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**System
Behavior**

Transfer Functions-- A review of how to derive transfer functions. We will almost exclusively used transfer function analysis, rather than resorting to differential equations or state space analysis.Time Domain Analysis-- How to analyze the time domain performance of circuits given the transfer function (using Laplace Transforms). In this course we will more often be concerned with frequency domain specifications, but time domain analysis will also come up regularly.Frequency Domain Analysis-- How to analyze the frequency domain performance of circuits. You should review methods of drawing Bode plots.First Order Systems-- A brief review of the time domain and frequency domain performance of first order systems: both lowpass and highpass.Second Order Systems-- A brief review of the time and frequency domain behavior of second order systems: lowpass, highpass, bandpass, bandstop. This material is much more complicated than first order systems, but it is at least as important.

**Transfer Functions**

Transfer functions can be used to analyze and understand both the time domain and frequency domain behavior of circuits. The transfer function is the "gain" of the function determined with the circuit analysis methods described earlier, with the impedances of capacitors and inductors given by 1/sC and sL, respectively (see the section on passive circuit elements, above). As an example, consider the circuit shown below:

By the voltage divider rule, we can write the ratio of output to input:

which is the transfer function.

**Time Domain Analysis**

We won't be doing much domain analysis from transfer functions, but the zero-state (input only, no initial conditions) can be easily determined from the transfer function by using Laplace Transforms (which are used via a Laplace Transform table - the methods won't be covered here) or by finding particular and homogenous solutions (or zero-input and zero-state). As an example, consider the step response of the transfer function given above. If v

_{i}(t) is a unit step function, then V_{i}(s)=1/s. Then V_{o}(s) is given by

The final result is achieved by inverse Laplace Transform:

You should know how to find the step response for any first order system. The analysis is based upon two facts:

The time constant for a first order system is either τ=RC (for a system with resistors and capacitors) or τ=L/R (for a circuit with inductors).

The response of a first order system (to a constant input, for t>0) is given by:

where v(0) is the quantity (typically current or voltage) at t=0+ (i.e., just after t=0), and v(∞) is the steady state value of the quantity.Typically v(0), v(∞) and τ are all easily determined. Click here for examples.

**Frequency Domain Analysis**

The frequency domain description of a system details how the system will behave for sinusoidal inputs at different frequencies. As the signal passes through our system both its magnitude and phase will, in general, be changed.

Consider the transfer function,

In order to study its frequency response we let s=jω, so that

(we could have also done a Bode plot). We can now plot H(jω) as a function ofjω. However, since this is a complex quantity we will generally plot the magnitude and phase of the function on two separate graphs. This is shown below:

Note that this is not a Bode plot. Bode plots have a logarithmic axis for frequency, and magnitude is expressed in deciBels (dBs). (You should review how to construct Bode Plots from your E12 or E58 textbook).

These graphs show that if the input to the system=3sin(10t) the output would be 6sin(10t-90°). This is determined from the graphs by determining the magnitude (a factor of 2) and the phase (90°) at 10 rad/sec. If the input is 4sin(21t) the output would be 2sin(21t-150°) because the gain of the system at 21 rad/sec is 0.5 and the phase is -150°. It is important to note that the frequency on these diagrams is typically given in rad/sec (ω), and not Hz (f). The conversion is ω=2πf.

**First Order Systems**

There are two generic types of first order systems, high pass and low-pass. We will first examine the low-pass system.

LowpassFor a lowpass system, the transfer function is of the form

where A

_{0}is called the DC gain, τ (=1/ω_{0}) is the time constant. The quantity ω_{0}(=1/τ) is called the "break" frequency, the 3dB frequency, or the bandwidth. The unit step response (time domain) is given in the following graph (with A_{0}=3, τ=10, ω_{0}=0.1).

Note that the time constant (as indicated by either the blue or green lines) is 10 seconds, and that the steady state value to a unit step input is A

_{0}=3.The frequency response of the system is shown below. Note that low frequencies are passed, while high frequencies are attenuated..

Note that the DC frequency (as ω goes to zero) is A

_{0}=3=9.5 dB. Also note that the break frequency (corner frequency, 3 dB frequency, bandwidth) is ω_{0}=0.1 rad/sec (at 0.1 rad/sec the gain is down 3 dB from the DC value).

HighpassFor a highpass system, the transfer function is of the form

where A

_{0}is called the high frequency gain, τ (=1/ω_{0}) is the time constant, and ω_{0}(=1/τ) is the "break", or 3dB, frequency. The unit step response (time domain) is given in the following graph (with A_{0}=3, τ=10, ω_{0}=0.1).

