Lesson 20Series AC Circuits

Learning Objectives Compute the total impedance for a series AC

circuit.

Apply Ohm’s Law, Kirchhoff’s Voltage Law and the voltage divider rule to AC series circuits.

Graph impedances, voltages and current as a function of phase.

Graph voltages and current as a function of time.

Impedance Impedance is a complex quantity that can be

made up of resistance (real part) and reactance (imaginary part).

Z = R + jX = Z ()

Unit of impedance is ohms ().

Review

Resistance For resistors, voltage and current are in phase.

0 0R R j R Z

Review

For inductors, voltage leads current by 90º.

0 90 90L L Lj L L X X j Z

Review

For capacitors, voltage lags current by 90º.

1 10 90 90C C Cj X X j

C C

Z

Review

Impedance Because impedance is a complex quantity, it can be

represented graphically in the complex plane.

ZR = R0º = R + j0 = R

ZL = XL90º = 0 + jXL = jXL

ZC = XC-90º = 0 - jXC = -jXC

ELI the ICE man

E leads I I leads E

When voltage is applied to an inductor, it resists the change of current. The current builds up more slowly, lagging in time and phase.

Since the voltage on a capacitor is directly proportional to the charge on it, the current must lead the voltage in time and phase to conduct charge to the capacitor plate and raise the voltage

Volta

geIn

duct

ance

Curre

nt

Volta

ge

Capac

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e

Curre

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Solving complex EE problems

1. Convert sine waves to phasors

2. Perform multiplication/division if needed

3. Convert phasors to complex numbers if needed to perform addition/subtraction

4. Convert back to phasor form for the answer to the problem

Important Notes

Peak values are useful for time domain representations of signals.

RMS values are the standard when dealing with phasor domain representations

If you need to represent something in the time domain, you will need to convert RMS->Peak voltage to obtain Em

2 ( ) sin(2 )RMS m RMS mV E V e t E ft E

Example Problem 1

For the circuit below:

a) Time Domain voltage and current v(t) and i(t)

b) Draw the sine waveforms for v and i

c) Draw the phasor diagram showing the relationship between V and I

Example Problem 1

i(t)=50 sin (20000t) mA

v(t)=25 sin (20000t -90°) V

314 μsec

AC Series Circuits Total impedance is sum of

individual impedances. Also note that current is the

same through each element.

1 2T n Z Z Z Z

Impedance example

Impedance example

3 4 5 3

3.2 18.4

TZ j j j

??????TZ

Special case of Impedance

Whenever a capacitor and inductor of equal reactances are placed in series, the equivalent circuit is a short circuit.

Impedance If the total impedance has only real

component, the circuit is said to be resistive (X = 0 or = 0°).

But since If > 0°, the circuit is inductive. ELI If < 0°, the circuit is capacitive. ICE

E IZ

Example Problem 2

A network has a total impedance of ZT=24.0kΩ-30˚ at a frequency of 2 kHz.

If the network consists of two series elements, what types of components are those and what are their R/L/C values?

The phasor sum of voltage drops and rise around a closed loop is equal to zero.

Kirchhoff’s Voltage Law (KVL)

10 0 8 36.87 12 53.13

6 126.87

c R LE V V

V

KVL ??????cV

Voltage Divider Rule (VDR)

310 0

4 6 3

6 126.87

Cc

R L C

VDR

ZE

Z Z Z

j

j j

V

EZ

ZV

T

xx

??????cV

Example Problem 3es(t)=170 sin (1000t + 0) V.

Determine ZTOT

Determine total current ITOT

Determine voltages VR, VL, and Vc

Verify KVL for this circuit Graph E, VL, VC, VR in the time domain

Example Problem 3

180° phase difference between L and C

Example Problem 4

es(t)=294 sin (377t + 0) V.

Use the voltage divider rule to find VL

Determine the value of inductance