Using the Impedance Method


 Clifton Little
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1 Using the Impedance Method The impedance method allows us to completely eliminate the differential equation approach for the determination of the response of circuits. In fact the impedance method even eliminates the need for the derivation of the system differential equation. Knowledge of the impedance of the various elements in a circuit allows us to apply any of the circuits analysis methods (KVL, KL, nodal, superposition Thevenin etc.) for the determination of the circuits characteristics: voltages across elements and current through elements. Before proceeding let s review the impedance definitions and properties of the capacitor and the inductor. Frequency ( ω ) limits Element Impedance Low ( ω 0) High ( ω ) apacitor Z Z Z 0 jω OPEN SHOT Inductor ZL 0 ZL ZL jωl SHOT OPEN Let s now continue with the analysis of the series L circuit shown on Figure. We would like to calculate the voltage across the capacitor. V v v cos( ωt) s o L + Vc  Figure. Series L circuit In order to gain a deeper perspective into the power of the impedance method we will first derive the differential equation for V and then solve it using the algebraic procedure derived previously. In turn we will proceed with the application of the impedance method. The equation for V is obtained as follows: KVL for the circuit mesh gives 6.07/.07 Spring 006, haniotakis and ory
2 di() t vocos( ωt) i( t) + L + V (.) dt The current flowing in the circuit is dv it () (.) dt And Equation (.) becomes d V dv o V v Note that this is a second order differential equation. + + cos( ωt) (.3) dt L dt L L For a source term of the form The solution is j ( t ) ve ω o (.4) V () t Ae j( ωt+ φ ) (.5) Substituting into Equation (.3) we obtain j v j e φ o Α ω + ω+ L L L (.6) jφ vo Α e L ω + jω L + L (.7) vo ω L + jω Which may be simplified as follows j v Α φ e o + ( ω L) ( ω) e ω j tan ω L (.8) Therefore the amplitude A of V is And the phase is Α φ v o ( ω L) + ( ω) tan (.9) ω ω L (.0) 6.07/.07 Spring 006, haniotakis and ory
3 Now we will calculate the voltage V by using the impedance method. In terms of the impedance the L circuit is V S Z ZLjLω Zc jω + Vc  Figure This is now a representation in the frequency domain since impedance is a frequency domain complex quantity The voltage V may now be determined by applying the standard voltage divider relation Z V Vs Z + Z L + Z jω Vs + jωl+ jω Vs ω L jω + (.) Which is the same as Equation (.7). Note that we never had to write down the differential equation. We may now complete the solution by writing again gives V Ae j( ωt+ φ ) and V S j( t ) voe ω which And Α v o ( ω L) + ( ω) (.) φ tan ω ω L (.3) 6.07/.07 Spring 006, haniotakis and ory 3
4 Similarly we can calculate the voltage V across resistor + V  Vs Z ZLjLω Zc jω + Vc  The voltage divider relationship gives Upon simplification it becomes V Vs Z + Z L + Z Vs Z Vs + jωl+ jω jω ω L jω + (.4) ω V Vs e ( ω L) + ( ω) Note the π/ phase difference between V and Vc. π ω j tan ω L (.5) Also, the voltage across the inductor becomes: L ω L j tan ω L ω V Vs e ( ω L) + ( ω) (.6) 6.07/.07 Spring 006, haniotakis and ory 4
5 Example: A frequency independent voltage divider onsider the voltage divider shown below for which the load may be modeled as a parallel combination of resistor and inductor L. vs L vo Figure 3 In terms of the impedance the circuit becomes Z Vs jlω ZL Z Vo Figure 4 The voltage Vo is given by ZL// Z Vo ZL// Z+ Z ZL Z ZL Z+ Z ZL+ Z ( ) jωl jωl+ j L+ ( ( ω )) jωl + jωl + ( ) (.7) Equation (.7) may also be written in polar form as follows 6.07/.07 Spring 006, haniotakis and ory 5
6 Vo ωl ( ) + L + ω ωτ e + + ωτ π j φ e π ωl( + ) j tan (.8) Where And L( + ) τ (.9) ωl( + ) φ tan tan ( ωτ ) (.0) The frequency dependence of the voltage divider is shown on Figure 5. Here we have plotted the amplitude of Vo as a function of ωτ for. Note the asymptotic value Vs indicated by the dotted line. At high frequencies, for which the inductor acts like an open circuit, the divider ratio reduces to that of the two resistors which in this case is ½ since both resistors are equal. At low frequencies, the low impedance of the inductor reduces the output voltage. At dc ( ω 0 ) the inductor acts like a short circuit and so Vo ωτ Figure 5 0 We would like to alter the design of the voltage divider so that it becomes independent of frequency for all frequencies. 6.07/.07 Spring 006, haniotakis and ory 6