Note that the time constant (as indicated by either the blue or green lines) is 10 seconds, and that the steady state value to a unit step input is A

_{0}=3.The frequency response of the system is shown below. Note that high frequencies are passed, while low frequencies are attenuated.

Note that the high frequency asymptote of the gain (as ω goes to infinity) is A

_{0}=3=9.5 dB, and that the gain goes to zero at low frequencies. Also note that the break frequency (corner frequency, 3 dB frequency) is ω_{0}=0.1 rad/sec (at 0.1 rad/sec the gain is down 3 dB from the high frequency).

**Second Order Systems**

LowpassFor a lowpass system, the transfer function is of the form

where A

_{0}is called the DC gain, ω_{0}is the natural, or resonant frequency. The quantity ζ is called the damping coefficient (or factor), Q is the called the quality, and B is called the bandwidth (more on Q and B when we discuss bandpass filters). The unit step response (time domain) is given in the following graph (with A_{0}=3, ω_{0}=10, and several values of ζ).

Note that the settling time (to within 5% of the final value) is given by (approximately) T

_{s}=2.5/(ζω_{0}), and that the ringing decreases with increasing ζ (i.e., increased damping). In all cases the steady state value is A_{0}times the magnitude of the step (here, A_{0}=3 and the step is unit magnitude).The frequency response of the system is shown below. Note that low frequencies are passed, while high frequencies are attenuated.

Note that the DC frequency (as ω goes to zero) is A

_{0}=3=9.5 dB. Also note that the break frequency (corner frequency, 3 dB frequency) is ω_{0}=10 rad/sec, and that the system becomes more resonant as the damping decreases (ζ decreases).

HighpassFor a highpass system, the transfer function is of the form

where A

_{0}is called the high frequency gain, ω_{0}is the natural, or resonant frequency and ζ is the damping coefficient. The unit step response (time domain) is given in the following graph (with A_{0}=3, ω_{0}=10, and several values of ζ).

Note that the ringing decreases with increasing ζ (i.e., increased damping), the steady state value is 0, and that the magnitude at t=0 is A

_{0}times the magnitude of the step (here, A_{0}=3 and the step is unit magnitude).The frequency response of the system is shown below. Note that high frequencies are passed, while low frequencies are attenuated.

Note that the high frequency asymptote (as ω goes to inifinity) is A

_{0}=3=9.5 dB, and that the gain goes to zero at low frequencies. Also note that the break frequency (corner frequency, 3 dB frequency) is ω_{0}=10 rad/sec, and that the system becomes more resonant as the damping decreases (ζ decreases).

BandpassFor a bandpass system, the transfer function is of the form

where A

_{0}is called the midband gain, and ω_{0}is the natural, or resonant, frequency. The quantity ζ is called the damping coefficient (or factor), Q is the called the quality, and B is called the bandwidth. The unit step response (time domain) is given in the following graph (with A_{0}=3, ω_{0}=10, and several values of ζ). We won't often be interested in the time domain behavior of bandpass systems, but the step response is presented here for completeness.

Note that the ringing decreases with increasing ζ (i.e., increased damping), the steady state value is 0, and that the magnitude at t=0 is A

_{0}times the magnitude of the step (here, A_{0}=3 and the step is unit magnitude).The frequency response of the system is shown below. Note that only a range (or band) of frequencies is passed, while most are attenuated.

Note that at high and low frequencies the gain goes to zero, while it is A

_{0}=3=9.5 dB at ω_{0}=10 rad/sec. Note that the system becomes more resonant as the damping decreases (ζ decreases). The bandwidth, B, of the circuit (where the gain is down by 3 dB from the maximum value) also decreases with decreasing ζ (B=2ζω_{0}), while the quality Q, increases (B=ω_{0}/Q).

Bandstop (or Bandreject)For a bandstop system, the transfer function is of the form

where A

_{0}is called the DC gain, ω_{0}is the natural, or resonant frequency, and the quantity ζ is called the damping coefficient (or factor). The unit step response (time domain) is given in the following graph (with A_{0}=3, ω_{0}=10, and several values of ζ). We won't often be interested in the time domain behavior of bandstop systems, but the step response is presented here for completeness.

The frequency response of the system is shown below. Note that only a range (or band) of frequencies is attenuated (or rejected), while most are passed without attenuated.

Note that at high and low frequencies the gain goes to A

_{0}=3=9.5 dB, while it is zero at ω_{0}=10 rad/sec. Note that the system becomes more selective as the damping decreases (ζ decreases).

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