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Spring 18
Lecture 6: Impedance (frequency dependent
resistance in the s-world), Admittance (frequency
dependent conductance in the s-world), and
Consequences Thereof
1. Professor Ray, what’s an impedance?
Answers: 1. derived from the word impede, “impedance” is a generalized frequency dependent resistance that lives only in the s-world. 2. In 201, 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.
Spring 18
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!!!!!
1. DEFINITION. Impedance, denoted Zin(s), 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) Zin(s) =
Vin(s)Iin(s)
, or more generally
(ii) Vin(s) = Zin(s)Iin(s) which avoids all that division
by zero stuff.
Spring 18
DEFINITION. Admittance, Yin(s) = 1
Zin(s), the inverse
of impedance, is a generalized frequency dependent
conductance.
2. Resistor Impedance/Admittance. Remember back
in the good old days of 201 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 201
back in the good old days? Now, the dreaded Pirate
Roberts uses the Laplace transform and “as you
wish”:
V (s) = R I (s) ! ZR(s) I (s) and
I (s) = 1
RV (s) ! YR(s) V (s)
Spring 18
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: iC = C
dvCdt
(ii) s-world: IC (s) = Cs VC (s) ! YC (s) VC (s) or
equivalently, in the usual Ohm’s law form:
VC (s) = 1
CsIC (s) ! ZC (s) IC (s)
Remarks: 1. Now this is different.
ZC (s) = 1
Cs is an s-dependent resistance
that makes up an s-dependent Ohm’s
law. Most things never stay the same.
Some do.
Spring 18
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: vL = L
diLdt
(ii) s-world: VL(s) = Ls IL(s) ! ZL(s) IC (s) which
is in the usual Ohm’s law form, and its admittance,
the converse is:
IL(s) = 1
LsVL(s) ! YL(s) VL(s)
Spring 18
Remarks: 1. ZL(s) = 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.
Spring 18
4. Manipulation RULES, i.e., the rules that govern
the manipulation of Z and Y.
Rule 1. Impedances (generalized resistances) are
manipulated like resistances.
Series LC circuit: Zcircuit (s) = Ls+ 1
Cs.
Rule 2. Admittances are manipulated like
conductances.
Parallel RC circuit: Ycircuit (s) = Cs+ 1
R.
Rule 3. Ohm’s Law in s-world: V (s) = Z(s)I (s) or
I (s) = Y (s)V (s).
Product Rule: if Z1(s) and Z2(s) are two impedances
in parallel, then
Spring 18
Zeq (s) = 1Y1(s)+Y2(s)
= 11
Z1(s)+ 1
Z2(s)
=Z1(s)Z2(s)
Z1(s)+ Z2(s)= Product
Sum
Multi-Parallel Admittance Rule:
Zeq (s) = 1
Y1(s)+Y2(s)+ ...+YN (s)
Multi-Series Impedance Rule:
Zeq = Z1 + Z2 + ...+ Zn
Remark: all other 201 rules apply. Use
them. Source transformations work.
Thevenin and Norton equivalents work
etc.
Spring 18
5. Series Circuits and Voltage Division
Example 1. Consider the circuit below.
(i) Zin = Z3 + Z4
(ii) Vout =
Z4Z3 + Z4
Vin (Voltage Division)
(iii) Iout =
VinZin
=Vin
Z3 + Z4 (Ohm’s law)
Example 2. Find the input impedance seen by the
source. Assume all parameter values are 1.
Spring 18
Zin(s) =R1
1Cs
R1 +1
Cs
+R2Ls
R2 + Ls= 1
s+1+ s
s+1= 1Ω
6. Parallel Circuits and Current/Voltage Division
Example 3. Consider the circuit below
(i) Yout =
1Z3 + Z4
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(ii) Yin = Y1 +Y2 +Yout
(iii) Zin =
1Y1 +Y2 +Yout
(iv) Iout =
YoutYin
Iin =Yout
Y1 +Y2 +YoutIin (Current Division)
(v) Vout = Z4Iout (Ohm’s law)
Example 4. Find the input admittance and
impedance of the circuit below. Suppose L = 1 H,
C = 0.5 F, and R1 = R2 = 1 Ω. Also, find Iout (s) .
Part 1.
Yin(s) = 1R1 + Ls
+ 1
R2 +1
Cs
=
1L
s+R1L
+
sR2
s+ 1R2C
Spring 18
= 1
s+1+ s
s+ 2= s2 + 2s+ 2
(s+1)(s+ 2)
Hence,
Zin(s) = (s+1)(s+ 2)
(s+1)2 +12
Part 2. By current division,
Iout (s) =
1s+1
s2 + 2s+ 2(s+1)(s+ 2)
Iin(s) = s+ 2s2 + 2s+ 2
Iin(s)
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 Z1 Y1 Z2 Y2 Zin Yin Vout Iout Iin
>> R1 = 1; R2 = 1; C = 0.5; L = 1;
>> Z1 = R1 + L*s
Z1 =
s + 1
>> Z2 = R2 + 1/(C*s)
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Z2 =
2/s + 1
>> Y1 = 1/Z1
Y1 =
1/(s + 1)
>> Y2 = collect(1/Z2)
Y2 =
s/(s + 2)
>> Yin = collect(Y1 + Y2)
Yin =
(s^2 + 2*s + 2)/(s^2 + 3*s + 2)
>> Zin = 1/Yin
Zin =
(s^2 + 3*s + 2)/(s^2 + 2*s + 2)
>> % By current division
>> Iout = Y1/Yin * Iin
Iout =
(Iin*(s^2 + 3*s + 2))/((s + 1)*(s^2 + 2*s + 2))
>> % By Ohm's law
>> Vout = Zin * Iin
Vout =
(Iin*(s^2 + 3*s + 2))/(s^2 + 2*s + 2)
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7. The 201/202 Twins: Thevenin and Norton
dressed in the s-world
(a) The equation of a Thevenin equivalent below is:
Vin(s) = Zth(s)Iin(s)+Voc(s)
(b) The Norton equivalent equation is:
Iin(s) = Yth(s)Vin(s)− Isc(s)
Spring 18
Relationship: Given Iin(s) = Yth(s)Vin(s)− Isc(s) we can rearrange and divide by Yth(s) :
Vin(s) = 1
Yth(s)Iin(s)+ 1
Yth(s)Isc(s)
or equivalently
Vin(s) = Zth(s)Iin(s)+Voc(s) where Voc(s) = Zth(s)Isc(s) . Example 6. Find the Thevenin equivalent of the
circuit below. We first find the Norton equivalent
and then convert to the Thevenin form.
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