.697 z－1 0.295 z－2 0.124 z－3 0.053 z－4 0.022 z－5 0.009 z－6 0.004 z－7
Kb(s) -1.073z-1 -0.453 z－2 -0.191 z－3 -0.081 z－4 -0.034 z－5 -0.014 z－6 -0.006 z－7 -0.003 z－8
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Each term of the above polynomial equations represents the amplitude of the transient voltage wave as it makes a round trip along the unit element. Therefore, the equations are considered to determine a transient response. Table 3.1 shows the values for the first eight terms of the polynomial equations. According to the table, the transient response that occurs until the circuit using a unit element reaches the steady state is due to multiple reflections within the unit element, which are expressed by the reflected and transmitted waves, which change discretely and discontinuously. This is different from the transient response of the lumped constant circuit, which depends on a time constant and is expressed as an exponential function. When the numerical values shown in Table 3.1 are used, it is possible to demonstrate the transient phenomenon, starting from (1) the initial state in which the input is zero and the voltage is also 0 V at any point in the unit element to (2) the transient state in which the voltage gradually increases with time as more power is input, and then to (3) the steady state in which resonance is achieved, as shown in Fig. 3.3. However, the use of complex voltage waves is required and the demonstration is very complicated. Therefore, we omit the demonstration of the transient response of the unit element here. Next, we determine the reflected wave at the internal resistance of the voltage source and the transmitted wave at the load resistance using the numerical values shown in Table 3.1 to obtain the transient phenomenon at the input and output terminals. To obtain the resonance that satisfies Eq. (3.17), it is not necessary to consider the complex power because only the active power is related to this resonance. In the following, we examine the transient phenomenon that occurs when a single rectangular pulse or a single-wavelength wave that can be expressed by actual numbers, such as a cosine wave, is input. As shown in Table 3.1, the voltages of the forward wave Ka(s) and the backward wave Kb(s) that make a round trip along the unit element are obtained. When these values are multiplied by the instantaneous voltage transmission coefficient (= 0.35), the voltage of the wave transmitted to the input/output line is obtained. The values obtained by multiplying the voltage of Ka(s) by 0.35 are summarized in Table 3.2 as the amplitudes of the transmitted wave. Table 3.2
Reflection and transmission amplitudes for the circuit shown in Fig.3.2 Amplitude of Amplitude of Reflection Transmission First response *0.65 0.578 time T lag 0.244 －0.375 time 2T lag 0.103 －0.159 time 3T lag 0.044 －0.067 time 4T lag 0.018 －0.028 time 5T lag 0.008 －0.012 time 6T lag 0.003 －0.005 time 7T lag 0.00 －0.002
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In addition, the values obtained by multiplying the voltage of Kb(s) by 0.35 are summarized in Table 3.2 as the amplitudes of the reflected wave. Note that the first wave that returns to the input line does not enter the unit element. Hence, the amplitude of the first response is calculated as the product of the amplitude (= 1) and the reflection coefficient (= 0.65). This is denoted as *0.65 in Table 3.2. The next response occurs after the wave makes a round trip along the unit element, i.e., after T s. The next response is obtained by multiplying the amplitude of each term in the polynomial equation of Kb(s) by 0.35. The transient response of the circuit using the unit element is explained by giving the responses using the numerical values in Table 3.2. The circuit used in the explanation is shown in Fig. 3.5(a). Semi-infinite-length lines with a characteristic resistance of 1 Ω are used for the left-side input and right-side output terminals. A unit element with a characteristic resistance of 4.714 Ω is inserted between the lines. The time required for the wave to travel the unit element is assumed to be T/2. Because the unit element with a characteristic resistance different from that of the semi-infinite-length lines exists between the lines, the reflected wave Vr, in addition to the incident wave Vi, exists at the input terminal, whereas the transmitted wave Vt exists at the output terminal. These voltage waves can also be regarded as the current waves or the normalized waves because the characteristic resistance of the lines is 1 Ω. Figure 3.5(b) shows the reflected wave Vr1 and the transmitted wave Vt1 obtained when a single rectangular pulse with an amplitude of 1 and a pulse width of
