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First Order System Rise Time
by Robert L Rauck Analog designers are often called upon to design an amplifier to meet a specific rise time requirement. Amplifiers with feedback loops that exhibits a first order rolloff are the easiest to characterize. One of the rules of thumb often used in such cases is the following equation:
This expression is used all the time but many engineers do not know where it came from or how to derive it. The origin of this expression arises from the behavior of a RC network when excited by a step input. Let's write an expression for the behavior of this circuit at switch closure using Laplace Transform notation. The circuit input is a step function due to the presence of the switch. We will assume the capacitor is initially uncharged. Now let's take the inverse transform. From here forward we will refer to the input voltage as Vin since it is a constant. In this circuit, the capacitor voltage is the output voltage. Therefore:
Here a specific corner frequency has been picked to stimulate the discussion.
Clearly the rise time of an RC circuit can be expressed as approximately 2.2•τ which, as we can see, is equal to approximately 0.35 devided by the RC corner frequency (f_{c}). Now we would like to extend this discussion to include the feedback amplifier case where the RC corner frequency is replaced in the expression by the loop crossover frequency.
Assume that G is a gain block with an embedded RC filter to give it a first order rolloff at a frequency more than one decade below loop crossover. Let's examine this expression at loop crossover where G•H = 1 at an angle of 90 degrees since we are more than a decade beyond the RC corner embedded in G. The term G•H/(1+G•H) is an error term that determines the departure of closed loop gain from the ideal gain term (flat with frequency) defined above.
Let's focus on when the magnitude of G*H is 1 (loop crossover frequency):
The loop crossover frequency will converge to K•f; (where f is the RC corner freq. of the embedded RC filter) when K is >> 1. The corner frequency of the closed loop transfer function represented by this amplifier circuit will be shown to be (K+1)•f which is ~ K•f when K is >> 1. Therefore we have demonstrated that the amplifier will behave like a simple RC circuit (multiplied by a constant to account for amplifier DC gain) where the effective corner frequency is essentially the loop crossover frequency.
The complete transfer function is therefore: Now let's take the inverse transform: This expression has the same form as the transfer function of the simple RC circuit defined above (except for the DC gain term ((1H) / H)•(K/(K+1)). Therefore the rise time of the amplifier will have the same form as the RC circuit. Stated another way, the loop crossover frequency is the effective corner frequency of the amplifier closed loop transfer function. One equivalent circuit would be a frequency independant gain block in series with an equivalent RC circuit. In this case the effective RC corner frequency would be shifted to the loop crossover frequency. The important thing to remember is that G•H / (1 + G•H) of a first order rolloff loop looks like the frequency response of a simple RC network when the ideal closed loop gain is flat with frequency. In the schematic that follows, I have modeled an amplifier that has the characteristics we have discussed. Here B1, B2 and the RC network are used to model an Op Amp that has an open loop gain of 10,000 and a gain rolloff at 10 Hz. B1 and B2 are user definable gain blocks that I have defined as indicated on the schematic. If we examine the closed loop gain of this amplifier, we see that it is flat at 10 until crossover where it begins to roll off. The DC loop gain is G•H = 80dB + 20•log(R2/(R2+R3)) = 59.172dB and therefore the ratio of corner freq to crossover freq is 10^{59.172/20} = 909.076 times the 10 Hz corner frequency of R1•C1 or 9.09 kHz crossover freq.
Now we will simulate the above circuit and compare the rise time result with the predicted performance. Success! Simulation matches prediction. 

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