Physics
Complex Impedance
Complex impedance is a measure of the opposition that a circuit presents to the flow of alternating current. It is represented as a complex number and takes into account both resistance and reactance. In a circuit, impedance is the combination of resistance and reactance, and it is used to analyze and design electrical circuits.
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10 Key excerpts on "Complex Impedance"
eBook - PDFCircuit Analysis with PSpice
A Simplified Approach
- Nassir H. Sabah(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
Then, Definition: Impedance Z is the ratio of the voltage phasor V m ∠θ v across a subcircuit to the current phasor I m ∠θ I flow- ing through the subcircuit in the direction of a voltage drop. The subcircuit could be a single R, L, or C, or a combination of these elements. The subcircuit should not contain independent sources but could contain dependent sources, as long as the con- trolling variables of these sources are all within the subcircuit. Z V I V I m v m i m m = = Ð Ð = Ð V I q q q (8.40) where θ = θ v − θ i (Figure 8.16a). When V is in volts and I is in amperes, Z is in ohms. Since Z is in general com- plex, it can be expressed as Z R jX = + (8.41) (Figure 8.16b). It should be emphasized that although Z is in general complex, it is not a phasor, because it is not a complex sinusoidal function of time in which the time variation has been suppressed. In fact, the exponential time variation cancels out in Equation 8.40. Z is there- fore drawn on a separate diagram from that of V and I. Moreover, the reason independent sources are excluded from the subcircuit in the definition of impedance is that in order to make the concept of impedance useful, V + I – a b FIGURE 8.15 Definition of impedance. 212 Circuit Analysis with PSpice: A Simplified Approach impedance should depend only on the values of the passive circuit elements and their configuration, inde- pendently of any applied excitation due to independent sources. Dependent sources can be included, as long as their controlling voltages or currents are within the sub- circuit. The effect of the dependent sources in this case is to alter values of R, L, and C. In the case of an ideal resistor, V and I are in phase (Equation 8.22), which means that Z is real and equal to R. It follows that R in Equation 8.41 is the usual resis- tance that we have considered in previous chapters, and that for an ideal resistor, X = 0.
eBook - PDF- R A Ashen(Author)
- 2016(Publication Date)
- Butterworth-Heinemann(Publisher)
Unfortunate-ly, at present, stability problems preclude the widespread use of ferro-electrics in capacitors. Since current is the rate of charge movement, Equation (2.24) may be differentiated with respect to time to give the following relation-ship between voltage and current for a capacitor. Q = CV (2.24) dt (2.27) (2.25) (b) Figure 2.9 Voltage-current relationship for capacitance (a) time variation (b) phasor diagram For sinusoidal supply conditions, a similar procedure may be carried out for capacitance as was performed for inductance, and this is illustrated in Figure 2.9. Here, voltage may be conveniently represen-ted by the expression y /lV sin (cot-cp), and the resulting current is y /lcoCV cos (cot — cp). These waveforms are shown in Figure 2.9(a). In this case, the current leads the voltage by n/2 rad, as opposed to the equivalent lag in the inductive case. The corresponding phasor dia-gram is shown in Figure 2.9(b). It should now be self-evident that V = — I (2.28) coC 1 /(coC) is termed capacitive reactance and is given the symbol X c . The presence of —j in the equation indicates the n/2 rad lag of the voltage behind the current in the capacitor. Capacitance and capacitive reactance 25 (a) 26 Use of complex numbers in a.c. circuit analysis 2.7 Impedance The impedance of a circuit, or any individual circuit component, is defined as the ratio of potential difference between the ends of the circuit to current flowing through it. Under sinusoidal a.c. conditions, impedance can be expressed using complex notation as has been previously developed. Hence it may be expressed as RJX L , and —jX c for resistance, inductance and capacitance respectively. The benefits of complex notation become apparent when the im-pedance of circuits containing many elements is to be evaluated. Under sinusoidal supply conditions, elements may be combined in series and parallel, using complex notation, in a manner essentially identical to that employed in d.c.
