Introduction
Following up on I calculated the AC gain of the Fuzz Face, I will now try to calculate the input impedance.
Circuit Diagram and Equivalent Circuit
I will use the same circuit diagram and equivalent circuit as in the previous article.
Typical Fuzz Face circuit diagram (NPN)
Simplified equivalent circuit of the Fuzz Face
Calculation
Input impedance Z_{in} is defined as the input voltage v_{in} divided by the input current i_{1}.
\begin{equation}
Z_{in} = \frac{v_{in}}{i_{1}} \tag{1}
\end{equation}
Looking at the equivalent circuit, the current i_1 flowing into the input terminal is the sum of the current i_{b1} flowing into the base of Q_1 and the current i_f returning from the emitter of Q_2 through the feedback resistor R_F. That is,
\begin{equation}
i_1 = i_{b1} + i_f \tag{2}
\end{equation}
. The goal is to find i_{b1} and i_f respectively and express i_1 in terms of v_{in}.
Quoting equations from the previous article
We will utilize the equations derived in I calculated the AC gain of the Fuzz Face. The base current i_{b1} of Q_1 can be expressed as:
\begin{equation}
i_{b1} = \frac{v_{in}}{r_{ie1}} \tag{3}
\end{equation}
Also, since the feedback current i_f is determined by the potential difference between v_{in} and v_{e2}:
\begin{equation}
i_f = \frac{v_{in} - v_{e2}}{R_F} \tag{4}
\end{equation}
To express i_f in terms of v_{in}, we need to express v_{e2} in terms of v_{in}.
Derivation of the voltage ratio v_{e2}/v_{in}
Let's restate equations (6) and (7) derived in the previous article:
\begin{equation}
v_{e2} = - \left[ \frac{h_{fe1} R_{C1}}{r_{ie1}} v_{in} + i_{b2} (R_{C1} + r_{ie2}) \right] \tag{1-6}
\end{equation}
\begin{equation}
(1 + h_{fe2}) i_{b2} + \frac{v_{in}}{R_F} = v_{e2} \frac{R_F + R_{E2}}{R_F R_{E2}} \tag{1-7}
\end{equation}
Here, since we want to express v_{e2} in terms of v_{in} by eliminating i_{b2}, solving equation (1-6) for i_{b2} gives:
i_{b2} = -\frac{1}{R_{C1} + r_{ie2}} \left[ v_{e2} + \frac{h_{fe1} R_{C1}}{r_{ie1}} v_{in} \right]
Substituting this into equation (1-7):
(1 + h_{fe2}) \left( -\frac{1}{R_{C1} + r_{ie2}} \right) \left[ v_{e2} + \frac{h_{fe1} R_{C1}}{r_{ie1}} v_{in} \right] + \frac{v_{in}}{R_F} = v_{e2} \frac{R_F + R_{E2}}{R_F R_{E2}}
Moving the terms containing v_{e2} to the left side and terms containing v_{in} to the right side:
v_{e2} \left[ \frac{R_F + R_{E2}}{R_F R_{E2}} + \frac{1 + h_{fe2}}{R_{C1} + r_{ie2}} \right] = \frac{v_{in}}{R_F} - \frac{(1 + h_{fe2}) h_{fe1} R_{C1}}{(R_{C1} + r_{ie2}) r_{ie1}} v_{in}
Organizing the right side:
v_{e2} \left[ \frac{R_F + R_{E2}}{R_F R_{E2}} + \frac{1 + h_{fe2}}{R_{C1} + r_{ie2}} \right] = v_{in} \left[ \frac{1}{R_F} - \frac{(1 + h_{fe2}) h_{fe1} R_{C1}}{(R_{C1} + r_{ie2}) r_{ie1}} \right]
Now, let \alpha be the coefficient of the left side and \beta be the coefficient of the right side:
\begin{equation}
\alpha = \frac{R_F + R_{E2}}{R_F R_{E2}} + \frac{1 + h_{fe2}}{R_{C1} + r_{ie2}} \tag{5}
\end{equation}
\begin{equation}
