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LECTURE NO. 3
INCREMENTAL MODEL OF A PN JUNCTION DIODE
Fig.1. A dc diode circuit.
V _{DD} = V _{D} + I _{D} .R _{D} _____(1) This describes DC load line.
I _{D} = I _{D0} exp(V _{D} /V _{T} )_____(2) This is the device characteristics.
Q-point or the quiescent point is the DC operating point and is obtained as the intersection of DC load line and the device characteristics.
Figure 2. The dc load line , the device characteristics and the Q point.
Tan(α)= slope of the load line= (-1/R _{D} )
Under signal conditions:
Figure 3. Signal is being coupled with the diode circuit.
i _{D} = I _{D} + i _{d} where i _{D} is the instantaneous diode current.
I _{D} is the DC diode current.
i _{d} is the incremental diode current.
And v _{D} (instantaneous diode voltage) = V _{D} (DC Diode Voltage) + v _{d} _(incremental diode voltage)
Now the loop or the mesh equation is:-
(V _{DD} +v _{s} )=v _{D} +i _{D} R _{D} _______(3)
Rewriting the above equation we get:-
(V _{DD} +v _{s} )= (V _{D} + v _{d} )+( I _{D} + i _{d} ) R _{D} _____(4)
Now we have (Instantaneous – DC) = Incremental part,
that is Eq.(4)-Eq.(1) :
v _{s} = v _{d} + i _{d} R _{D} = v _{d} + v _{o} ;
Incremental circuit will be:-
Figure 4. The incremental circuit of the Diode Circuit with signal.
A diode under instantaneous conditions has two parts :-
The above is a relation between the diode current and the diode voltage.
We know that
e ^{Ɵ} = 1+Ɵ+(Ɵ ^{2} /2!)+ (Ɵ ^{3} /3!)+………….
If Ɵ<<1; e ^{Ɵ} = 1+Ɵ This now becomes a linear equation.
If incremental voltage across the diode is less than 5 mV then Ɵ<<1.
To maintain linearity, we maintain all the signals small.
So under small signal approximations,
e ^{Ɵ} = 1+Ɵ will hold good.
Now we note that V _{T} =26mV and the room temperature=300K and if V _{d} <5 mV then we satisfy the small signal condition and we obtain:
Then we have;
Thus;
Let:
Putting V _{T} =26mV and I _{D} =1mA we get r _{d} =26Ω.
Thus the incremental part of the diode circuit was determined as follows:
Incremental resistance r _{d} = (Thermal Resistance V _{T} / Quiescent Current through the diode)
This is small signal approximation. Thus Diode equivalent circuit is composed of linear elements only under small signal condition. Hence the circuit in Figure 4 is incremental circuit or small signal equivalent circuit. The incremental resistance r _{d} offered by the diode under small signal condition is a linear resistance and is included in the circuit only under incremental condition.
Table 1. Values of incremental resistance at 300K offered by a diode under various DC diode currents.
I _{D} (mA) | r _{d} (ohms)=V _{T} /I _{D} at Room Temerature(300K |
1µA | 26 kohms |
10µA | 2.6 kohms |
100µA | 260 ohms |
1 mA | 26 ohms |
10mA | 2.6ohms |
100mA | 0.26 ohms. |
Amplitude or Harmonic Distortion.
As can be seen from Figure 3, input sinusoidal voltage v _{S} appears as addition and subtraction to V _{DD} . Hence under signal condition, the load line is being shifted as shown in Figure 5. In doing so Q pint also shifts generating sinusoidal current swing. As can be seen in the figure, a small segment of I-V curve of the diode(which is the case under small signal condition) is essentially a straight line. Hence Q moves along a straight segment and in the process generates a sinusoidal swing in the current.
Figure 5. Diode Current Sinusoidal Swing under Sinusoidal Input Voltage under small signal condition.
But as seen in Figure 6, if input voltage is a large signal then Q traces a non-linear segment of I-V curve. This results in non-sinusoidal current swing in the diode. This means a sinusoidal voltage is not giving rise to a sinusoidal current in the output load resistance R _{L} . Hence the output voltage will be non-sinusoidal and its Fourier Series Expansion will contain Fundamental and Harmonics. This is known as Amplitude or Harmonic Distortion.
Figure 6. Non-sinusoidal diode current swing when input voltage is a large voltage.
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