Electronic Instrumentation Experiment 5 * AC Steady State Theory * Part A: RC and RL Circuits * Part B: RLC Circuits

AC Steady State Theory w Transfer Functions w Phasors w Complex Impedance w Complex Transfer Functions

Transfer Functions w The transfer function describes the behavior of a circuit at V out for all possible V in.

Simple Example

More Complicated Example What is H now?  H now depends upon the input frequency (  = 2  f) because the capacitor and inductor make the voltages change with the change in current.

Influence of Resistor on Circuit w Resistor modifies the amplitude of the signal by R w Resistor has no effect on the phase

Influence of Capacitor on Circuit  Capacitor modifies the amplitude of the signal by 1/  C  Capacitor shifts the phase by -  /2

Influence of Inductor on Circuit  Inductor modifies the amplitude of the signal by  L  Inductor shifts the phase by +  /2

How do we model H? w We want a way to combine the effect of the components in terms of their influence on the amplitude and the phase. w We can only do this because the signals are sinusoids cycle in time derivatives and integrals are just phase shifts and amplitude changes

Phasors w Phasors allow us to manipulate sinusoids in terms of amplitude and phase changes w Phasors are based on complex polar coordinates w The influence of each component is given by Z, its complex impedance

Phasor References w ion.html ion.html w pACCircuits/ACCircuits.pdf pACCircuits/ACCircuits.pdf w hasors/Phasors.html hasors/Phasors.html

Phasor Applet

Complex Polar Coordinates

Review of Polar Coordinates point P is at ( r p cos  p, r p sin  p )

Introduction to Complex Numbers w z p is a single number represented by two numbers w z p has a “real” part (x p ) and an “imaginary” part (y p )

Now we can define Phasors w The real part is our signal. w The two parts allow us to determine the influence of the phase and amplitude changes mathematically. w After we manipulate the numbers, we discard the imaginary part.

Magnitude and Phase w Phasors have a magnitude and a phase derived from polar coordinates rules.

Euler’s Formula

Phasor Applet

Manipulating Phasors (1)  Note  t is eliminated by the ratio This gives the phase change between signal 1 and signal 2

Manipulating Phasors (2)

Understanding the influence of Phase

Complex Impedance w Z defines the influence of a component on the amplitude and phase of a circuit Resistors: Z R = R change the amplitude by R Capacitors: Z C =1/j  C change the amplitude by 1/  C shift the phase -90 (1/j=-j) Inductors: Z L =j  L change the amplitude by  L shift the phase +90 (j)

Capacitor Impedance Proof Prove:

Complex Transfer Functions w If we use phasors, we can define H for all circuits in this way. w If we use complex impedances, we can combine all components the way we combine resistors.  H and V are now functions of j and 

Simple Example

Simple Example (continued)

Using H to find V out

Simple Example (with numbers)

Adding Phasors & Other Applets

Part A -- RC and RL Circuits w High and Low Pass Filters w H and Filters

High and Low Pass Filters High Pass Filter H = 0 at  0 H = 1 at    at  c Low Pass Filter H = 1 at  0 H = 0 at    at  c  c =2  f c fcfc fcfc

Corner Frequency w The corner frequency of an RC or RL circuit tells us where it transitions from low to high or visa versa. w We define it as the place where w For RC circuits: w For RL circuits:

Corner Frequency of our example

H(j  ),  c, and filters  We can use the transfer function, H(j  ), and the corner frequency,  c, to easily determine the characteristics of a filter.  If we consider the behavior of the transfer function as  approaches 0 and infinity and look for when H nears 0 and 1, we can identify high and low pass filters. w The corner frequency gives us the point where the filter changes:

Our example at low frequencies

Our example at high frequencies

Our example is a low pass filter What about the phase?

Our example has a phase shift

Taking limits  At low frequencies, (ie.  =10 -3 ), lowest power of  dominates  At high frequencies (ie.  =10 +3 ), highest power of  dominates

Capture/PSpice Notes w Showing the real and imaginary part of the signal in Capture: PSpice->Markers->Advanced ->Real Part of Voltage ->Imaginary Part of Voltage in PSpice: Add Trace real part: R( ) imaginary part: IMG( ) w Showing the phase of the signal in Capture: PSpice->Markers->Advanced->Phase of Voltage in PSPice: Add Trace phase: P( )

Part B -- RLC Circuits w Band Filters w Another example w A more complex example

Band Filters f0f0 f0f0 Band Pass Filter H = 0 at  0 H = 0 at    at  0 =2  f 0 Band Reject Filter H = 1 at  0 H = 1 at    at  0 =2  f 0

Resonant Frequency w The resonant frequency of an RLC circuit tells us where it reaches a maximum or minimum. w This can define the center of the band (on a band filter) or the location of the transition (on a high or low pass filter). w We are already familiar with the equation for the resonant frequency of an RLC circuit:

Another Example

At Very Low Frequencies At Very High Frequencies

At the Resonant Frequency if L=10uH, C=1nF and R=1K     rad/sec  f   Hz 

Our example is a low pass filter Phase  = 0 at  0  = -  at   Magnitude  = 1 at  0  = 0 at   f   Hz  

A more complex example w Even though this filter has parallel components, we can still handle it. w We can combine complex impedances like resistors

Determine H

At Very Low Frequencies At Very High Frequencies

At the Resonance Frequency

Our example is a band pass filter Phase  = 90 at  0  = 0 at    = -  at   Magnitude  = 0 at  0 H=1 at    = 0 at   ff

Simple Filter Design w Step One: Pick a High Pass or Low Pass Filter Configuration This is done by looking at the high or low frequency limit of various simple circuits. For example, for an RC low pass filter, we choose the following configuration:

Simple Filter Design w At very high frequencies, the capacitor becomes a short circuit. (Impedance goes to zero.) w At very low frequencies, the capacitor becomes an open circuit. (Impedance goes to infinity.) The output appears fully across the capacitor. w Thus, if the output voltage is taken across the capacitor, this is a low pass filter.

Simple Filter Design w Step Two: Pick values for R1 and C1. The impedance of C1 decreases from infinity to zero as frequency increases from zero to infinity. At the corner frequency of the circuit, the magnitude of the impedance of C1 equals the impedance of R1

Simple Filter Design w At the corner frequency,

Simple Filter Design w The magnitude of H at the corner frequency is

Simple Filter Design w Thus, for frequencies less than the corner frequency, V out is comparable to V in. w For frequencies greater than the corner frequency, the output is much smaller than the input.

Example w For this example, we select components from our boxes of parts to pass most frequencies below 100Hz. w For R=50 and C=33uF, the corner frequency is about 96Hz.