\beta = \frac{1}{R_F} - \frac{(1 + h_{fe2}) h_{fe1} R_{C1}}{(R_{C1} + r_{ie2}) r_{ie1}} \tag{6}
\end{equation}
The ratio v_{e2}/v_{in} can be found as:
\begin{equation}
\frac{v_{e2}}{v_{in}} = \frac{\beta}{\alpha} \tag{7}
\end{equation}
Substituting the result of equation (7) into equation (4), i_f can be expressed as:
i_f = \frac{v_{in} - v_{e2}}{R_F} = \frac{v_{in}}{R_F} \left( 1 - \frac{v_{e2}}{v_{in}} \right) = \frac{v_{in}}{R_F} \left( 1 - \frac{\beta}{\alpha} \right) = \frac{v_{in}}{R_F} \cdot \frac{\alpha - \beta}{\alpha}
Substituting equation (3) and the above equation into equation (2):
i_1 = i_{b1} + i_f = \frac{v_{in}}{r_{ie1}} + \frac{v_{in}}{R_F} \cdot \frac{\alpha - \beta}{\alpha}
Factoring out v_{in}:
i_1 = v_{in} \left[ \frac{1}{r_{ie1}} + \frac{\alpha - \beta}{\alpha R_F} \right]
Therefore, the input impedance Z_{in} is:
\begin{equation}
Z_{in} = \frac{v_{in}}{i_1} = \frac{1}{\frac{1}{r_{ie1}} + \frac{\alpha - \beta}{\alpha R_F}} \tag{8}
\end{equation}
This is equivalent to the parallel combination of r_{ie1} and \dfrac{\alpha R_F}{\alpha - \beta}, so it can be expressed as:
\begin{equation}
Z_{in} = r_{ie1} \,//\, \frac{\alpha R_F}{\alpha - \beta} \tag{9}
\end{equation}
Finally, substituting \alpha, \beta into equation (9) and organizing it, we get:
\begin{equation}
Z_{in} = \frac{r_{ie1} \left[ (R_F + R_{E2})(R_{C1} + r_{ie2}) + (1 + h_{fe2}) R_F R_{E2} \right]}{(R_F + R_{E2})(R_{C1} + r_{ie2}) + (1 + h_{fe2}) R_F R_{E2} + r_{ie1}(R_{C1} + r_{ie2}) + R_{E2}(1 + h_{fe2})(r_{ie1} + h_{fe1} R_{C1})} \tag{10}
\end{equation}
This is the exact solution for the input impedance.
Now, recalling the section Thinking a bit deeper from the previous article,
.MODEL 2SC8888-W NPN(
+ IS=2.544E-14 BF=380.43 NF=1.002 VAF=125.5 IKF=0.385
+ ISE=4.88E-15 NE=1.35 BR=8.22 NR=1.01 VAR=18.5 IKR=0.18
+ ISC=3.25E-14 NC=1.25
+ RB=12.8 IRB=5.0E-05 RBM=2.0 RE=0.185 RC=0.88
+ CJE=15.32p VJE=0.75 MJE=0.34
+ CJC=5.12p VJC=0.55 MJC=0.32 FC=0.5
+ TF=445.0p XTF=8.0 VTF=12.0 ITF=0.3 TR=65.0n
+ EG=1.11 XTI=3.0 XTB=1.5
if we represent r_{ie} and h_{fe} using various transistor parameters given by SPICE model parameters like these, along with the thermal voltage V_T \,(\approx 26\,\mathrm{mV}):
\begin{equation}
r_{ie} \approx r_{b} + \frac{ \beta_{eff} \cdot N_F \cdot V_T}{I_C} + (1+\beta_{eff}) \cdot r_e \tag{1-14}
\end{equation}
\beta_{eff} = \frac{I_C}{I_B} \approx \frac{B_F}{1 + \dfrac{B_F \cdot I_{SE}}{I_S} \cdot \exp\left(\dfrac{V_{BE}(N_F - N_E)}{N_F \cdot N_E \cdot V_T}\right)}
h_{fe} \approx \beta_{eff} \approx \frac{B_F}{1 + \dfrac{B_F \cdot I_{SE}}{I_S} \cdot \exp\left(\dfrac{V_{BE}(N_F - N_E)}{N_F \cdot N_E \cdot V_T}\right)}
Applying a slightly coarser approximation:
By substituting these, we can express the input impedance using only SPICE model parameters, collector current, thermal voltage, and various circuit constants. The key point to note here is that r_{ie} is proportional to the inverse of the collector current.