eBook - PDF- Johann Kraft(Author)
- 2024(Publication Date)
- Future Managers(Publisher)
A new concept called impedance is now introduced. The impedance of the network in this instance (RL-series networks) is the opposition offered to the flow of an alternating current and is the vector sum of the reactance of the inductor and the resistance of the resistor. Definitions Impedance – the vector sum or combination of resistance and reactance quantised as Z and measured in ohm Vector sum – the result of adding two or more vectors together This impedance is mathematically given by: Z = √ R 2 + X L 2 Observe Figure 2.6 and you will notice that the above expression is derived by applying the Pythagorean Theorem to the triangle. Similarly, the supply voltage expression can be derived by applying Pythagoras rule using Figure 2.5 (b) and is mathematically given by: V S = √ V R 2 + V L 2 V R = I × R and V L = I × X L The current in the network is mathematically given by: I = V S Z You will further notice that there is a phase angle θ between the current flowing in the network and the supply voltage. This is termed the phase angle and is mathematically given by: θ = tan –1 X L R or θ = tan –1 V L V R Where: θ = angle in degrees X L = reactance in ohm V L = voltage in volt V R = voltage in volt R = resistance in ohm I = current in ampere V S = voltage in volt Z = impedance in ohm 75 Module 2 • Alternating current theory Definitions Phase angle – the characteristic or angular component of a periodic wave Pythagorean Theorem –a rule in geometry which states that, in a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides eLink Visit this link to learn more about applying the Pythagoras theorem: futman.pub/PythagorasTheorem1 Example 2.6 Consider the given network diagram. R V L V R L I V S = 250 V/50 Hz 25 mH 10 ohm Figure 2.7 Determine by calculation from the given information on the network diagram the: 1. Inductive reactance of the inductor. 2. Impedance of the network.
eBook - PDFElectronics
A Course Book for Students
- G. H. Olsen(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Illustrations of the effect of two operators viz (-1) and j voltage v R is equal to i R , and the phasor representing it is in the reference direction. The voltage v L is equal to icoL, but to show that it is advanced by 90° we have v L = jioeL. It is customary to keep the j and ω together so we would write v L = ijoeL. The supply voltage ν is the phasor sum of v R and v L , i.e. ν = v Ä + v L = iR + i )0 )L = i(R + ]ù)L). Dividing by i we have v/i = R + )ct )L. This is the imped-ance Z. The expression for Ζ is, therefore, a complex one made up of 82 RESPONSE OF CIRCUITS CONTAINING PASSIVE COMPONENTS two parts, viz. a real part R and an imaginary part jcoL. Such terms, although frequently used in textbooks on electronics, are not, in the author's opinion, satisfactory, since some confusion can arise in the mind of a beginner when confronted with the statement that the reactive component is imaginary. If any reader cares to connect himself to a high inductance coil, and then have a large current in the Figure 3.21. The replacement of a phasor diagram by complex numbers coil suddenly discontinued, the resulting severe electric shock should convince him that there is nothing imaginary about the voltage (i)oeL) involved. The j term is not imaginary but does have a phase shift of 90° from some reference direction. It is, therefore, better to call a complex quantity, or number, a general number, and to regard such a number as having an ordinary component and a quadrature compo-nent. The terms 'ordinary' and 'quadrature' replace 'real' and 'imaginary'. The quadrature component is so called because of the 90° shift from the reference direction. The magnitude of a general number is found by applying Pythagoras' Theorem. Thus, if Ζ is expressed as the general number R + ](oL then the magnitude, i.e. modulus, of Ζ is written with two lines bracketing the symbol, and is given by |Z| = V[R 2 + (coL) 2 ].
