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431/531 Class Notes 2
1.5 Thevenin Theorem (contd.)
Recall that the Thevenin Theorem states that any collection of resistors and EMFs is equiv- alent to a circuit of the form shown within the b ox lab elled \Circuit A" in the gure b elow. As b efore, the load resistor RL is not part of the Thevenin circuit. The Thevenin idea, however, is most useful when one considers two circuits or circuit elements, with the rst circuit's output providing the input for the second circuit. In Fig. 6, the output of the rst circuit (A), consisitng of VTH and RTH , is fed to the second circuit element (B ), which consists simply of a load resistance (RL ) to ground. This simple con guration represents, in a general way, a very broad range of analog electronics.
R TH
V TH Vout R L
Circuit A
Circuit B
Figure 6: Two interacting circuits.
1.5.1 Avoiding Circuit Loading
VTH is a voltage source. In the limit that RTH! 0 the output voltage delivered to the load RL remains at constant voltage. For nite RTH , the output voltage is reduced from VTH by an amount I RTH , where I is the current of the complete circuit, which dep ends up on the value of the load resistance RL : I = VTH =(RTH + RL ). Therefore, RTH determines to what extent the output of the rst circuit b ehaves as an ideal voltage source. An approximately ideal b ehavior turns out to b e quite desirable in most cases, as Vout can b e considered constant, indep endent of what load is connected. Since our combined equivalent circuit (A + B ) forms a simple voltage divider, we can easily see what the requirement for RTH can b e found from the following:
Vout = VTH
R
L RTH + RL
VTH
1 + (RTH =RL )
Thus, we should try to keep the ratio RTH =RL smal l in order to approximate ideal b ehavior and avoid \loading the circuit". A maximum ratio of 1 = 10 is often used as a design rule of thumb. A go o d p ower supply will have a very small RTH , typically much less than an ohm. For a battery this is referred to as its internal resistance. The dimming of one's car headlights when the starter is engaged is a measure of the internal resistance of the car battery.
1.5.2 Input and Output Imp edance
Our simple example can also b e used to illustrate the imp ortant concepts of input and output resistance. (Shortly, we will generalize our discussion and substitute the term \imp edance" for resistance. We can get a head start by using the common terms \input imp edance" and \output imp edance" at this p oint.)
The output imp edance of circuit A is simply its Thevenin equivalent resistance RTH. The output imp edance is sometimes called \source imp edance".
The input imp edance of circuit B is its resistance to ground from the circuit input. In this case it is simply RL.
It is generally p ossible to reduce two complicated circuits, which are connected to each other as an input/output pair, to an equivalent circuit like our example. The input and output imp edances can then b e measured using the simple voltage divider equations.
2 RC Circuits in Time Domain
2.0.3 Capacitors
Capacitors typically consist of two electro des separated by a non-conducting gap. The quantitiy capacitance C is related to the charge on the electro des (+Q on one and Q on the other) and the voltage di erence across the capacitor by
C = Q=VC
Capacitance is a purely geometric quantity. For example, for two planar parallel electro des each of area A and separated by a vacuum gap d, the capacitance is (ignoring fringe elds) C = 0 A=d, where 0 is the p ermittivity of vacuum. If a dielectric having dielectric constant is placed in the gap, then 0! 0 . The SI unit of capacitance is the Farad. Typical lab oratory capacitors range from 1pF to 1 F. For DC voltages, no current passes through a capacitor. It \blo cks DC". When a time varying p otential is applied, we can di erentiate our de ning expression ab ove to get
I = C
dVC dt
for the current passing through the capacitor.
Note that in this case Vin can b e any function of time. Also note from our solution Eqn. 3 that the limit Vout Vin corresp onds roughly to t RC. Within this approximation, we see clearly from Eqn. 4 why the circuit ab ove is sometimes called an \integrator".
2.0.7 RC Di erentiator
Let's rearrange our RC circuit as shown in Fig. 8.
Vin Vout
C
I R
Figure 8: RC circuit | di erentiator.
Applying Kircho 's second Law, we have Vin = VC + VR , where we identify VR = Vout. By Ohm's Law, VR = I R, where I = C (dVC =dt) by Eqn. 1. Putting this together gives
Vout = RC
d dt
(Vin Vout )
In the limit Vin Vout , we have a di erentiator:
Vout = RC
dVin dt
By a similar analysis to that of Section 2.0.6, we would see the limit of validity is the opp osite of the integrator, i.e. t RC.
