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Small-Signal Analysis#

Small-signal analysis is a technique for analyzing circuits which contain time-variant components, provided that currents / voltages deviate only slightly from some fixed base values.

Given such a time-variant component, the idea is to represent its signal \(S(t)\) (where \(S(t)\) is a label for either the current \(I(t)\) or the voltage \(V(t)\)) into the sum of this constant base value and a time-dependent part:

\[ S(t) = S_Q + \Delta S(t) \]

Algorithm: Small-Signal Analysis

  1. Write the signals \(S_1, \dotsc, S_n\) of all time-variant components as sums like above:
\[ \begin{aligned} S_1 &= S_{1,Q} + \Delta S_1(t) \\ &\vdots \\ S_n &= S_{n, Q} + \Delta S_n(t) \end{aligned} \]

  1. Analyze the circuit under the assumption that \(\Delta S_1(t), \dotsc, \Delta S_n(t)\) are always zero. The solutions for the circuit under this assumption are known as the circuit's DC inputs, DC bias point or quiescent point (Q point).

  2. Analyze the circuit under the assumption that \(S_{1,Q}, \dotsc, S_{n, Q}\) are all zero.

When we perform small-signal analysis on a circuit, we get two descriptions of its behavior:
- The DC bias point tells us how the circuit would behave when there are no temporal deviations from the constant signals \(S_{1,Q}, \dotsc, S_{n, Q}\).
- The second description tells us how the circuit reacts to the deviations \(S_1(t), \dotsc, S_n(t)\).

Tip

Small-signal analysis becomes really powerful when combined with load line analysis and linearization.

Example: Small-Signal Analysis

We want to perform small-signal analysis on the following circuit:

Small-Signal Analysis Example

We are given that \(R = 100 \mathop{\mathrm{\Omega}}\) and that the I-V characteristic of the p-n diode is

\[ I(t) = I_S \cdot \exp \left({\frac{V_D(t)}{V_T}}\right) - I_S, \]

where \(I_s = 10 \mathop{\mathrm{pA}}\) and \(V_T = 25 \mathop{\mathrm{mV}}\).

  1. We write out the signals of a time-independent and a time-dependent part:
\[ \begin{aligned} V(t) &= V_Q + \Delta V(t) \\ \\ V_R(t) &= V_{R,Q} + \Delta V_R(t) \\ \\ V_D(t) &= V_{D, Q} + \Delta V_D(t) \\ \\ I(t) &= I_Q + \Delta I(t) \end{aligned} \]
  1. We perform load line analysis on the circuit under the assumption that \(\Delta V(t)\), \(\Delta V_R(t)\), \(\Delta V_D(t)\) and \(\Delta I(t)\) are all zero:

Small-Signal Analysis Example Bias Point

The series circuit of the Ohmic resistor and the ideal voltage source is equivalent to a linear source whose I-V characteristic is

\[ V' = RI_Q' + V_Q, \]

where \(I_Q' = -I_Q\) because of the sign convention. This equivalent linear source and the diode are connected in parallel, so we know that \(V' = V_{D,Q}\). Then, the operating points must obey the following system of equations:

\[ \begin{aligned} I_Q &= 10 \mathop{\mathrm{pA}} \cdot \exp \left({\frac{V_{D,Q}}{25 \mathop{\mathrm{mV}}}}\right) - 10 \mathop{\mathrm{pA}} \text{ (from the diode's I-V characteristic)} \\ \\ V_{D, Q} &= -100\mathop{\mathrm{\Omega}} \cdot I_Q + V_Q \text{ (from the source's I-V characteristic)} \end{aligned} \]

For example, if we were given that \(V_Q = 1 \mathop{\mathrm{V}}\) and solved the system, we would obtain the following operating point:

\[ \begin{aligned} I_Q &= 10 \mathop{\mathrm{mA}} \\ V_{D, Q} &= V' = 0.5 \mathop{\mathrm{V}} \\ V_{R, Q} &= -0.5 \mathop{\mathrm{V}} \end{aligned} \]
  1. We analyze the circuit under the assumption that \(V_Q\), \(V_{R,Q}\), \(V_{D,Q}\) and \(I_Q\) are all zero:

Small-Signal Analysis Example Time

We can apply the same method of simplification as we did in step 2 in order to obtain the following system of equations:

\[ \begin{aligned} \Delta I(t) &= 10 \mathop{\mathrm{pA}} \cdot \exp \left({\frac{\Delta V_D(t)}{25 \mathop{\mathrm{mV}}}}\right) - 10 \mathop{\mathrm{pA}} \text{ (from the diode's I-V characteristic)} \\ \\ \Delta V_D(t) &= -100 \mathop{\mathrm{\Omega}} \cdot \Delta I(t) + \Delta V(t) \text{ (from the source's I-V characteristic)} \end{aligned} \]

If we were given an expression for \(\Delta V(t)\), we would be able to obtain expressions for the other signals as well.