As an example, in addition to each circuit constant (shown below), if we roughly calculate with h_{fe1}, h_{fe2} = 100, and I_{C1}=50\,\mathrm{\mu A}, I_{C2}=500\,\mathrm{\mu A} according to the collector current:
Typical Fuzz Face circuit constants
(Note that the operating point is not optimized)
The input impedance is found to vary in the range of approximately 2k (at maximum gain) to 33k (at minimum gain) depending on the value of RV_1.
I want to provide a physical interpretation, so let's organize it a bit more.
If we define:
\Delta = (R_F + R_{E2})(R_{C1} + r_{ie2}) + (1 + h_{fe2}) R_F R_{E2}
then we get:
\begin{equation}
Z_{in} = \frac{r_{ie1} \Delta}{\Delta + r_{ie1}(R_{C1} + r_{ie2}) + R_{E2}(1 + h_{fe2})(r_{ie1} + h_{fe1} R_{C1})} \ tag{11}
\end{equation}
Physical meaning of \Delta
Let's consider the physical meaning of \Delta. By combining equations (1-6) and (1-7) from I calculated the AC gain of the Fuzz Face, we can formulate it in matrix form as:
\underbrace{
\begin{pmatrix}
-1 & -(R_{C1} + r_{ie2}) \\
\frac{R_F + R_{E2}}{R_F R_{E2}} & -(1 + h_{fe2})
\end{pmatrix}
}_{\text{Circuit Matrix } \mathbf{A}}
\begin{pmatrix}
v_{e2} \\
i_{b2}
\end{pmatrix}
=v_{in}
\underbrace{
\begin{pmatrix}
\frac{h_{fe1}R_{C1}}{r_{ie1}} \\
\frac{1}{R_F}
\end{pmatrix}
}_{\text{Input Vector}}
The determinant of this circuit matrix \mathbf{A} is:
\det(\mathbf{A}) = (1 + h_{fe2}) + (R_{C1} + r_{ie2}) \cdot \frac{R_F + R_{E2}}{R_F R_{E2}}
Multiplying the entire equation by R_F R_{E2} shows that it is exactly represented by \Delta. In other words, \Delta is a quantity proportional to the determinant of the circuit matrix and can be interpreted as a parameter that determines the overall responsiveness of the circuit.
Physical Interpretation
By further organizing equation (11), a more intuitive form emerges:
\begin{align}
Z_{in} &= r_{ie1} \cdot \frac{1}{1 + \frac{1}{\Delta} \left[ r_{ie1}(R_{C1} + r_{ie2}) + R_{E2}(1 + h_{fe2})(r_{ie1} + h_{fe1} R_{C1}) \right]} \notag \\
&= r_{ie1} \cdot \frac{1}{1+\gamma}
\tag{12}
\end{align}
A form with better insight has appeared. Here, we have set:
\gamma = \frac{1}{\Delta} \left[ r_{ie1}(R_{C1} + r_{ie2}) + R_{E2}(1 + h_{fe2})(r_{ie1} + h_{fe1} R_{C1}) \right]
From this equation, we can see that the base component of the input impedance is r_{ie1}, and the input impedance is pulled down by \gamma, which corresponds to the loop gain.
Recalling that the rough approximation for r_{ie}
r_{ie} \approx \frac{V_T}{I_C}
holds, it becomes clear that the input impedance is inversely proportional to the collector current of Q_1. That is, there is a direct relationship where increasing the collector current of Q_1 causes the input impedance to decrease.
Now, let's consider examining the R_{E2} dependence of the input impedance. Here, I aim to organize the expression by making bold approximations to clarify the contents of \gamma.
From the SPICE model parameters, the following relationships hold:
\begin{equation}
r_{ie,k} \approx \frac{V_T}{I_{C,k}}, \quad h_{fe,k} \approx B_{F,k}
\end{equation}
By further applying:
\begin{align}
R_F + R_{E2} &\approx R_F \notag{} \\
R_{C1} + \frac{V_T}{I_{C,k}} &\approx R_{C1} \quad \left( \because \frac{V_T}{I_{C,k}} \ll R_{C1} \right) \notag{} \\
1 + B_{F,k} &\approx B_{F,k} \notag{}
\end{align}
we obtain the following approximation for the input impedance:
\begin{equation}
Z_{in} \approx \frac{V_T}{I_{C1}} \cdot \frac{1}{1 + \frac{B_{F1} B_{F2} R_{E2} R_{C1}}{R_F (R_{C1} + B_{F2} R_{E2})}} \tag{13}
\end{equation}
In particular, when R_{E2} \rightarrow 0:
Z_{in} \rightarrow \frac{V_T}{I_{C1}} \cdot \frac{1}{1+0} = \frac{V_T}{I_{C1}}
which shows that it is determined almost uniquely by the collector current alone.
