Lesson 20 Series 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. Review

For inductors, voltage leads current by 90º. Review

For capacitors, voltage lags current by 90º. Review

Impedance Because impedance is a complex quantity, it can be represented graphically in the complex plane. Z R = R  0º = R + j0 = R Z L = X L  90º = 0 + jX L = jX L Z C = X C  -90º = 0 - jX C = -jX C

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 Voltage Inductance Current Voltage Capacitance Current

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 E m

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.

Impedance example

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

Example Problem 2 A network has a total impedance of Z T =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) KVL

Voltage Divider Rule (VDR)

Example Problem 3 e s (t)=170 sin (1000t + 0) V. Determine Z TOT Determine total current I TOT Determine voltages V R, V L, and V c Verify KVL for this circuit Graph E, V L, V C, V R in the time domain

Example Problem 3 180° phase difference between L and C

Example Problem 4 e s (t)=294 sin (377t + 0) V. Use the voltage divider rule to find V L Determine the value of inductance