- Kuldip S. Rattan, Nathan W. Klingbeil, Craig M. Baudendistel(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
5.10, the impedance of the resistor is given by Z 1 = R. The total impedance of the inductor and capacitor in series is given by Z 2 = j X L + 1 j X C , where j = √ −1. Suppose R = 10, X L = 10, and X C = 20, all measured in ohms: (a) Express the impedance Z 1 and Z 2 in both rectangular and polar forms. (b) Suppose the source voltage is V = 100 √ 2 ∠45 ∘ V. Compute the voltage V 1 given by V 1 = Z 2 Z 1 + Z 2 V. (5.5) V R C L V 1 + − Figure 5.10 Voltage divider circuit for example 5-2. 5.7 Further Examples of Complex Numbers in Electric Circuits 145 Solution (a) The impedance Z 1 can be written in rectangular form as Z 1 = R = 10 + j 0 Ω. The impedance Z 1 can be written in polar form as Z 1 = √ 10 2 + 0 2 ∠atan2(0, 10) = 10 ∠0 ∘ Ω. The impedance Z 2 can be written in rectangular form as Z 2 = j X L + 1 j X C = j 10 + 1 j 20 ( j j ) = j 10 − j 20 = −j 10 = 0 − j 10 Ω. The impedance Z 2 can be written in polar form as Z 2 = √ 0 2 + (−10) 2 ∠atan2(−10, 0) = 10 ∠−90 ∘ Ω. (b) V 1 = Z 2 Z 1 + Z 2 V = ( 10 ∠−90 ∘ (10 + j 0) + (0 − j 10) ) (100 √ 2 ∠45 ∘ ) = ( 10 ∠−90 ∘ 10 − j 10 ) (100 √ 2 ∠45 ∘ ) = ( 10 ∠−90 ∘ 10 √ 2 ∠−45 ∘ ) (100 √ 2 ∠45 ∘ ) = ( 1 √ 2 ∠−45 ∘ ) (100 √ 2 ∠45 ∘ ) = 100 ∠0 ∘ V = 100 + j 0 V. 146 Chapter 5 Complex Numbers in Engineering Example 5-3 In the circuit shown in Fig. 5.11, the impedances of the various components are Z R = R, Z L = j X L , and Z C = 1 j X C , where j = √ −1. Suppose R = 10, X L = 10, and X C = 10, all measured in ohms. (a) Express the total impedance Z = Z C + Z R Z L Z R +Z L in both rectangular and polar forms. (b) Suppose a voltage V = 50 √ 2 ∠45 ∘ V is applied to the circuit shown in Fig. 5.11. Find the current I flowing through the circuit if I is given by I = V Z (5.6) Solution (a) The impedance Z R can be written in rectangular and polar forms as Z R = R = 10 + j 0 Ω = 10 ∠0 ∘ Ω. C Z R L Figure 5.11 Total impedance of the circuit for example 5-3.
eBook - PDF- A. Henderson(Author)
- 2014(Publication Date)
- Arnold(Publisher)
7.5 Extension of the meaning of impedance Now consider an inductor through which a damped sinusoidal current flows (see Figure 7.8). The current is i = III e<* cos (cot + φ), (7.5) so the complex current added is I = III ei + j(uLIei, or V = (σ + jco) L III ei
eBook - PDF- Giuseppe Petrone, Giuliano Cammarata, Giuseppe Petrone, Giuliano Cammarata(Authors)
- 2008(Publication Date)
- IntechOpen(Publisher)
The Recent Advances in Modelling and Simulation 102 following material properties and geometric parameters were used for computing the specimen impedance: σ =8.34 x 10 5 S/m, μ r =1.004, r o =6.35mm, r c =0.5mm and t o =2mm. The Complex Impedance ( Z ) of a specimen for an alternating current flow situation consists of real ( ) ' Z and imaginary ( ) Z components, viz. a resistance ( R ) and a reactance ( X ). jX R jZ ' Z Z + = + = (35) In the case of a metallic specimen, the reactance is primarily due to the contribution from the internal inductance ( L i ) of the specimen. While the resistance is related directly to energy loss due to ohmic heating, the inductance describes the ability of a conductor to store magnetic energy (Hallen 1962). The expression for the Complex Impedance by incorporating the inductance is written as follows: i L ω j R Z + = (36) The computation of R and L i of a metallic cylindrical disk specimen using the analytical electric field expressions was conducted via energy methods (Kelekanjeri 2007). The resistance and the inductance are obtained by calculating the Joule heat-loss and the total internal magnetic energy respectively, the expressions for which, are listed as follows (Hallen 1962): ∫ = V c rms 2 dV E . E σ I 1 R and (37) dV H . H μ I 1 dV H . B I 1 L V c rms 2 V c rms 2 i ∫ = ∫ = (38) In the above expressions E and H are the total complex electric and magnetic fields respectively, while the subscript c denotes the complex conjugate. The total magnetization-B is related to the magnetic field H by the magnetic permeability μ . The root mean square value of the current of amplitude o I is given by 2 I I o rms = . The total electric field E and the magnetic field H are given in terms of the electric field components r E and z E as follows: z ˆ E r ˆ E E z r + = and (39) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ = z E r E ωμ j 1 H r z (40) The analytical expressions for r E and z E are given in section 3.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The capacitive reactance, like resistance, is measured in ohms and determines how much rms current exists in a capacitor in response to a given rms voltage across the capacitor. It is found experimentally that the capacitive reactance X C is inversely proportional to both the frequency f and the capacitance C, according to the following equation: X C 5 1 2p fC (23.2) For a fixed value of the capacitance C, Figure 23.2 gives a plot of X C versus frequency, according to Equation 23.2. A comparison of this drawing with Figure 23.1 This halftime performance by the Who was part of the festivities at Super Bowl XLIV. Without the aid of alternating current (ac) circuits it would not be possible to stage such entertainment spectac- ulars. Ac circuits lie at the heart of all the audio systems used in the performance. 