Introduction:



Transfer Functions:

Note in the last example of the previous unit, [pic] is entirely dependent on the circuit elements and defines the relationship between the specified input vs and output vo. Therefore, if the input magnitude and phase changes, the output can be determined by multiplying the magnitude of the input by .33 ([pic]) and adding -9.46( ([pic]) to the input phase. In this sense, [pic] transfers the input value to the output value.

Definition:

A transfer function (TF) is a complex-valued function of (, such that the TF magnitude indicates the scaling between the magnitudes of the circuit input and output, and the TF phase indicates the shifting between the phases of the circuit input and output. The transfer function defines this relationship for every possible input frequency.

To find a transfer function, convert to impedance circuit, but leave the impedance values as functions of j(=p. Then solve for the ratio of the phasor output divided by the phasor input.

Transfer Function Example:

Determine the transfer function for the circuit below, where the input is vi(t) and the output is io(t).

[pic]

Show transfer function is given by:

[pic]

Transfer Function Example:

Determine the transfer function for the circuit below, where the input is vs(t) and the output is io(t).

[pic]

Show transfer function is given by:

[pic]

Transfer Function Example:

Determine the transfer function for the circuit below, where the input is vi(t) and the output is vo(t).

[pic]

Show transfer function is given by:

[pic]

Determining Transfer Functions in SPICE:

If the input has unit amplitude and zero phase, then the ratio of the phasor output over phasor input equals the phasor output. Thus, set input equal to the source:

>> V1 1 0 AC 1 0

>> (Insert rest of circuit description)

Command SPICE to evaluate the phasor analysis over a frequency range (in Hz):

>>.AC {Point spacing =>LIN or DEC} { # of evaluation points} {start point (0} {end point}

|Command SPICE to print a result to a table: |Command SPICE to plot result: |

|>>.PRINT AC VM(node), VP(node), VDB(node) |>>.PLOT AC VM(node), VP(node), VDB(node) |

SPICE Example:

Determine the transfer function using SPICE:

[pic]

Hint: Use to the following model for Op Amps:

[pic]

[pic]

* SPICE Code for OP amp circuit Example in

* Unit 1 to create a Transfer Function

* Circuit Components Outside of OP Amp

* Define input with unit magnitude and 0 phase

* so output will be identically equal to the

* transfer function

V1 1 0 AC 1 0

C1 1 2 10UF

R1 2 0 50K

R3 3 0 5K

R4 3 4 45K

* Circuit Components for OP Amp Model

R2 2 3 2MEG

G1 0 5 2 3 .1M

R5 5 0 10K

C2 5 0 1.519UF

E1 0 6 5 0 2E5

R6 6 4 75

* Analysis and Output Statements

.AC DEC 150 .01 150k

.PRINT AC VM(4) VP(4)

.PROBE

.END

* Select Trace on Probe

* menu to add trace dB(abs(V(4))

* to plot TF magnitude.

* Then for a new plot add trace

* 180*atan(V(4) / 3.14159

* to plot TF phase.

Poles and Zeros:

The transfer function of a linear system can be written as a ratio of polynomials:

[pic]

For all real-time systems m ( n.

The order of the denominator polynomial defines the order of the system.

The roots of the numerator polynomial are referred to as zeros of the system.

The roots of the denominator polynomial are referred to as poles of the system.

The transfer function can be rewritten as:

[pic]

Identify the poles and zeros for the above system with the transfer function shown above.

A pole-zero diagram of a transfer function is the complex number plane where the zero positions are indicated by the 0 symbol and pole positions by the X symbol.

Pole-Zero Diagram Example:

Pole-zero diagrams are useful for rough sketches of the transfer function magnitude.

For values of ( where the j(-axis is near a pole, a relative maximum occurs.

For values of ( where the j(-axis is near a zero, a relative minimum occurs.

The closer the pole or zero is to the j(-axis, the sharper the max or min point.

Determine the pole-zero diagram and sketch the transfer function magnitude.

[pic]

[pic]

-----------------------

Phasor Magnitude

Phasor Phase (optional, default = 0)

Source Label

Node Location

Source Type

Voltage Magnitude in Decibels

Voltage Phase

Voltage Magnitude

for the (A741 op-amp A0=2x105, Ri=2M(, R0=75(, RL=10k(, CL=1.1519 (F

[pic]

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