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: As I have mentioned when we developed interpretations of the capacitor and inductor in the s- world, impedance is a generalized resistance that is frequency dependent. 2. Professor Ray, no or is that know equations? Equation definition below and you do need to KNOW them!!!!!

2 . Definition. Impedance, denoted Z in, in the s- world, and only in the s- world, forever and ever and ever, in the total absence of initial conditions, is (i) Z in V in I in, or more generally (ii) V in Z in I in which avoids all that division by zero stuff. Admittance, Y in. The inverse of impedance, i.e., Y in Z in. Clearly Z in Y in. 2. Resistor Impedance/Admittance. Remember back in the good old days of 20 when resistors, denoted R,

3 were resistors and Ohm s law, V = RI, was Ohm s law in the time world. They were the good old days when class was easy. And now, after taking Laplace transforms: V R I! Z R I and I R V! Y R V Remark: 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

4 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. 2. At s = 0, the impedance (generalized resistance) of the capacitor is infinite meaning the capacitor looks like an open circit, meaning that 0-frequency current, which is dc, does not get through. 3. Inductance Impedance/Admittance. (i) t- world: v L = L di L dt

5 (ii) s- world: V L Ls I L! Z L I C which is in the usual Ohm s law form, and its admittance converse I L Ls V L! Y L V L 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.

6 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. Admittancs are manipulated like conductances. Parallel RC circuit: Y circuit Cs + R. Product Rule: if Z and Z 2 are two impedances in parallel, then

7 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- Parallel Impedance Rule: Z eq = Z + Z Z n Remark: all other 20 rules apply. Use them.

8 Rule 3. Ohm s Law: V ZI or I YV. 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.

9 Z in R Cs R + Cs + R 2 Ls R 2 + Ls = s + + s s + = 6. Parallel Circuits and Current/Voltage Division Example 3. Consider the circuit below

10 (i) Y out = Z 3 + Z 4 (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.

11 Part. Y in R + Ls + R 2 + Cs = L s + R L + s R 2 s + R 2 C = 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 7. The 20/202 Twins: Thevenin and Norton (a) The equation of a Thevenin equivalent below is: V in Z th I in +V oc

12 (b) The Norton equivalent equation is: I in Y th V in I sc 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

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

14 V in αcs + Cs I in + αcs + Cs I s Here: V oc Z th I sc. = Z th I in +V oc