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

Spring 18

(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)

Spring 18

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)

Spring 18

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.

Spring 18

(a) Iin = IC − Is

(b) IC = Cs Vin −α IC⎡⎣ ⎤⎦ implies IC = Cs

αCs+1Vin .

(c) Therefore, the Norton equivalent is:

Iin(s) = Cs

αCs+1Vin(s)− Is(s) = Yth(s)Vin(s)− Is(s)

(d) Equivalently,

Vin(s) = αCs+1Cs

Iin(s)+ αCs+1Cs

Is(s)

= Zth(s)Iin(s)+Voc(s)

Here: Voc(s) = Zth(s)Isc(s) .