23 | Alternating Current Circuits 583 Chapter | 23 LEARNING OBJECTIVES After reading this module, you should be able to... 23.1 | Calculate capacitive reactance. 23.2 | Calculate inductive reactance. 23.3 | Calculate impedance in an RCL circuit. 23.4 | Calculate the resonance frequency of an RCL circuit. 23.5 | Describe how semiconductor devices operate. Larry French/Stringer/Getty Images R Frequency, f (Hz) Resistance, R (ohms) V 0 sin 2 ft π Figure 23.1 The resistance in a purely resistive circuit has the same value at all frequencies. The maximum emf of the generator is V 0 . 584 Chapter 23 | Alternating Current Circuits reveals that a capacitor and a resistor behave differently. As the frequency becomes very large, Figure 23.2 shows that X C approaches zero, signifying that a capacitor offers only a negligibly small opposition to the alternating current. In contrast, in the limit of zero frequency (i.e., direct current), X C becomes infinitely large, and a capacitor pro- vides so much opposition to the motion of charges that there is no current.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The capacitive reactance, like resistance, is measured in ohms and determines how much rms current exists in a capacitor in response to a given rms voltage across the capacitor. It is found experimentally that the capacitive reactance X C is inversely proportional to both the frequency f and the capacitance C, according to the following equation: X C 5 1 2p fC (23.2) For a fixed value of the capacitance C, Figure 23.2 gives a plot of X C versus frequency, according to Equation 23.2. A comparison of this drawing with Figure 23.1 This halftime performance by The Who was part of the festivities at Super Bowl XLIV. Without the aid of alternating current (ac) circuits it would not be possible to stage such entertainment spectac- ulars. Ac circuits lie at the heart of all the audio systems used in the performance. 23 | Alternating Current Circuits 651 Chapter | 23 LEARNING OBJECTIVES After reading this module, you should be able to... 23.1 | Calculate capacitive reactance. 23.2 | Calculate inductive reactance. 23.3 | Calculate impedance in an RCL circuit. 23.4 | Calculate the resonance frequency of an RCL circuit. 23.5 | Describe how semiconductor devices operate. Larry French/Stringer/Getty Images R Frequency, f (Hz) Resistance, R (ohms) V 0 sin 2 ft π Figure 23.1 The resistance in a purely resistive circuit has the same value at all frequencies. The maximum emf of the generator is V 0 . 652 Chapter 23 | Alternating Current Circuits reveals that a capacitor and a resistor behave differently. As the frequency becomes very large, Figure 23.2 shows that X C approaches zero, signifying that a capacitor offers only a negligibly small opposition to the alternating current. In contrast, in the limit of zero frequency (i.e., direct current), X C becomes infinitely large, and a capacitor pro- vides so much opposition to the motion of charges that there is no current.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
However, since the current lags the voltage by 90° in an inductor, the current phasor is behind the voltage phasor by 90° in the direction of rotation. The instantaneous values of the voltage and current are equal to the vertical components of the corresponding phasors. 23.3 Circuits Containing Resistance, Capacitance, and Inductance When a resistor, a capacitor, and an inductor are con- nected in series, the rms voltage across the combination is related to the rms current according to Equation 23.6, where Z is the impedance of the combination. The impedance is measured in ohms (Ω) and is given by Equation 23.7, where R is the resistance, and X L and X C are, respec- tively, the inductive and capacitive reactances. V rms = I rms Z (23.6) Z = √ ___________ R 2 + (X L − X C ) 2 (23.7) The tangent of the phase angle ϕ between current and voltage in a series RCL circuit is given by Equation 23.8. tan ϕ = X L − X C _ R (23.8) Only the resistor in the RCL combination consumes power, on average. The average power ¯ P consumed in the circuit is given by Equation 23.9, where cos ϕ is called the power factor of the circuit. ¯ P = I rms V rms cos ϕ (23.9) 23.4 Resonance in Electric Circuits A series RCL circuit has a resonant frequency f 0 that is given by Equation 23.10, where L is the inductance and C is the capacitance. At resonance, the impedance Check Your Understanding (The answer is given at the end of the book.) 13. CYU Figure 23.2 shows a full-wave rectifier circuit, in which the direction of the current through the load resistor R is the same for both positive and negative halves of the generator’s voltage cycle.
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