7 One way to address this problem is to add a compensating inductor L as shown on the following schematic. L vs L vo The equivalent circuit in terms of impedance is Z Vs Z + Vo  And the voltage divider ratio becomes Z Vo Vs Vs Z+ Z Z + Z Frequency independence implies that the ratio of impedances of frequency. This ratio is given by jωl Z + jωl Z jωl + jωl L + jωl L + jωl Z Z (.) must be independent (.) 6.07/.07 Spring 006, haniotakis and ory 7
8 Equation (.) becomes independent of ω if L L (.3) Which results in a voltage divider ratio of Vo Vs + (.4) 6.07/.07 Spring 006, haniotakis and ory 8
9 A close look at frequency response. (Frequency selection) As we have discussed previously, the frequency response of a circuit or a system refers to the change in the system characteristics with frequency. A convenient way to represent this response is to plot the ratio of the response signal to the source signal. For the generic representation shown on Figure 6, the response may be given as the ratio of the output Y(ω) to the input X(ω). This ratio is called the transfer function of the system and it is labeled H(ω) Y ( ω) H ( ω) (.5) X ( ω) Linear X ( ω ) Y ( ω ) system Figure 6 The output and input (Y(ω) and X(ω) ) may represent the amplitude or the phase of the signals. As an example let s consider the circuit shown on Figure vs(t) vc(t)  Vs jω Vc  The transfer function for this circuit is Figure 7 Z jω H ( ω) Z + Z + j ω + jω (.6) The magnitude and the phase of H(ω) are 6.07/.07 Spring 006, haniotakis and ory 9
10 H ( ω) + ( ω) (.7) φ ω tan ( ) (.8) In practice the range of frequencies that is used in plotting H ( ω ) is very wide and thus a linear scale for the frequency axis is often not suitable. In practice H ( ω ) is plotted versus the logarithm of the frequency. In addition it is common to plot the transfer function in db, where H( ω) 0log H( ω) (.9) db 0 The plot of H( ω ) db versus log( ω ) is called the Bode plot. For our example circuit with 0kΩ and 47nF Figure 8(a) and (b) show the plot of H ( ω ) versus ω and log( ω ) respectively. Note that the semi logarithmic plot presents the information in a more visual way. (a) (b) Figure /.07 Spring 006, haniotakis and ory 0
11 When H ( ω ) is calculated in db the plot versus the logarithm of frequency is shown on Figure 9. Figure 9 From the above plot we see the strong dependence of the magnitude of the output signal on the frequency. Figure 0 shows the plot of the phase as a function of frequency. Figure 0 At low ω for which the capacitor acts like an open circuit the phase is zero. At high frequencies the capacitor acts like a short circuit ad the phase goes to 90 o. 6.07/.07 Spring 006, haniotakis and ory
12 Now let s continue by graphically exploring the response of L circuits. Vs L + Vc  The amplitude and phase of Vc are given by Equations (.) and (.3) which we rewrite here for convenience. Vc Vs H ( ω) ( ω L) + ( ω) (.30) φ tan ω ω L (.3) Figure shows the plot of H ( ω ) as a function of frequency for 300Ω, L47mH and 47nF (the values we also used in laboratory). Figure In the limit as ω 0, H ( ω). Note also that there is a peak at a certain frequency which by inspection of Equation (.30) occurs when ω L /.07 Spring 006, haniotakis and ory
13 By increasing the value of the resistor the peak becomes less pronounced. Figure shows the transfer function for 300Ω and.5kω. Figure The phase plot is shown on Figure 3. Note that the transition happens again when ω L 0 Figure /.07 Spring 006, haniotakis and ory 3
14 The following Plots show the normalized transfer function for V and the corresponding phase. 6.07/.07 Spring 006, haniotakis and ory 4
15 Similarly the normalized transfer function for VL is In the next two classes we will explore this behavior further and develop their physical significance with regard to their frequency selectivity characteristics. 6.07/.07 Spring 006, haniotakis and ory 5
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