On the other hand, at the minimum value of RV_1, if we assume B_{F2} R_{E2} = m R_{C1} \quad (1 < m < 10) for convenience:
\begin{equation}
Z_{in} \approx \frac{V_T}{I_{C1}} \cdot \frac{1}{1 + \frac{m B_{F1} R_{C1}}{(m+1) R_F}} \tag{13.1}
\end{equation}
Depending on the design, in most cases, R_{C1} is about 2 to 10 times larger than R_F. Considering B_{F1} is 100 or more, the input impedance decreases to as little as one-hundredth compared to when R_{E2} is at its maximum.
To speak qualitatively, the input current is sucked into two branches: the base of transistor Q_1 and R_F. At the other end of R_F, the inverted voltage from Q_1 appears via the base-emitter of Q_2. As a result, a potential difference greater than just the input voltage appears across R_F, causing more current to be sucked in.
In other words, when R_{E2} is large, the current output from the Q_2 emitter is converted into voltage by R_{E2}. Since the resulting voltage is out of phase with the input, from the perspective of the input side, the potential difference across R_F appears larger. Conversely, when R_{E2} is small, the potential difference is kept small, which leads to the difference in input impedance.
Given this, it is more natural to think that RV_1 is controlling the input impedance rather than the gain.
This is the reason why the Fuzz Face's input impedance is extremely low, and this low input impedance is the factor that enhances the volume cleanup (responsiveness) of the guitar.
Considering interaction with the guitar
Let's consider the interaction between the extremely low input impedance of the Fuzz Face and the guitar.
Impedance characteristics of guitar pickups
(Quoted from https://www.ishibashi.co.jp/academic/super_manual2/electricity)
Generally, since a guitar pickup is a coil, its impedance tends to increase at higher frequencies (strictly speaking, due to the parasitic capacitance of the coil, self-resonance occurs and the impedance tends to drop after peaking around 10 kHz). Due to this characteristic and the Fuzz Face's extremely low input impedance, a phenomenon occurs where high frequencies cannot be fully received and are rolled off (high-frequency loss) due to the difference between low and high frequency impedance. On the other hand, lowering the guitar's volume increases the impedance across all frequency bands. As a result, the relative difference narrows, high-frequency loss is suppressed, leading to the unique sound of the Fuzz Face.
Extracting only the inductance and typical resistance components of the guitar's electrical system along with the input impedance of the Fuzz Face, it can be represented as shown below.
Diagram showing only the inductance and resistance components of the guitar's electrical system
This can be organized as a voltage divider as shown in the figure below.
Organized as a voltage divider
Since the reactance caused by the inductance component of the coil increases in proportion to the frequency, the impedance of the upper resistor tends to increase as the frequency increases.
On the other hand, since the lower resistor is dominated by the input impedance (except when near silence), it can be practically regarded as Z_{in}.
When the volume is close to 10, the upper resistor under high-frequency conditions can be considered as:
\sqrt{(R_{ccw}+R_{DC})^2+(2\pi f L_s)^2} \approx 2\pi f L_s
Since the lower resistor is constant, we can see that the gain decreases as the frequency increases.
v_{fuzz} = \frac{Z_{in}}{Z_{in}+ 2\pi f L_s}
In contrast, when the volume is sufficiently low, we can approximate that R_{ccw} is dominant compared to the coil's reactance (though this is a bit of a simplification):
\sqrt{(R_{ccw}+R_{DC})^2+(2\pi f L_s)^2} \approx R_{ccw}
This results in a frequency-independent gain, as shown in the following equation:
v_{fuzz} = \frac{Z_{in}}{Z_{in}+ R_{ccw}}
This is what leads to the "bell-like chime" (sparkling cleanup) where the high end remains even as the volume is rolled back.
Conclusion
The calculation of input impedance was quite complex, but we found that the input impedance is pulled down by feedback. And this low input impedance is what leads to the sparkling cleanup of the Fuzz Face.
In the next article, I would like to introduce how to calculate the optimal operating point. It's easier than calculating gain or input impedance, so stay tuned.
Discussion