Lecture 6: Impedance (frequency dependent. resistance in the s-world), Admittance (frequency. dependent conductance in the s-world), and

Transcription

1 Lecture 6: Impedance (frequency dependent resistance in the s-world), Admittance (frequency dependent conductance in the s-world), and Consequences Thereof. Professor Ray, what s an impedance? Answers:. derived from the word impede, impedance is a generalized frequency dependent resistance that lives only in the s-world. 2. In 20, resistance was a t-world thing because it is constant over frequency, ideally speaking. Its DNA roots are in the s-world. 3. When we developed interpretations of the capacitor and inductor in the s-world, we saw frequency dependent resistance/conductance. And to distinguish it from a distinguished or is that extinguished professor, capacitors and inductors have a frequency dependent impedance whose website and face book page is in the s-world.

2 2. Professor Ray, no or is that know equations? Answer: Ahhhh, no equations is a no-way in 202. And, you do need to KNOW them!!!!!. DEFINITION. Impedance, denoted Z in, living only in the s-world, forever and ever and ever, in the total absence of initial conditions in the circuit with ALL sources set to zero, is (i) Z in V in, or more generally I in (ii) V in Z in I in which avoids all that division by zero stuff.

3 DEFINITION. Admittance, Y in, the inverse Z in of impedance, is a generalized frequency dependent conductance. 2. Resistor Impedance/Admittance. Remember back in the good old days of 20 when resistors, denoted R, were resistors and Ohm s law, V = RI, was Ohm s law in the t-world. Weren t things easy back in 20 back in the good old days? Now, the dreaded Pirate Roberts uses the Laplace transform and as you wish : V R I! Z R I and I R V! Y R V

4 Remark: Inconceivable. Looks the same as in the time world and so it is. Some things never change. Most do. 3. Capacitance Impedance/Admittance. (i) t-world: i C = C dv C dt (ii) s-world: I C Cs V C! Y C V C or equivalently, in the usual Ohm s law form: V C Cs I C! Z C I C Remarks:. Now this is different. Z C Cs is an s-dependent resistance that makes up an s-dependent Ohm s law. Most things never stay the same. Some do.

5 2. At s = 0, the impedance (generalized resistance) of the capacitor is infinite meaning the capacitor looks like an open circuit, meaning that 0-frequency current, which is dc, does not get through a capacitor. 3. Inductance Impedance/Admittance. (i) t-world: v L = L di L dt (ii) s-world: V L Ls I L! Z L I C which is in the usual Ohm s law form, and its admittance, the converse is: I L Ls V L! Y L V L

6 Remarks:. Z L Ls is an s-dependent resistance that makes up an s-dependent Ohm s law. Wow, really cool. Can t wait to tell my date next weekend; being in lower case ee (elementary education) he/she is going to be so excited. 2. At s = 0, the impedance (generalized resistance) of the inductor is zero meaning the inductor looks like a short circuit, meaning that 0-frequency current, which is dc, goes right through like an Ipass toll booth.

7 4. Manipulation RULES, i.e., the rules that govern the manipulation of Z and Y. Rule. Impedances (generalized resistances) are manipulated like resistances. Series LC circuit: Z circuit Ls + Cs. Rule 2. Admittances are manipulated like conductances. Parallel RC circuit: Y circuit Cs + R. Rule 3. Ohm s Law in s-world: V ZI or I YV. Product Rule: if Z and Z 2 are two impedances in parallel, then

8 Z eq Y + Y 2 Z + Z 2 = Z Z 2 Z + Z 2 Product Sum Multi-Parallel Admittance Rule: Z eq Y + Y Y N Multi-Series Impedance Rule: Z eq = Z + Z Z n Remark: all other 20 rules apply. Use them. Source transformations work. Thevenin and Norton equivalents work etc.

9 5. Series Circuits and Voltage Division Example. Consider the circuit below. (i) Z in = Z 3 + Z 4 (ii) V out = Z 4 Z 3 + Z 4 V in (Voltage Division) (iii) I out = V in Z in = V in Z 3 + Z 4 (Ohm s law) Example 2. Find the input impedance seen by the source. Assume all parameter values are.

10 Z in R Cs R + Cs + R 2 Ls R 2 + Ls = s + + s s + = Ω 6. Parallel Circuits and Current/Voltage Division Example 3. Consider the circuit below (i) Y out = Z 3 + Z 4

11 (ii) Y in = Y + Y 2 + Y out (iii) Z in = Y + Y 2 + Y out (iv) I out = Y out Y in I in = (v) V out = Z 4 I out (Ohm s law) Y out Y + Y 2 + Y out I in (Current Division) Example 4. Find the input admittance and impedance of the circuit below. Suppose L = H, C = 0.5 F, and R = R 2 = Ω. Also, find I out. Part. Y in R + Ls + R 2 + Cs = L s + R L + s R 2 s + R 2 C

12 = s + + s s + 2 = s2 + 2s + 2 (s +)(s + 2) Hence, Z in (s +)(s + 2) (s +) Part 2. By current division, I out s + s 2 + 2s + 2 (s +)(s + 2) I in s + 2 s 2 + 2s + 2 I in Remark: How might we do Example 4 in MATLAB so that we can not let our academics interfere with our social education. Here is the code: >> syms s t Z Y Z2 Y2 Zin Yin Vout Iout Iin >> R = ; R2 = ; C = 0.5; L = ; >> Z = R + L*s Z = s + >> Z2 = R2 + /(C*s)

13 Z2 = 2/s + >> Y = /Z Y = /(s + ) >> Y2 = collect(/z2) Y2 = s/(s + 2) >> Yin = collect(y + Y2) Yin = (s^2 + 2*s + 2)/(s^2 + 3*s + 2) >> Zin = /Yin Zin = (s^2 + 3*s + 2)/(s^2 + 2*s + 2) >> % By current division >> Iout = Y/Yin * Iin Iout = (Iin*(s^2 + 3*s + 2))/((s + )*(s^2 + 2*s + 2)) >> % By Ohm's law >> Vout = Zin * Iin Vout = (Iin*(s^2 + 3*s + 2))/(s^2 + 2*s + 2)

14 7. The 20/202 Twins: Thevenin and Norton dressed in the s-world (a) The equation of a Thevenin equivalent below is: V in Z th I in +V oc (b) The Norton equivalent equation is: I in Y th V in I sc

15 Relationship: Given I in Y th V in I sc we can rearrange and divide by Y th : V in Y th I in + Y th I sc or equivalently V in Z th I in +V oc where V oc Z th I sc. Example 6. Find the Thevenin equivalent of the circuit below. We first find the Norton equivalent and then convert to the Thevenin form.

16 (a) I in = I C I s (b) I C = Cs V in α I C implies I C = Cs αcs + V in. (c) Therefore, the Norton equivalent is: I in Cs αcs + V in I s Y th V in I s (d) Equivalently, V in αcs + Cs I in + αcs + Cs I s = Z th I in +V oc Here: V oc Z th I sc.